Fortune magazine cover designed by Arthur Lidov, depicting Maxwell’s thermodynamic surface of an “imaginary substance” similar to (though not quite) water based on J. W. Gibbs's work, alongside Gibbs's formula for the phase rule, bottom left

To describe a closed, homogeneous system — that is one of constant composition, such as an ideal gas — we need two parameters of state (e.g. T and p). For a heterogeneous system in equilibrium, consisting of one component (e.g. water) and two phases (e.g. liquid and vapour), we require only one parameter of state (e.g. T).
This can be expressed with Gibbs’s phase rule, which specifically describes the number of possible degrees of freedom (or variance) of a chemical system (where C = number of components, P = number of phases in the system): F = C ‒ P + 2
The number 2 is specified because this formulation assumes both T and p can be varied.

The thermodynamic surface for a typical substance is shown in this diagram, with the x axis (width) indicating volume, y axis (height) indicating pressure and z axis (depth) indicating temperature.
Water is thermodynamically atypical, as is readily observed from icebergs that float on liquid water — this can be seen by comparing this diagram to the illustration above, after Maxwell’s 1874 sculpture (itself based on Gibbs’s papers). Maxwell used coordinates of volume (x), entropy (y) and energy (z) — plotted from surrogate measures of pressure and temperature.
Maxwell stated that this model allowed “the principal features of known substances [to] be represented on a convenient scale”.
The construction of this was far more interesting than that of any automatist dream painting (though here the cover art is clearly trying to conjure up the surrealist landscapes of Magritte and contemporaries)

The numerical data about entropy can only be obtained by integration from data which are for most bodies very insufficient, and besides it would require a very unwieldy model to get all the features, say of CO2, well represented, so I made no attempt at accuracy, but modelled a fictitious substance, in which the volume is greater when solid than when liquid; and in which, as in water, the saturated vapour becomes superheated by compression. When I had at last got a plaster cast I drew on it lines of equal pressure and temperature, so as to get a rough motion of their forms. This I did by placing the model in sunlight, and tracing the curve when the rays just grazed the surface…


A superb summary of the two scientists’ graphical methods was put together by Ron Kriz at Virginia Tech (view full size image here). The melée of multi-coloured lines is a bit perplexing, and bringing a physical sculpture in to demonstrate the concept — stepping away from the 2D triple point plots still used in undergraduate lectures today — was a stroke of genius in a time long before the advent of sophisticated computer visualisations.
This general graphic method was not just to plot existing thermodynamic data, but rather to envision total derivatives — related to the work on vector calculus Gibbs was renowned for (his lectures on the subject were collected at the start of the 20th century to form an influential textbook).
Dr Kriz feels this object should provoke reflection on how we consider visualisation methods in science:

The development of the thermodynamic theory of state is a rare but excellent example that demonstrates how scientists combine analytic and graphical methods together with how they understand science. How scientists combine analytical and graphical models into new knowledge exemplifies a cognitive processes that includes visual thinking or what Dr. Daniel Coy describes as “geometric reasoning”. This new knowledge was reported and documented by Gibbs as a graphical method, so that others could reproduce and build on that understanding. As the graphical method was being developed by Gibbs the intent was not to use graphics for presentation but rather to develop the theory. This is contrary to the popular belief that imaging in science is used for presentation which can at times be insightful. 
After reading and studying Gibbs and Maxwell, perhaps the reader would agree that neither Gibbs nor Maxwell developed their graphical method for presentation, a metaphor, or as an intriguing anecdotal experience that could not be scientifically reproduced. Rather the graphical method was sufficiently developed and described by Gibbs to be inclusive with developing the thermodynamic theory of state, which was reproduced and further developed graphically by Maxwell. Recall in summary Gibbs states,

In the foregoing discussion, the equations which express the fundamental principles of thermodynamics in an analytical form have been assumed, and the aim has only been to show how the same relations may be expressed geometrically. It would, however, be easy, starting from the first and second laws of thermodynamics as usually enunciated, to arrive at the same results without the aid of analytical formulae, to arrive, for example, at the conception of energy, of entropy, of absolute temperature, in the construction of the diagram without the analytical definitions of these quantities, and to obtain the various properties of the diagram without the analytical expression of the thermodynamic properties which they involve.

This is not a subjective process, e.g. what visual tools were used, how were they used, or how were the tools designed. The integrity of Gibbs’ and Maxwell’s graphical method is a well established, scientific, objective, and a reproducible process that has nothing to do with the subjective use of tools. This graphical method is inclusive with the developement of the thermodynamic theory of state where Gibbs demonstrates that understanding this theory can be accomplished “...without the aid of analytic formulae”, e.g. his equation of state. In fact Gibbs thought his graphical method was so important that,

Such a course would have been better fitted to show the independence and sufficiency of a graphical method, but perhaps less suitable for an examination of the comparative advantages or disadvantages of different graphical methods.

Hopefully the independence and sufficiency of a graphical method, as proposed by Gibbs, was developed and demonstrated here by envisioning energy as a surface defined as a scalar function of two independent variables, e.g. entropy and volume, where the gradient of the scalar function are slopes tangent to this surface and equal to temperature and negative pressure, as defined in Figs. 5 and 8. However since neither this surface nor the gradient lines tangent to this surface are not associated with a specific set of physical properties, this general graphical method is indeed coextensive in its application.

Further reading:
◉  Ronald D. Kriz (2007) Thermodynamic Case Study: Gibbs’s Thermodynamic Graphical Method — Envisioning total derivatives of scalar functions with two independent variables as raised surfaces and tangent planes. Virginia Tech.
◉  Preface to Elementary principles in statistical mechanics. JW Gibbs, 1902 and full text transcribed at Wikisource
◉  Open University PHYS 7.3 (1996) Internal energy, heat and energy transfer. University of Reading◉  Desmond Fearnley-Sander (1979) Hermann Grassmann and the creation of linear algebra.

Fortune magazine cover designed by Arthur Lidov, depicting Maxwell’s thermodynamic surface of an “imaginary substance” similar to (though not quite) water based on J. W. Gibbs's work, alongside Gibbs's formula for the phase rule, bottom left

To describe a closed, homogeneous system — that is one of constant composition, such as an ideal gas — we need two parameters of state (e.g. T and p). For a heterogeneous system in equilibrium, consisting of one component (e.g. water) and two phases (e.g. liquid and vapour), we require only one parameter of state (e.g. T).

This can be expressed with Gibbs’s phase rule, which specifically describes the number of possible degrees of freedom (or variance) of a chemical system (where C = number of components, P = number of phases in the system): F = C ‒ P + 2

The number 2 is specified because this formulation assumes both T and p can be varied.

image

The thermodynamic surface for a typical substance is shown in this diagram, with the x axis (width) indicating volume, y axis (height) indicating pressure and z axis (depth) indicating temperature.

Water is thermodynamically atypical, as is readily observed from icebergs that float on liquid water — this can be seen by comparing this diagram to the illustration above, after Maxwell’s 1874 sculpture (itself based on Gibbs’s papers). Maxwell used coordinates of volume (x), entropy (y) and energy (z) — plotted from surrogate measures of pressure and temperature.

Maxwell stated that this model allowed “the principal features of known substances [to] be represented on a convenient scale”.

The construction of this was far more interesting than that of any automatist dream painting (though here the cover art is clearly trying to conjure up the surrealist landscapes of Magritte and contemporaries)

The numerical data about entropy can only be obtained by integration from data which are for most bodies very insufficient, and besides it would require a very unwieldy model to get all the features, say of CO2, well represented, so I made no attempt at accuracy, but modelled a fictitious substance, in which the volume is greater when solid than when liquid; and in which, as in water, the saturated vapour becomes superheated by compression. When I had at last got a plaster cast I drew on it lines of equal pressure and temperature, so as to get a rough motion of their forms. This I did by placing the model in sunlight, and tracing the curve when the rays just grazed the surface…

image

A superb summary of the two scientists’ graphical methods was put together by Ron Kriz at Virginia Tech (view full size image here). The melée of multi-coloured lines is a bit perplexing, and bringing a physical sculpture in to demonstrate the concept — stepping away from the 2D triple point plots still used in undergraduate lectures today — was a stroke of genius in a time long before the advent of sophisticated computer visualisations.

This general graphic method was not just to plot existing thermodynamic data, but rather to envision total derivatives — related to the work on vector calculus Gibbs was renowned for (his lectures on the subject were collected at the start of the 20th century to form an influential textbook).

Dr Kriz feels this object should provoke reflection on how we consider visualisation methods in science:

The development of the thermodynamic theory of state is a rare but excellent example that demonstrates how scientists combine analytic and graphical methods together with how they understand science. How scientists combine analytical and graphical models into new knowledge exemplifies a cognitive processes that includes visual thinking or what Dr. Daniel Coy describes as “geometric reasoning”. This new knowledge was reported and documented by Gibbs as a graphical method, so that others could reproduce and build on that understanding. As the graphical method was being developed by Gibbs the intent was not to use graphics for presentation but rather to develop the theory. This is contrary to the popular belief that imaging in science is used for presentation which can at times be insightful. 

After reading and studying Gibbs and Maxwell, perhaps the reader would agree that neither Gibbs nor Maxwell developed their graphical method for presentation, a metaphor, or as an intriguing anecdotal experience that could not be scientifically reproduced. Rather the graphical method was sufficiently developed and described by Gibbs to be inclusive with developing the thermodynamic theory of state, which was reproduced and further developed graphically by Maxwell. Recall in summary Gibbs states,

In the foregoing discussion, the equations which express the fundamental principles of thermodynamics in an analytical form have been assumed, and the aim has only been to show how the same relations may be expressed geometrically. It would, however, be easy, starting from the first and second laws of thermodynamics as usually enunciated, to arrive at the same results without the aid of analytical formulae, to arrive, for example, at the conception of energy, of entropy, of absolute temperature, in the construction of the diagram without the analytical definitions of these quantities, and to obtain the various properties of the diagram without the analytical expression of the thermodynamic properties which they involve.

This is not a subjective process, e.g. what visual tools were used, how were they used, or how were the tools designed. The integrity of Gibbs’ and Maxwell’s graphical method is a well established, scientific, objective, and a reproducible process that has nothing to do with the subjective use of tools. This graphical method is inclusive with the developement of the thermodynamic theory of state where Gibbs demonstrates that understanding this theory can be accomplished “...without the aid of analytic formulae”, e.g. his equation of state. In fact Gibbs thought his graphical method was so important that,

Such a course would have been better fitted to show the independence and sufficiency of a graphical method, but perhaps less suitable for an examination of the comparative advantages or disadvantages of different graphical methods.

Hopefully the independence and sufficiency of a graphical method, as proposed by Gibbs, was developed and demonstrated here by envisioning energy as a surface defined as a scalar function of two independent variables, e.g. entropy and volume, where the gradient of the scalar function are slopes tangent to this surface and equal to temperature and negative pressure, as defined in Figs. 5 and 8. However since neither this surface nor the gradient lines tangent to this surface are not associated with a specific set of physical properties, this general graphical method is indeed coextensive in its application.

Further reading:

◉  Ronald D. Kriz (2007) Thermodynamic Case Study: Gibbs’s Thermodynamic Graphical Method — Envisioning total derivatives of scalar functions with two independent variables as raised surfaces and tangent planesVirginia Tech.

◉  Preface to Elementary principles in statistical mechanics. JW Gibbs, 1902 and full text transcribed at Wikisource

◉  Open University PHYS 7.3 (1996) Internal energy, heat and energy transfer. University of Reading◉  Desmond Fearnley-Sander (1979) Hermann Grassmann and the creation of linear algebra.

image/svg+xml
9
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A bioinformatics walk-through: Accessing protein-protein interaction interfaces for all known protein structures with PDBe PISA

If this summer’s posting became a little infrequent, part of the blame lies with computational research I’ve been working on, regarding the systems biology of chromosomal translocations and the ensuing chimeric proteins at the Medical Research Council Laboratory of Molecular Biology in Cambridge.

A sizeable part of bioinformatics ‘dry lab’ work falls into what has been described in the NYT as ‘data wrangling’ (or the work of a ‘data janitor’). This post is about accessing the data held in the Protein Databank in Europe's repository of Proteins, Interfaces, Structures and Assemblies (PDBe PISA).

Sent out onto the web to find a source of structural protein-protein interaction data with amino acid-level resolution, my first port of call was the Nucleic Acids Research Molecular Biology Online Database Collection (which I’d read of in the opening chapters of Arthur Lesk’s Introduction to Bioinformatics) where I found a sizeable list of PPI databases.

Not wanting to click through each, I chose to browse this programmatically, using Javascript-automated AJAX requests (effectively asking the website to give me web pages but without displaying them) and just ‘scrape’ what I wanted (full workings here), as follows:

From these results, here’s a little background info on PDBe:

  {
    "name": "PDBe",
    "url": "http://www.ebi.ac.uk/pdbe/",
    "entryurl": "http://www.oxfordjournals.org/nar/database/summary/456",
    "desc": "EMBL-EBI's Protein Data Bank in Europe (PDBe) is the European resource for the collection, organization and dissemination of data about biological macromolecular structures. PDBe is one of four partners in the worldwide Protein Data Bank (wwPDB), the consortium entrusted with the collation, maintenance and distribution of the global repository of macromolecular structure data. PDBe uses a relational database that presents the data derived from the Protein Data Bank (PDB) in a consistent way and allows users to retrieve meaningful data using complex and sophisticated searches including simple textual queries or more complex 3D structure-based queries. PDBe has also developed a number of advanced tools for analysis of macromolecules. The \"Structure Integration with Function, Taxonomy and Sequence\" (SIFTS) initiative integrates data from a number of bioinformatics resources that is used by major global sequence, structure and protein-family resources. Furthermore, PDBe works actively with the X-ray crystallography, Nuclear Magnetic Resonance (NMR) spectroscopy and cryo-Electron Microscopy (EM) communities and is a partner in the Electron Microscopy Data Bank (EMDB). The active involvement with the scientific communities has resulted in improved tools for structure deposition and analysis.",
    "ref": null,
    "absurl": "http://nar.oxfordjournals.org/cgi/content/abstract/42/D1/D285",
    "email": "pdbe@ebi.ac.uk"
  },

Web scraping can feel quite kludgy, and there are doubtless better ways to do the above. Having said that, it’s great for prototyping: you can use Javascript within a web browser console, i.e. without littering your computer with temporary files. What’s more, dedicated communities like the ScraperWiki forum are around to support and develop the associated tools, and in its more elaborate incarnations ‘scraping’ features in journals like Briefings in Bioinformatics (“Web scraping technologies in an API World” was published there just this week).

After having decided on PDBe PISA thanks to my scraped-together report, and finding no guidance on how to tackle the task, I turned to the bioinformatician’s equivalent of [computing/programming Q&A site] Stack Overflow known as Biostars. My question got a grand total of 0 answers(!), so what follows is my approach — which may either be of interest as a peek into the work going under the banner of ‘bioinformatics’ or as a guide to other scientists seeking to access the same information.

First off, a Python script parcelled up a list of every PDB code (the unique identifier to an author-deposited structure from X-ray crystallography, NMR etc.) in PDB into comma-separated chunks of 50, which were stuck onto the end of a web-service query as recommended. The server would process these queries, understood through its “API”: the CGI of cgi-bin in the URL means it’s invoking a script on the server, which in turn expects interfaces.pisa? to be followed by comma-separated PDB codes. Given these expectations, the API will respond in a regular manner each time, enabling reliable scripting.

With over 2000 such queries for interface data (each of them requesting 50 PDB-code-identified structures), this isn’t something you want to be doing manually. It wasn’t clear exactly which PDB entries were needed at the time, so the full complement was downloaded.

This download script just works for one query, putting the received XML in one file - to handle all 2029 queries, a bit of lateral thinking was required. 50 queries (each containing 50 PDB codes) were executed to make up a single interfacesi-j.xml file, where i is an integer 1 to 4, and likewise j from 1 to 10 (plus a bonus 4-11 to get those final 29). Download scripts (named similarly as getxmli-j.py) were written individually by another script — code writing code…

With download scripts written, the task of running each of them consecutively fell to yet another Python script, playing the sound of Super Mario picking up a coin when each file finished downloading, or the Mario pause-game sound upon encountering an error, because I could because clear feedback becomes necessary on something taking days across multiple computers.

Inevitably a minority of the queries failed, and had to be obtained separately.

Once downloaded, various pattern matching text-processing programs were run on the XML from within a shell script — readers unfamiliar with programming may have heard of these this week thanks to the 22 year old security bug(s) being referred to as shellshock. Shell scripts make looping through files in this manner a simple task, and are becoming essential for everyday file manipulation now that I’m a reformed Windows user. For the 41 XML files, a function runprocessor was called, with instructions to:

  1. Split each file successively at every <pisa_interfaces> tag through to the closing </pisa_interfaces> tag, the line numbers of which were stored together in an ordered list (an “array variable”) pisapairs
  2. Write each of these sections to a cache file xmlcache.xml, of suitable size for parsing by a Python XML parser.
  3. Reduce the time spent by the parser by in turn splitting this cache into just the PDB entries in the shortlist of interest with a function extractsubsets
  4. Initiate a Python script to read the entire cachesubset.xml file into memory, and write the pertinent structural data into a report formatted as tab-separated values (TSV). This file is a mere few hundred megabytes compared to the 120 GB grand total for the XML.

Clicking Details for an interface on the list of all interfaces for a given protein structure, e.g. for the only one in spider silk precursor protein spidroin, shows the interfacial residues in yellow:
image

The output threads together all interfacial residues and the associated statistical figures for each on a single line for every interface, but it’s simple enough to separate out each according to commas (then colons) to get a longform residue-per-line output once all XML is processed.

Progress is indicated in terminal output, where the current i and j values are printed followed by the pisapair (i.e. which of the 50 pisa_interfaces tags) is being worked through:

image

As shown in the logfile, there are inevitable errors, such as Entry not found: it’s simple enough to find the difference between the output report file’s list of PDB codes and the input ‘shortlist’, which can be mapped back to the constituent files for any follow-up investigation (the “wrangling” facet of computational science alluded to earlier) since the order of the original 2029 queries is known:

I’m putting these together in a code repository on GitHub, with a disclaimer that it’s not fit for all purposes (for instance if you’re interested in H-bonds, in brown on the PISA website residue table, above).

A lot of this was painfully slow — there’s nothing to be done about the speed of downloading the files, given that its rate is limited by the server. Yes there was a lot of data to get through, but Python’s sluggishness at the final step makes me wonder if I could implement some leaner algorithm, parallelise, etc., but with term recommenced code optimisation on a successfully completed task isn’t top priority. Advice on improvements would be appreciated if you have any.

I’m currently reading Jones & Pevzner’s An Introduction to Bioinformatics Algorithms which gives insight into how you can analyse and improve these types of operations (the book is core reading for a Coursera.org lecture series which kicks off next month), and have been recommended Goldwasser & Tamassia’s Data Structures and Algorithms in Python (a few online resources in a similar vein are available here).

I’ve also been fiddling with Julia, an R-like language with C-like speeds — in a 2012 blog post its creators say they “created Julia, in short, because we are greedy”. Fernando Perez is overseeing its incorporation into IPython Notebooks as ‘Project Jupyter’ and a port of R’s ggplot2 library has recently emerged for Julia under the name of Gadfly (a tutorial IPy NB is up here).

I’m starting a final year undergraduate project as of this week, on mapping small RNA-seq data to miRNAs, under the supervision of the founder of the database central to cataloguing this class of non-coding RNA ‒ super exciting stuff! :¬)

If you’ve got questions on PISA you think I could help with, feel free to ask here, or shoot me an email.

PDBe PISA homepage

✣ Peter Briggs, a scientific programmer at STFC Daresbury Laboratory, has a nice little guide to the service here.

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Pairing and stacking
Everybody knows GC base pairs are stronger than AT base pairs because they form three hydrogen bonds rather than 2, right?

The specificity of Watson-Crick base pairing is crucial to DNA duplex formation and covered early on in any good introductory genetics course. I&#8217;d always thought that these were what primarily led to the stability of the duplex, and was sure this is what was taught in first year too, but earlier this year I noted base stacking effects were in fact the primary determinants of this fundamental molecular phenomenon.
Taking a look around online it&#8217;s easy to find contradicting opinions on the subject, which makes me feel a bit less foolish over my uncertainty.
Hydrophobic, electrostatic base-stacking interactions depend on base aromaticity and dipole moments
Maxim Frank-Kamenetskii (whose father also worked in bio-/chemical physics) and colleagues at Boston University published a paper resolving the ambiguity as late as 2006(!) in which they studied DNA molecules with ‘solitary nicks and gaps [to] measure temperature and salt-dependence of the stacking free energy of the DNA double helix’

For the first time, DNA stacking parameters are obtained directly (without extrapolation) for temperatures from below room temperature to close to melting temperature.
From stacking parameters of individual contacts, we calculate base-stacking contribution to the stability of A•T- and G•C-containing DNA polymers. We find that temperature and salt dependences of the stacking term fully determine the temperature and the salt dependence of DNA stability parameters. For all temperatures and salt concentrations employed in present study, base-stacking is the main stabilizing factor in the DNA double helix. A•T pairing is always destabilizing and G•C pairing contributes almost no stabilization. Base-stacking interaction dominates not only in the duplex overall stability but also significantly contributes into the dependence of the duplex stability on its sequence.

This might seem trivial, but the significance comes from the dependence of a base pair&#8217;s stability on its neighbours when stacking is more important.

Nearest-neighbor stability parameters have been introduced to account for sequence effects in DNA stability. These parameters are obtained from the analysis of the melting data for DNA polymers, DNA oligomers and DNA dumbbells, and present the cumulative (base pairing and stacking) contribution of each dinucleotide stack to the overall stability of the molecule. In fact, DNA melting experiments do not allow separation of the two contributions.

I&#8217;d definitely heard the Frank-Kamenetskii family name somewhere before, but rather than his father it&#8217;s more likely to have been from readings in DNA topology.

Partitioning of base pairing and stacking contributions to DNA stability not only delivers a new aspect in the fundamental understanding of DNA structure and energetics, but also it has significant implications in a number of biological processes. Fluctuations in local helical conformation of DNA, the phenomenon known as DNA breathing, lead to infrequent events of base pair opening thus making normally buried groups available for modification and interaction with proteins. Fluctuational base pair opening implies disruption of hydrogen bonds between the complementary bases and flipping the base out of the helical stack disrupting two contacts. Heterogeneous stacking at these contacts determine sequence dependence of the base pair fluctuational motility.
Moreover, single-stranded break (a nick) in the DNA double helix is stabilized by stacking interactions between base pairs flanking the lesion; these interactions are sequence-dependent. In the cell, DNA nicks are substrates for DNA damage-detecting and DNA-repair proteins.

Frank-Kamenetskii reviewed DNA breathing, or ‘fraying’ last year in Nucleic Acids Research, noting how the issue of ‘strong bending of the double helix has attracted a lot of attention’ over the last decade.

It was suggested by Crick and Klug that sharp kinks of the double helix, which maintain base pairing but disrupt the stacking interaction between two adjacent base pairs, can be energetically favorable way to make strong bends of the double helix. Although such kinks have to be energetically costly, they can essentially reduce the bends and, correspondingly, the bending energy in the neighboring stacks of base pairs.
Similar, but not identical, to Crick &amp; Klug kinks are kinks associated with base pair openings. Probably, energetic cost of such disruptions should be even higher than for Crick &amp; Klug kinks. However, the disruptions that include open base pairs should provide even more local flexibility and could be a preferable way of changing twist of the double helix, if a stressed conformation requires it.

To divert from the topic a little here, the conclusions regarding DNA breathing were fascinating:

The free energy of a base pair opening is close to 7 kcal/mol for opening AT base pairs. If we assume that this free energy corresponds to formation of flexible hinges… we find that the average number of kinks approaches 1 for DNA circles of 70 bp.
[These data] make it very possible that the real nature of the double helix disruption, when short fragments are circularized, consists in the single base pair openings. However, there are no conformational data on DNA segments with opened base pairs detected by the cited methods. Thus, at present time we do not have sufficient information on structural and energetic properties of DNA disruptions to solve this challenging problem solely by the computations.


Essentially, this breathing/fraying phenomenon can lead to circularisation, which helps explain the abundance of these compared to predictions made from the wormlike chain model of DNA. In unstressed DNA, the probability of these openings is around 1 in 10,000 but shoots up in small DNA circles to 1 open bp per circle of 70bp.
Stacking is mentioned fleetingly once more in the final paragraph, with reference to Hoogsteen (certain non-canonical i.e. non-CG/AT) base pairs, which appear in ~1% of linear DNA.

Although it looks like stacking with adjacent pairs is well preserved when a Hoogsteen base pair is formed, it may still produce a point of higher flexibility that can facilitate strong bending of the double helix.

The experiments of the 2006 paper were pretty elegant ‒ nick a strand of DNA at a set position (making it migrate more slowly on a gel) and match up stacking interactions to changes in migration distance

Differential retardation of nicked DNA is due to specific interactions characteristic to each nicked dinucleotide stack. Quantitatively, equilibrium between stacked/closed and unstacked/open conformations at the nick site is governed by stacking free energy, ΔGST, so that:
Nclosed / Nopen = e − ΔGST / RT
where Nclosed and Nopen are occupancies of stacked and unstacked conformations at the DNA nick, respectively, R is the universal gas constant and T is the absolute temperature.

To get really esoteric here, the authors note that while “presence of a single-stranded gap precludes stacking interactions between two helical interfaces… this is not the case for 1 nt gaps”.

…it appears that stacking between the base pairs flanking the gap is restored to some degree leading to anisotropic, directional bending of the molecule reducing the size of the gapped cavity 48,55.
Molecules with longer gaps however, have been shown to possess isotropic bending flexibility which manifests itself in the absence of helical periodicity in electrophoretic mobility and cyclization kinetics measurements 50,53. Molecules with gaps 2, 3 and 4 nt in length migrate very closely during PAGE revealing similarity of their effective conformations. Additional factors [i.e. sequence of single-stranded linker] come into effect once the gap size is longer than persistent length of single chains—in this case, the gap is likely to act as a hinge.

Upon going still further into their own results, the authors concede that while this represents very strong evidence for stacking contribution to stability predominating over base pairing, the model used is still simplistic.
Briefly, H-bond breaking between complementary bases and base stacking along the helical axis are disrupted upon melting, increasing conformational entropy, releasing counterions and changing solvent interactions — since as stated above, base stacking effects are hydrophobic, a model which doesn&#8217;t examine these solvation effects isn&#8217;t wholly conclusive.

Strictly speaking, partitioning of these effects between ΔGST and ΔGBP terms is not known.

This isn&#8217;t challenging the main result, rather commenting on a finer point. The extended discussion however is approaching my limit for one night; if you&#8217;re interested to read more, as usual the links are below.

GC pairing does not contribute to stabilization of DNA duplex, while A•T pairing is always destabilizing. This finding presents a paradigm shift in the understanding of the interplay of the forces stabilizing DNA double helix. For all temperatures heterogeneity of stacking interactions in A•T- and G•C-containing contacts accounts for at least half of heterogeneity in the stability of A•T- and G•C-polymers; the other half is due to the difference in the energetics of A•T and G•C base pairing. The data on separation of stacking and base pairing contributions have made it possible to describe sequence-dependent fluctuational opening of the DNA double helix

• Yakovchuk et al (2006) Base-stacking and base-pairing contributions into thermal stability of the DNA double helix. Nucleic Acids Res, 34(2)•• Krueger and Frank-Kamenetskii (2006) Sequence-dependent base pair opening in DNA double helix. Biophys J, 90(9)

Pairing and stacking

Everybody knows GC base pairs are stronger than AT base pairs because they form three hydrogen bonds rather than 2, right?

image

The specificity of Watson-Crick base pairing is crucial to DNA duplex formation and covered early on in any good introductory genetics course. I’d always thought that these were what primarily led to the stability of the duplex, and was sure this is what was taught in first year too, but earlier this year I noted base stacking effects were in fact the primary determinants of this fundamental molecular phenomenon.

Taking a look around online it’s easy to find contradicting opinions on the subject, which makes me feel a bit less foolish over my uncertainty.

Hydrophobic, electrostatic base-stacking interactions depend on base aromaticity and dipole moments

Maxim Frank-Kamenetskii (whose father also worked in bio-/chemical physics) and colleagues at Boston University published a paper resolving the ambiguity as late as 2006(!) in which they studied DNA molecules with ‘solitary nicks and gaps [to] measure temperature and salt-dependence of the stacking free energy of the DNA double helix

For the first time, DNA stacking parameters are obtained directly (without extrapolation) for temperatures from below room temperature to close to melting temperature.

From stacking parameters of individual contacts, we calculate base-stacking contribution to the stability of A•T- and G•C-containing DNA polymers. We find that temperature and salt dependences of the stacking term fully determine the temperature and the salt dependence of DNA stability parameters. For all temperatures and salt concentrations employed in present study, base-stacking is the main stabilizing factor in the DNA double helix. A•T pairing is always destabilizing and G•C pairing contributes almost no stabilization. Base-stacking interaction dominates not only in the duplex overall stability but also significantly contributes into the dependence of the duplex stability on its sequence.

This might seem trivial, but the significance comes from the dependence of a base pair’s stability on its neighbours when stacking is more important.

Nearest-neighbor stability parameters have been introduced to account for sequence effects in DNA stability. These parameters are obtained from the analysis of the melting data for DNA polymers, DNA oligomers and DNA dumbbells, and present the cumulative (base pairing and stacking) contribution of each dinucleotide stack to the overall stability of the molecule. In fact, DNA melting experiments do not allow separation of the two contributions.

I’d definitely heard the Frank-Kamenetskii family name somewhere before, but rather than his father it’s more likely to have been from readings in DNA topology.

Partitioning of base pairing and stacking contributions to DNA stability not only delivers a new aspect in the fundamental understanding of DNA structure and energetics, but also it has significant implications in a number of biological processes. Fluctuations in local helical conformation of DNA, the phenomenon known as DNA breathing, lead to infrequent events of base pair opening thus making normally buried groups available for modification and interaction with proteins. Fluctuational base pair opening implies disruption of hydrogen bonds between the complementary bases and flipping the base out of the helical stack disrupting two contacts. Heterogeneous stacking at these contacts determine sequence dependence of the base pair fluctuational motility.

Moreover, single-stranded break (a nick) in the DNA double helix is stabilized by stacking interactions between base pairs flanking the lesion; these interactions are sequence-dependent. In the cell, DNA nicks are substrates for DNA damage-detecting and DNA-repair proteins.

Frank-Kamenetskii reviewed DNA breathing, or ‘fraying’ last year in Nucleic Acids Research, noting how the issue of ‘strong bending of the double helix has attracted a lot of attention’ over the last decade.

It was suggested by Crick and Klug that sharp kinks of the double helix, which maintain base pairing but disrupt the stacking interaction between two adjacent base pairs, can be energetically favorable way to make strong bends of the double helix. Although such kinks have to be energetically costly, they can essentially reduce the bends and, correspondingly, the bending energy in the neighboring stacks of base pairs.

Similar, but not identical, to Crick & Klug kinks are kinks associated with base pair openings. Probably, energetic cost of such disruptions should be even higher than for Crick & Klug kinks. However, the disruptions that include open base pairs should provide even more local flexibility and could be a preferable way of changing twist of the double helix, if a stressed conformation requires it.

To divert from the topic a little here, the conclusions regarding DNA breathing were fascinating:

image

The free energy of a base pair opening is close to 7 kcal/mol for opening AT base pairs. If we assume that this free energy corresponds to formation of flexible hinges… we find that the average number of kinks approaches 1 for DNA circles of 70 bp.

[These data] make it very possible that the real nature of the double helix disruption, when short fragments are circularized, consists in the single base pair openings. However, there are no conformational data on DNA segments with opened base pairs detected by the cited methods. Thus, at present time we do not have sufficient information on structural and energetic properties of DNA disruptions to solve this challenging problem solely by the computations.

Essentially, this breathing/fraying phenomenon can lead to circularisation, which helps explain the abundance of these compared to predictions made from the wormlike chain model of DNA. In unstressed DNA, the probability of these openings is around 1 in 10,000 but shoots up in small DNA circles to 1 open bp per circle of 70bp.

Stacking is mentioned fleetingly once more in the final paragraph, with reference to Hoogsteen (certain non-canonical i.e. non-CG/AT) base pairs, which appear in ~1% of linear DNA.

Although it looks like stacking with adjacent pairs is well preserved when a Hoogsteen base pair is formed, it may still produce a point of higher flexibility that can facilitate strong bending of the double helix.

The experiments of the 2006 paper were pretty elegant ‒ nick a strand of DNA at a set position (making it migrate more slowly on a gel) and match up stacking interactions to changes in migration distance

Differential retardation of nicked DNA is due to specific interactions characteristic to each nicked dinucleotide stack. Quantitatively, equilibrium between stacked/closed and unstacked/open conformations at the nick site is governed by stacking free energy, ΔGST, so that:

Nclosed / Nopen = e − ΔGST / RT

where Nclosed and Nopen are occupancies of stacked and unstacked conformations at the DNA nick, respectively, R is the universal gas constant and T is the absolute temperature.

To get really esoteric here, the authors note that while “presence of a single-stranded gap precludes stacking interactions between two helical interfaces… this is not the case for 1 nt gaps”.

…it appears that stacking between the base pairs flanking the gap is restored to some degree leading to anisotropic, directional bending of the molecule reducing the size of the gapped cavity 48,55.

Molecules with longer gaps however, have been shown to possess isotropic bending flexibility which manifests itself in the absence of helical periodicity in electrophoretic mobility and cyclization kinetics measurements 50,53. Molecules with gaps 2, 3 and 4 nt in length migrate very closely during PAGE revealing similarity of their effective conformations. Additional factors [i.e. sequence of single-stranded linker] come into effect once the gap size is longer than persistent length of single chains—in this case, the gap is likely to act as a hinge.

Upon going still further into their own results, the authors concede that while this represents very strong evidence for stacking contribution to stability predominating over base pairing, the model used is still simplistic.

Briefly, H-bond breaking between complementary bases and base stacking along the helical axis are disrupted upon melting, increasing conformational entropy, releasing counterions and changing solvent interactions — since as stated above, base stacking effects are hydrophobic, a model which doesn’t examine these solvation effects isn’t wholly conclusive.

Strictly speaking, partitioning of these effects between ΔGST and ΔGBP terms is not known.

This isn’t challenging the main result, rather commenting on a finer point. The extended discussion however is approaching my limit for one night; if you’re interested to read more, as usual the links are below.

GC pairing does not contribute to stabilization of DNA duplex, while A•T pairing is always destabilizing. This finding presents a paradigm shift in the understanding of the interplay of the forces stabilizing DNA double helix. For all temperatures heterogeneity of stacking interactions in A•T- and G•C-containing contacts accounts for at least half of heterogeneity in the stability of A•T- and G•C-polymers; the other half is due to the difference in the energetics of A•T and G•C base pairing. The data on separation of stacking and base pairing contributions have made it possible to describe sequence-dependent fluctuational opening of the DNA double helix

• Yakovchuk et al (2006) Base-stacking and base-pairing contributions into thermal stability of the DNA double helix. Nucleic Acids Res, 34(2)
•• Krueger and Frank-Kamenetskii (2006) Sequence-dependent base pair opening in DNA double helix. Biophys J90(9)

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Molecular motions inside the cell
A paper in Science this week describes the use of carbon nanotubes to pinpoint the movements of the living cell in fine detail, making for a really nice study in quantitative/mathematical biology.

Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random “stirring” that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.

The &#8220;mesoscopic" scale is more often seen in the context of pure and applied physics (microelectronics, nanofabrication and nanotechnology), though journals such as Soft Matter present research articles giving the same &#8216;condensed matter&#8217; treatment to biological systems (“Where physics meets chemistry meets biology”).
From ancient Greek μέσος it refers simply to a &#8216;middle&#8217;/intermediate between the molecular and macroscopic scale, where neither atomistic/quantum nor classical physics/bulk models best describe observed behaviour, and novel effects may be described — from interference effects, to quantum confinement (giving rise to band gaps) and charging effects (such as the Coulomb blockade/staircase).
Although often presented as a water-based solvent, the cytosol is more accurately described as a “highly dynamic composite material ” with mechanical properties dominated by microtubules (MTs), F-actin and intermediate filaments; all driven by metabolism-energised polymerisation of actin and tubulin and from motor proteins (specifically nucleotide triphosphate hydrolysis).
The traditional technique to observe cells in motion is fluorescence microscopy, though long-term tracking of single molecules has been hindered by fluorophores&#8217; instabilities and the fluorescence background in cells.
Though biological networks have been termed ‘scale-free’ or ‘-invariant’, and metabolic rate for example is well known to follow a power law, the internal structure of the cell itself is far from self-similar across scales. 

At short times (microseconds to milliseconds), thermal motions should dominate. Between milliseconds and seconds, thermal diffusion might still be relevant, but there is mounting evidence, both in vitro and in vivo, that the motion of larger objects couples to myosin-driven stress fluctuations in the cytoskeleton.

»  Mizuno (2007) Nonequilibrium mechanics of active cytoskeletal networks.
»  Brangwynne (2008) Cytoplasmic diffusion: Molecular motors mix it up.

Here, temporal fluctuations, reminiscent of thermal diffusion in liquids, can arise from nonequilibrium dynamics in the viscoelastic cytoskeleton. On longer time scales, from minutes to hours, directed transport and larger-scale collective motions typically dominate. The motion of probe particles tracked inside cells has been classified as subdiffusive, diffusive, or superdiffusive. Such classifications, however, obscure the distinction between thermally driven and nonequilibrium fluctuations and are inadequate to identify intracellular material properties.

Motor proteins direct a whole host of molecular motions, kinesins and myosins being among the most heavily studied in vitro. Using fluorescence microscopy to track individual motor proteins is not only limited by instability of fluorophores, but the quality of the images taken (&#8220;signal to noise&#8221;) and efficiency of targetting probes to specific molecules.
Modern optical equipment and carefully designed fluorescent dyes have enabled experiments tracking single molecules at a time, though in living cells the authors note these experiments&#8217; timeframes have been limited to around a second.

Their solution was to use single-walled carbon nanotubes (SWNTs), “stiff quasi–one-dimensional tubular all-carbon nanostructures with diameters of ~1 nm and persistence lengths above 10 μm” — which have the convenient property of luminescence in the near-infrared, a region ‘virtually free of autofluorescence in biological tissues’. Not only this, but the excitation time is ~100ps, such that high excitation can give ~1&#160;ms resolution (1&#160;ms = 109 ps).
The nanotubes were dispersed throughout the cell wrapped in short DNA oligonucleotides, with HaloTag protein fusion tags covalently attaching them specifically onto kinesin motor proteins (see Fig. 1, above).

Besides observing directed kinesin-driven transport on MTs, it is possible to directly observe fluctuations of the MT network because a moving kinesin must be bound to a MT. The MT tracks are embedded in the viscoelastic actin cytoskeleton, which in turn fluctuates as a result of stresses generated by cytoplasmic myosins.


With just 100 per cell, the group could track kinesin for up to an hour and a half, observing ~30% of them moving with some sense of direction; the rest locally constrained and moving in a random (stochastic) manner.
Some of the kinesins moved the whole length of the cell, suggesting they had cargo vesicles [along with other motor proteins] attached. Calculating mean squared displacement, MSD, of the molecules&#8217; trajectories showed it grew over time following a power law which could be used to characterise the motion, 〈Δr2(τ)〉∝ τα(where r is distance travelled in the focal plane and τ the lag time). The exponent α shifted from ¼ to 1 between 5&#160;ms and 2.5&#160;s, indicating clear scale variance to the motion.
After this, the group acquired the nanotubes&#8217; fluorescent signal at a rate of four frames per second ‒ using this 250&#160;ms window to observe an intermediate between the thermal diffusion seen on the short timescales and directed motor activity.
With a well-designed control or two, they showed that the transverse motion of the nanotube-marked microtubules was not due to kinesin motors, but reflecting intrinsic dynamics of the cytoskeleton.

The way the relatively rigid MTs report these dynamics depends on two restoring forces: the elastic force of bent MTs and the force exerted by the strained cytoskeletal matrix in which the MTs are embedded. Because it is hard to bend an elastic rod on short length scales, the surrounding matrix yields to the MT when it is deformed on short length scales. By contrast, the MT yields to matrix forces for deflections of wavelength larger than ~1 μm. The shorter-wavelength MT deflections relax faster than our 5-ms frame rate. Therefore, we assume that the transverse MT motion we observe reflects the (active or passive) strain fluctuations of the surrounding matrix.
The MSD power-law exponent α generally reflects the randomness of motion. More precisely, in any medium, the MSD of an embedded probe particle is governed both by the material properties of the medium and the temporal characteristics of the forces driving the particle. For thermally driven Brownian motion in simple liquids, the MSD exponent α = 1. For thermal motion in viscoelastic media, which exhibit time- and frequency-dependent viscosity and elasticity, α &lt; 1 strictly holds. For viscoelastic materials, the stiffness G(ω) typically increases with a power of frequency ω: G(ω) ∝ ωβ. This is observed in polymer solutions, where the viscoelastic exponent β ≈ 0.5 to 0.8, as well as in cells, where β ≈ 0.1 to 0.2 on time scales on the order of seconds. This value of the exponent is close to what is expected for purely elastic materials, where β = 0. 
The nearly elastic behavior of cells can be understood as a consequence of strong cross-linking in the cytoskeleton.
 Knowing the driving forces, it is possible to construct a relation between MSD exponent α and viscoelastic exponent β. For thermal driving forces, the MSD exponent α = β. Thermal fluctuations can therefore never appear as “superdiffusive” motion with α &gt; 1. Nonthermal driving, by contrast, can result in superdiffusive motion. Theory provides a specific prediction for motion in nearly elastic solids driven by random stress fluctuations with long correlation times and sudden transitions: α = 1 + 2β. This prediction is expected to apply for cytoskeletal stress fluctuations caused by randomly distributed cytoplasmic myosin minifilaments. Myosin locally contracts the actin network with an attachment time of several seconds, followed by sudden release. Some hints of this predicted scaling have been reported for cells and reconstituted acto-myosin model systems. When β = 0 (i.e., in the elastic limit), the resulting MSDs can look deceptively like Brownian motion in a simple liquid, although the physical reason is entirely different. For observation times τ longer than the correlation time of the driving forces, the MSD is predicted to level off, as we observed. In our experiments, the stress correlation time should correspond to typical cytoplasmic myosin motor engagement times, which are indeed reported to be ~10&#160;s in cells.

Still attached to microtubules, the kinesin molecules exhibit vigorous random (Brownian-like) motion as they are buffeted by myosins as described ‒ likely thrusting MTs into the path of other cellular particles. Tubulin forms strong tubular filaments embedded in a more flexible actin network. Nonmuscle myosin II exerts mechanical stress on it, which is released ‘suddenly’ as random stirring of the whole filament network, including the microtubules.

We observed a transition between thermal dynamics in the dominantly elastic cytoskeleton at short times to strongly nonequilibrium power-law dynamics, likely driven by myosin activity, at intermediate times. When the time exceeded the correlation time of the random stress generators, the intermediate regime was followed by a saturation to a maximum MSD, nearly constant over time. Note that in this regime, the MSD amplitude corresponds to a root mean square displacement of ~500 nm, which is larger than the estimated mesh size of the actin network, and thus larger than the expected spacing of obstacles in the crowded cytoplasm.

The authors lastly used myosin inhibitor blebbistatin to block myosin from the actin network, confirming their hypothesis with a dose-dependent reduction in what they call the amplitude of active stirring, and exponent α, “establishing nonmuscle myosin II as the dominant driving factor for random cytoskeletal stirring”.

We can explain the regimes we observe by a quantitative model of cytoskeletal fluctuations and directed motor motion that describes the transition from thermal motion to nonequilibrium stirring dynamics driven by myosin, as well as the transition from stirring dynamics to directed transport driven by kinesin. Our observations were made possible by the use of SWNT labels for broadband molecular tracking in cells. Many questions concerning motor transport in cells will now be addressable using this approach. We have focused here on the stirring dynamics, which constitute an important mode of active intracellular transport between the limits of random thermal diffusion and directed transport, accelerating nonspecific transport through the nanoporous cytoskeleton.

Lead author Nikta Fakhri will soon leave the Göttingen Institute for Biophysics to join the faculty at MIT as assistant professor of physics. Fakhri gave a talk in Massachussets last year on the topic, to the Chemical Engineering department in which some of the details of this paper made their debut:

The discovery of fullerenes provided exciting insights into how highly symmetrical structures of pure carbon can have remarkable physical properties. Single-walled carbon nanotubes (SWNTs) are the vanguard of such architectures. The organization of the hexagonal honeycomb carbon lattice into high-aspect-ratio cylinders with a variety of helical symmetries creates very unusual macromolecular structures representing an emerging research area in condensed matter physics and materials science: traditionally hard materials appearing in new soft matter applications and environments.
… the dynamics of SWNTs in liquids are essentially polymer-like. By exploiting the intrinsic near-infrared fluorescence of semiconducting SWNTs, we have imaged the Brownian motion of individual nanotubes in water and have measured directly the bending stiffness of SWNTs. The semiflexible chain model represents accurately the configurational dynamics of SWNTs suspended in water. Interestingly, the persistence length of SWNTs is comparable to that of biopolymers. This finding paves the way for using SWNTs as a model system for semiflexible polymers to answer long-standing fundamental questions in polymer physics.
… the confined dynamics of stiff macromolecules in crowded environments [are] a common feature of polymer composites and the cell cytoskeleton. In fixed porous networks, we find that even a small bending flexibility strongly enhances SWNTs&#8217; motion. This ends a 30-year-old debate in polymer physics: the rotational diffusion constant is proportional to the filament bending compliance and counter-intuitively, independent of the network porosity. The dynamics of SWNTs in equilibrium and non-equilibrium biopolymer networks is more complex.
At long times, SWNTs reptate in the networks. At short times SWNTs can sample the spectrum of local stresses in equilibrium networks. In the non-equilibrium networks we observe strong local shape fluctuations driven by force generating molecular motors. I will discuss a newly developed microrheology technique in which we use nanotubes as “stealth probes” to measure viscoelastic properties of the host media. Finally, I will introduce a new single-molecule technique based on ultra-stable near-infrared fluorescence of short SWNTs, to study intracellular transport dynamics in living cells and in whole organisms. The combination of long-time stability and high signal-to-noise ratio enables the accurate long-term tracking of single motor proteins tagged with SWNTs traversing the entire cell. Remarkably, we can distinguish the motor protein’s motion along its microtubule track from the track’s underlying random non-thermal fluctuations.

She envisions the technology as applicable beyond probing biophysical questions, in the design of 'active' technical materials.
“Imagine a microscopic biomedical device that mixes tiny samples of blood with reagents to detect disease or smart filters that separate squishy from rigid materials.”
Fakhri will join the Physics of Living Systems group, seemingly on such a bio-materials science project. MIT lab colleague Jeremy England, known for work showing that E. coli reproduction is close to thermodynamic limits of efficiency, spoke of common interest in the cytosol and diffusive processes.

“We&#8217;re interested in the non-equilibrium thermodynamics of biological organization, so that could be construed to be about evolution and the origins of life or just about how you make or design self-replicators with desired properties.”
“Increasingly there are now instruments where you can make quantitative measurements on fluorescently labeled proteins in live cells,” England explains. “The cell biologists have their language and their frame of analysis that they&#8217;re most comfortable with for describing the phenomenon, but if there are interesting phenomena that are only going to be identifiable if you do the right quantitative analysis on all these numbers that you can now measure in the cell, then it&#8217;s useful to have people who are a bit more theoretically minded or physics minded who are there, when rubber meets road, when the data is being generated and helping to influence what kind of experiments get done.”
“We&#8217;re looking, for example, at diffusion of proteins in cells. Diffusion as a qualitative phenomenon is just things spreading out over space, but as a quantitative phenomenon, you can look at things like how rapidly a protein that&#8217;s labeled over here in the cell will wander over to another region of the cell that&#8217;s a certain distance away, and if you can make measurements of that, then you can start to say things that are more specific about characteristics of the diffusion that you are observing than simply seeing it spread out. And in those quantitative measurements, you can sometimes then see differences perhaps between different cells, or different conditions for the same type of cell, that may have biological relevance but that you wouldn&#8217;t have necessarily identified without the quantitative analysis,” England says.

⌇  Fakhri et al. (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science, 344(1687), 1031-5
See also:⌇  Levine and MacKintosh (2009) The mechanics and fluctuation spectrum of active gels. J Phys Chem B, 113, 3820–3830⌇  MacKintosh and Levine (2008) Nonequilibrium mechanics and dynamics of motor-activated gels. Phys Rev Lett, 100, 018104⌇  Lau et al. (2003) Microrheology, stress fluctuations, and active behavior of living cells. Phys Rev Lett, 91, 198101⇢  Related post : water&#8217;s SED failure in molecular orientational diffusion
Molecular motions inside the cell
A paper in Science this week describes the use of carbon nanotubes to pinpoint the movements of the living cell in fine detail, making for a really nice study in quantitative/mathematical biology.

Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random “stirring” that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.

The &#8220;mesoscopic" scale is more often seen in the context of pure and applied physics (microelectronics, nanofabrication and nanotechnology), though journals such as Soft Matter present research articles giving the same &#8216;condensed matter&#8217; treatment to biological systems (“Where physics meets chemistry meets biology”).
From ancient Greek μέσος it refers simply to a &#8216;middle&#8217;/intermediate between the molecular and macroscopic scale, where neither atomistic/quantum nor classical physics/bulk models best describe observed behaviour, and novel effects may be described — from interference effects, to quantum confinement (giving rise to band gaps) and charging effects (such as the Coulomb blockade/staircase).
Although often presented as a water-based solvent, the cytosol is more accurately described as a “highly dynamic composite material ” with mechanical properties dominated by microtubules (MTs), F-actin and intermediate filaments; all driven by metabolism-energised polymerisation of actin and tubulin and from motor proteins (specifically nucleotide triphosphate hydrolysis).
The traditional technique to observe cells in motion is fluorescence microscopy, though long-term tracking of single molecules has been hindered by fluorophores&#8217; instabilities and the fluorescence background in cells.
Though biological networks have been termed ‘scale-free’ or ‘-invariant’, and metabolic rate for example is well known to follow a power law, the internal structure of the cell itself is far from self-similar across scales. 

At short times (microseconds to milliseconds), thermal motions should dominate. Between milliseconds and seconds, thermal diffusion might still be relevant, but there is mounting evidence, both in vitro and in vivo, that the motion of larger objects couples to myosin-driven stress fluctuations in the cytoskeleton.

»  Mizuno (2007) Nonequilibrium mechanics of active cytoskeletal networks.
»  Brangwynne (2008) Cytoplasmic diffusion: Molecular motors mix it up.

Here, temporal fluctuations, reminiscent of thermal diffusion in liquids, can arise from nonequilibrium dynamics in the viscoelastic cytoskeleton. On longer time scales, from minutes to hours, directed transport and larger-scale collective motions typically dominate. The motion of probe particles tracked inside cells has been classified as subdiffusive, diffusive, or superdiffusive. Such classifications, however, obscure the distinction between thermally driven and nonequilibrium fluctuations and are inadequate to identify intracellular material properties.

Motor proteins direct a whole host of molecular motions, kinesins and myosins being among the most heavily studied in vitro. Using fluorescence microscopy to track individual motor proteins is not only limited by instability of fluorophores, but the quality of the images taken (&#8220;signal to noise&#8221;) and efficiency of targetting probes to specific molecules.
Modern optical equipment and carefully designed fluorescent dyes have enabled experiments tracking single molecules at a time, though in living cells the authors note these experiments&#8217; timeframes have been limited to around a second.

Their solution was to use single-walled carbon nanotubes (SWNTs), “stiff quasi–one-dimensional tubular all-carbon nanostructures with diameters of ~1 nm and persistence lengths above 10 μm” — which have the convenient property of luminescence in the near-infrared, a region ‘virtually free of autofluorescence in biological tissues’. Not only this, but the excitation time is ~100ps, such that high excitation can give ~1&#160;ms resolution (1&#160;ms = 109 ps).
The nanotubes were dispersed throughout the cell wrapped in short DNA oligonucleotides, with HaloTag protein fusion tags covalently attaching them specifically onto kinesin motor proteins (see Fig. 1, above).

Besides observing directed kinesin-driven transport on MTs, it is possible to directly observe fluctuations of the MT network because a moving kinesin must be bound to a MT. The MT tracks are embedded in the viscoelastic actin cytoskeleton, which in turn fluctuates as a result of stresses generated by cytoplasmic myosins.


With just 100 per cell, the group could track kinesin for up to an hour and a half, observing ~30% of them moving with some sense of direction; the rest locally constrained and moving in a random (stochastic) manner.
Some of the kinesins moved the whole length of the cell, suggesting they had cargo vesicles [along with other motor proteins] attached. Calculating mean squared displacement, MSD, of the molecules&#8217; trajectories showed it grew over time following a power law which could be used to characterise the motion, 〈Δr2(τ)〉∝ τα(where r is distance travelled in the focal plane and τ the lag time). The exponent α shifted from ¼ to 1 between 5&#160;ms and 2.5&#160;s, indicating clear scale variance to the motion.
After this, the group acquired the nanotubes&#8217; fluorescent signal at a rate of four frames per second ‒ using this 250&#160;ms window to observe an intermediate between the thermal diffusion seen on the short timescales and directed motor activity.
With a well-designed control or two, they showed that the transverse motion of the nanotube-marked microtubules was not due to kinesin motors, but reflecting intrinsic dynamics of the cytoskeleton.

The way the relatively rigid MTs report these dynamics depends on two restoring forces: the elastic force of bent MTs and the force exerted by the strained cytoskeletal matrix in which the MTs are embedded. Because it is hard to bend an elastic rod on short length scales, the surrounding matrix yields to the MT when it is deformed on short length scales. By contrast, the MT yields to matrix forces for deflections of wavelength larger than ~1 μm. The shorter-wavelength MT deflections relax faster than our 5-ms frame rate. Therefore, we assume that the transverse MT motion we observe reflects the (active or passive) strain fluctuations of the surrounding matrix.
The MSD power-law exponent α generally reflects the randomness of motion. More precisely, in any medium, the MSD of an embedded probe particle is governed both by the material properties of the medium and the temporal characteristics of the forces driving the particle. For thermally driven Brownian motion in simple liquids, the MSD exponent α = 1. For thermal motion in viscoelastic media, which exhibit time- and frequency-dependent viscosity and elasticity, α &lt; 1 strictly holds. For viscoelastic materials, the stiffness G(ω) typically increases with a power of frequency ω: G(ω) ∝ ωβ. This is observed in polymer solutions, where the viscoelastic exponent β ≈ 0.5 to 0.8, as well as in cells, where β ≈ 0.1 to 0.2 on time scales on the order of seconds. This value of the exponent is close to what is expected for purely elastic materials, where β = 0. 
The nearly elastic behavior of cells can be understood as a consequence of strong cross-linking in the cytoskeleton.
 Knowing the driving forces, it is possible to construct a relation between MSD exponent α and viscoelastic exponent β. For thermal driving forces, the MSD exponent α = β. Thermal fluctuations can therefore never appear as “superdiffusive” motion with α &gt; 1. Nonthermal driving, by contrast, can result in superdiffusive motion. Theory provides a specific prediction for motion in nearly elastic solids driven by random stress fluctuations with long correlation times and sudden transitions: α = 1 + 2β. This prediction is expected to apply for cytoskeletal stress fluctuations caused by randomly distributed cytoplasmic myosin minifilaments. Myosin locally contracts the actin network with an attachment time of several seconds, followed by sudden release. Some hints of this predicted scaling have been reported for cells and reconstituted acto-myosin model systems. When β = 0 (i.e., in the elastic limit), the resulting MSDs can look deceptively like Brownian motion in a simple liquid, although the physical reason is entirely different. For observation times τ longer than the correlation time of the driving forces, the MSD is predicted to level off, as we observed. In our experiments, the stress correlation time should correspond to typical cytoplasmic myosin motor engagement times, which are indeed reported to be ~10&#160;s in cells.

Still attached to microtubules, the kinesin molecules exhibit vigorous random (Brownian-like) motion as they are buffeted by myosins as described ‒ likely thrusting MTs into the path of other cellular particles. Tubulin forms strong tubular filaments embedded in a more flexible actin network. Nonmuscle myosin II exerts mechanical stress on it, which is released ‘suddenly’ as random stirring of the whole filament network, including the microtubules.

We observed a transition between thermal dynamics in the dominantly elastic cytoskeleton at short times to strongly nonequilibrium power-law dynamics, likely driven by myosin activity, at intermediate times. When the time exceeded the correlation time of the random stress generators, the intermediate regime was followed by a saturation to a maximum MSD, nearly constant over time. Note that in this regime, the MSD amplitude corresponds to a root mean square displacement of ~500 nm, which is larger than the estimated mesh size of the actin network, and thus larger than the expected spacing of obstacles in the crowded cytoplasm.

The authors lastly used myosin inhibitor blebbistatin to block myosin from the actin network, confirming their hypothesis with a dose-dependent reduction in what they call the amplitude of active stirring, and exponent α, “establishing nonmuscle myosin II as the dominant driving factor for random cytoskeletal stirring”.

We can explain the regimes we observe by a quantitative model of cytoskeletal fluctuations and directed motor motion that describes the transition from thermal motion to nonequilibrium stirring dynamics driven by myosin, as well as the transition from stirring dynamics to directed transport driven by kinesin. Our observations were made possible by the use of SWNT labels for broadband molecular tracking in cells. Many questions concerning motor transport in cells will now be addressable using this approach. We have focused here on the stirring dynamics, which constitute an important mode of active intracellular transport between the limits of random thermal diffusion and directed transport, accelerating nonspecific transport through the nanoporous cytoskeleton.

Lead author Nikta Fakhri will soon leave the Göttingen Institute for Biophysics to join the faculty at MIT as assistant professor of physics. Fakhri gave a talk in Massachussets last year on the topic, to the Chemical Engineering department in which some of the details of this paper made their debut:

The discovery of fullerenes provided exciting insights into how highly symmetrical structures of pure carbon can have remarkable physical properties. Single-walled carbon nanotubes (SWNTs) are the vanguard of such architectures. The organization of the hexagonal honeycomb carbon lattice into high-aspect-ratio cylinders with a variety of helical symmetries creates very unusual macromolecular structures representing an emerging research area in condensed matter physics and materials science: traditionally hard materials appearing in new soft matter applications and environments.
… the dynamics of SWNTs in liquids are essentially polymer-like. By exploiting the intrinsic near-infrared fluorescence of semiconducting SWNTs, we have imaged the Brownian motion of individual nanotubes in water and have measured directly the bending stiffness of SWNTs. The semiflexible chain model represents accurately the configurational dynamics of SWNTs suspended in water. Interestingly, the persistence length of SWNTs is comparable to that of biopolymers. This finding paves the way for using SWNTs as a model system for semiflexible polymers to answer long-standing fundamental questions in polymer physics.
… the confined dynamics of stiff macromolecules in crowded environments [are] a common feature of polymer composites and the cell cytoskeleton. In fixed porous networks, we find that even a small bending flexibility strongly enhances SWNTs&#8217; motion. This ends a 30-year-old debate in polymer physics: the rotational diffusion constant is proportional to the filament bending compliance and counter-intuitively, independent of the network porosity. The dynamics of SWNTs in equilibrium and non-equilibrium biopolymer networks is more complex.
At long times, SWNTs reptate in the networks. At short times SWNTs can sample the spectrum of local stresses in equilibrium networks. In the non-equilibrium networks we observe strong local shape fluctuations driven by force generating molecular motors. I will discuss a newly developed microrheology technique in which we use nanotubes as “stealth probes” to measure viscoelastic properties of the host media. Finally, I will introduce a new single-molecule technique based on ultra-stable near-infrared fluorescence of short SWNTs, to study intracellular transport dynamics in living cells and in whole organisms. The combination of long-time stability and high signal-to-noise ratio enables the accurate long-term tracking of single motor proteins tagged with SWNTs traversing the entire cell. Remarkably, we can distinguish the motor protein’s motion along its microtubule track from the track’s underlying random non-thermal fluctuations.

She envisions the technology as applicable beyond probing biophysical questions, in the design of 'active' technical materials.
“Imagine a microscopic biomedical device that mixes tiny samples of blood with reagents to detect disease or smart filters that separate squishy from rigid materials.”
Fakhri will join the Physics of Living Systems group, seemingly on such a bio-materials science project. MIT lab colleague Jeremy England, known for work showing that E. coli reproduction is close to thermodynamic limits of efficiency, spoke of common interest in the cytosol and diffusive processes.

“We&#8217;re interested in the non-equilibrium thermodynamics of biological organization, so that could be construed to be about evolution and the origins of life or just about how you make or design self-replicators with desired properties.”
“Increasingly there are now instruments where you can make quantitative measurements on fluorescently labeled proteins in live cells,” England explains. “The cell biologists have their language and their frame of analysis that they&#8217;re most comfortable with for describing the phenomenon, but if there are interesting phenomena that are only going to be identifiable if you do the right quantitative analysis on all these numbers that you can now measure in the cell, then it&#8217;s useful to have people who are a bit more theoretically minded or physics minded who are there, when rubber meets road, when the data is being generated and helping to influence what kind of experiments get done.”
“We&#8217;re looking, for example, at diffusion of proteins in cells. Diffusion as a qualitative phenomenon is just things spreading out over space, but as a quantitative phenomenon, you can look at things like how rapidly a protein that&#8217;s labeled over here in the cell will wander over to another region of the cell that&#8217;s a certain distance away, and if you can make measurements of that, then you can start to say things that are more specific about characteristics of the diffusion that you are observing than simply seeing it spread out. And in those quantitative measurements, you can sometimes then see differences perhaps between different cells, or different conditions for the same type of cell, that may have biological relevance but that you wouldn&#8217;t have necessarily identified without the quantitative analysis,” England says.

⌇  Fakhri et al. (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science, 344(1687), 1031-5
See also:⌇  Levine and MacKintosh (2009) The mechanics and fluctuation spectrum of active gels. J Phys Chem B, 113, 3820–3830⌇  MacKintosh and Levine (2008) Nonequilibrium mechanics and dynamics of motor-activated gels. Phys Rev Lett, 100, 018104⌇  Lau et al. (2003) Microrheology, stress fluctuations, and active behavior of living cells. Phys Rev Lett, 91, 198101⇢  Related post : water&#8217;s SED failure in molecular orientational diffusion
Molecular motions inside the cell
A paper in Science this week describes the use of carbon nanotubes to pinpoint the movements of the living cell in fine detail, making for a really nice study in quantitative/mathematical biology.

Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random “stirring” that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.

The &#8220;mesoscopic" scale is more often seen in the context of pure and applied physics (microelectronics, nanofabrication and nanotechnology), though journals such as Soft Matter present research articles giving the same &#8216;condensed matter&#8217; treatment to biological systems (“Where physics meets chemistry meets biology”).
From ancient Greek μέσος it refers simply to a &#8216;middle&#8217;/intermediate between the molecular and macroscopic scale, where neither atomistic/quantum nor classical physics/bulk models best describe observed behaviour, and novel effects may be described — from interference effects, to quantum confinement (giving rise to band gaps) and charging effects (such as the Coulomb blockade/staircase).
Although often presented as a water-based solvent, the cytosol is more accurately described as a “highly dynamic composite material ” with mechanical properties dominated by microtubules (MTs), F-actin and intermediate filaments; all driven by metabolism-energised polymerisation of actin and tubulin and from motor proteins (specifically nucleotide triphosphate hydrolysis).
The traditional technique to observe cells in motion is fluorescence microscopy, though long-term tracking of single molecules has been hindered by fluorophores&#8217; instabilities and the fluorescence background in cells.
Though biological networks have been termed ‘scale-free’ or ‘-invariant’, and metabolic rate for example is well known to follow a power law, the internal structure of the cell itself is far from self-similar across scales. 

At short times (microseconds to milliseconds), thermal motions should dominate. Between milliseconds and seconds, thermal diffusion might still be relevant, but there is mounting evidence, both in vitro and in vivo, that the motion of larger objects couples to myosin-driven stress fluctuations in the cytoskeleton.

»  Mizuno (2007) Nonequilibrium mechanics of active cytoskeletal networks.
»  Brangwynne (2008) Cytoplasmic diffusion: Molecular motors mix it up.

Here, temporal fluctuations, reminiscent of thermal diffusion in liquids, can arise from nonequilibrium dynamics in the viscoelastic cytoskeleton. On longer time scales, from minutes to hours, directed transport and larger-scale collective motions typically dominate. The motion of probe particles tracked inside cells has been classified as subdiffusive, diffusive, or superdiffusive. Such classifications, however, obscure the distinction between thermally driven and nonequilibrium fluctuations and are inadequate to identify intracellular material properties.

Motor proteins direct a whole host of molecular motions, kinesins and myosins being among the most heavily studied in vitro. Using fluorescence microscopy to track individual motor proteins is not only limited by instability of fluorophores, but the quality of the images taken (&#8220;signal to noise&#8221;) and efficiency of targetting probes to specific molecules.
Modern optical equipment and carefully designed fluorescent dyes have enabled experiments tracking single molecules at a time, though in living cells the authors note these experiments&#8217; timeframes have been limited to around a second.

Their solution was to use single-walled carbon nanotubes (SWNTs), “stiff quasi–one-dimensional tubular all-carbon nanostructures with diameters of ~1 nm and persistence lengths above 10 μm” — which have the convenient property of luminescence in the near-infrared, a region ‘virtually free of autofluorescence in biological tissues’. Not only this, but the excitation time is ~100ps, such that high excitation can give ~1&#160;ms resolution (1&#160;ms = 109 ps).
The nanotubes were dispersed throughout the cell wrapped in short DNA oligonucleotides, with HaloTag protein fusion tags covalently attaching them specifically onto kinesin motor proteins (see Fig. 1, above).

Besides observing directed kinesin-driven transport on MTs, it is possible to directly observe fluctuations of the MT network because a moving kinesin must be bound to a MT. The MT tracks are embedded in the viscoelastic actin cytoskeleton, which in turn fluctuates as a result of stresses generated by cytoplasmic myosins.


With just 100 per cell, the group could track kinesin for up to an hour and a half, observing ~30% of them moving with some sense of direction; the rest locally constrained and moving in a random (stochastic) manner.
Some of the kinesins moved the whole length of the cell, suggesting they had cargo vesicles [along with other motor proteins] attached. Calculating mean squared displacement, MSD, of the molecules&#8217; trajectories showed it grew over time following a power law which could be used to characterise the motion, 〈Δr2(τ)〉∝ τα(where r is distance travelled in the focal plane and τ the lag time). The exponent α shifted from ¼ to 1 between 5&#160;ms and 2.5&#160;s, indicating clear scale variance to the motion.
After this, the group acquired the nanotubes&#8217; fluorescent signal at a rate of four frames per second ‒ using this 250&#160;ms window to observe an intermediate between the thermal diffusion seen on the short timescales and directed motor activity.
With a well-designed control or two, they showed that the transverse motion of the nanotube-marked microtubules was not due to kinesin motors, but reflecting intrinsic dynamics of the cytoskeleton.

The way the relatively rigid MTs report these dynamics depends on two restoring forces: the elastic force of bent MTs and the force exerted by the strained cytoskeletal matrix in which the MTs are embedded. Because it is hard to bend an elastic rod on short length scales, the surrounding matrix yields to the MT when it is deformed on short length scales. By contrast, the MT yields to matrix forces for deflections of wavelength larger than ~1 μm. The shorter-wavelength MT deflections relax faster than our 5-ms frame rate. Therefore, we assume that the transverse MT motion we observe reflects the (active or passive) strain fluctuations of the surrounding matrix.
The MSD power-law exponent α generally reflects the randomness of motion. More precisely, in any medium, the MSD of an embedded probe particle is governed both by the material properties of the medium and the temporal characteristics of the forces driving the particle. For thermally driven Brownian motion in simple liquids, the MSD exponent α = 1. For thermal motion in viscoelastic media, which exhibit time- and frequency-dependent viscosity and elasticity, α &lt; 1 strictly holds. For viscoelastic materials, the stiffness G(ω) typically increases with a power of frequency ω: G(ω) ∝ ωβ. This is observed in polymer solutions, where the viscoelastic exponent β ≈ 0.5 to 0.8, as well as in cells, where β ≈ 0.1 to 0.2 on time scales on the order of seconds. This value of the exponent is close to what is expected for purely elastic materials, where β = 0. 
The nearly elastic behavior of cells can be understood as a consequence of strong cross-linking in the cytoskeleton.
 Knowing the driving forces, it is possible to construct a relation between MSD exponent α and viscoelastic exponent β. For thermal driving forces, the MSD exponent α = β. Thermal fluctuations can therefore never appear as “superdiffusive” motion with α &gt; 1. Nonthermal driving, by contrast, can result in superdiffusive motion. Theory provides a specific prediction for motion in nearly elastic solids driven by random stress fluctuations with long correlation times and sudden transitions: α = 1 + 2β. This prediction is expected to apply for cytoskeletal stress fluctuations caused by randomly distributed cytoplasmic myosin minifilaments. Myosin locally contracts the actin network with an attachment time of several seconds, followed by sudden release. Some hints of this predicted scaling have been reported for cells and reconstituted acto-myosin model systems. When β = 0 (i.e., in the elastic limit), the resulting MSDs can look deceptively like Brownian motion in a simple liquid, although the physical reason is entirely different. For observation times τ longer than the correlation time of the driving forces, the MSD is predicted to level off, as we observed. In our experiments, the stress correlation time should correspond to typical cytoplasmic myosin motor engagement times, which are indeed reported to be ~10&#160;s in cells.

Still attached to microtubules, the kinesin molecules exhibit vigorous random (Brownian-like) motion as they are buffeted by myosins as described ‒ likely thrusting MTs into the path of other cellular particles. Tubulin forms strong tubular filaments embedded in a more flexible actin network. Nonmuscle myosin II exerts mechanical stress on it, which is released ‘suddenly’ as random stirring of the whole filament network, including the microtubules.

We observed a transition between thermal dynamics in the dominantly elastic cytoskeleton at short times to strongly nonequilibrium power-law dynamics, likely driven by myosin activity, at intermediate times. When the time exceeded the correlation time of the random stress generators, the intermediate regime was followed by a saturation to a maximum MSD, nearly constant over time. Note that in this regime, the MSD amplitude corresponds to a root mean square displacement of ~500 nm, which is larger than the estimated mesh size of the actin network, and thus larger than the expected spacing of obstacles in the crowded cytoplasm.

The authors lastly used myosin inhibitor blebbistatin to block myosin from the actin network, confirming their hypothesis with a dose-dependent reduction in what they call the amplitude of active stirring, and exponent α, “establishing nonmuscle myosin II as the dominant driving factor for random cytoskeletal stirring”.

We can explain the regimes we observe by a quantitative model of cytoskeletal fluctuations and directed motor motion that describes the transition from thermal motion to nonequilibrium stirring dynamics driven by myosin, as well as the transition from stirring dynamics to directed transport driven by kinesin. Our observations were made possible by the use of SWNT labels for broadband molecular tracking in cells. Many questions concerning motor transport in cells will now be addressable using this approach. We have focused here on the stirring dynamics, which constitute an important mode of active intracellular transport between the limits of random thermal diffusion and directed transport, accelerating nonspecific transport through the nanoporous cytoskeleton.

Lead author Nikta Fakhri will soon leave the Göttingen Institute for Biophysics to join the faculty at MIT as assistant professor of physics. Fakhri gave a talk in Massachussets last year on the topic, to the Chemical Engineering department in which some of the details of this paper made their debut:

The discovery of fullerenes provided exciting insights into how highly symmetrical structures of pure carbon can have remarkable physical properties. Single-walled carbon nanotubes (SWNTs) are the vanguard of such architectures. The organization of the hexagonal honeycomb carbon lattice into high-aspect-ratio cylinders with a variety of helical symmetries creates very unusual macromolecular structures representing an emerging research area in condensed matter physics and materials science: traditionally hard materials appearing in new soft matter applications and environments.
… the dynamics of SWNTs in liquids are essentially polymer-like. By exploiting the intrinsic near-infrared fluorescence of semiconducting SWNTs, we have imaged the Brownian motion of individual nanotubes in water and have measured directly the bending stiffness of SWNTs. The semiflexible chain model represents accurately the configurational dynamics of SWNTs suspended in water. Interestingly, the persistence length of SWNTs is comparable to that of biopolymers. This finding paves the way for using SWNTs as a model system for semiflexible polymers to answer long-standing fundamental questions in polymer physics.
… the confined dynamics of stiff macromolecules in crowded environments [are] a common feature of polymer composites and the cell cytoskeleton. In fixed porous networks, we find that even a small bending flexibility strongly enhances SWNTs&#8217; motion. This ends a 30-year-old debate in polymer physics: the rotational diffusion constant is proportional to the filament bending compliance and counter-intuitively, independent of the network porosity. The dynamics of SWNTs in equilibrium and non-equilibrium biopolymer networks is more complex.
At long times, SWNTs reptate in the networks. At short times SWNTs can sample the spectrum of local stresses in equilibrium networks. In the non-equilibrium networks we observe strong local shape fluctuations driven by force generating molecular motors. I will discuss a newly developed microrheology technique in which we use nanotubes as “stealth probes” to measure viscoelastic properties of the host media. Finally, I will introduce a new single-molecule technique based on ultra-stable near-infrared fluorescence of short SWNTs, to study intracellular transport dynamics in living cells and in whole organisms. The combination of long-time stability and high signal-to-noise ratio enables the accurate long-term tracking of single motor proteins tagged with SWNTs traversing the entire cell. Remarkably, we can distinguish the motor protein’s motion along its microtubule track from the track’s underlying random non-thermal fluctuations.

She envisions the technology as applicable beyond probing biophysical questions, in the design of 'active' technical materials.
“Imagine a microscopic biomedical device that mixes tiny samples of blood with reagents to detect disease or smart filters that separate squishy from rigid materials.”
Fakhri will join the Physics of Living Systems group, seemingly on such a bio-materials science project. MIT lab colleague Jeremy England, known for work showing that E. coli reproduction is close to thermodynamic limits of efficiency, spoke of common interest in the cytosol and diffusive processes.

“We&#8217;re interested in the non-equilibrium thermodynamics of biological organization, so that could be construed to be about evolution and the origins of life or just about how you make or design self-replicators with desired properties.”
“Increasingly there are now instruments where you can make quantitative measurements on fluorescently labeled proteins in live cells,” England explains. “The cell biologists have their language and their frame of analysis that they&#8217;re most comfortable with for describing the phenomenon, but if there are interesting phenomena that are only going to be identifiable if you do the right quantitative analysis on all these numbers that you can now measure in the cell, then it&#8217;s useful to have people who are a bit more theoretically minded or physics minded who are there, when rubber meets road, when the data is being generated and helping to influence what kind of experiments get done.”
“We&#8217;re looking, for example, at diffusion of proteins in cells. Diffusion as a qualitative phenomenon is just things spreading out over space, but as a quantitative phenomenon, you can look at things like how rapidly a protein that&#8217;s labeled over here in the cell will wander over to another region of the cell that&#8217;s a certain distance away, and if you can make measurements of that, then you can start to say things that are more specific about characteristics of the diffusion that you are observing than simply seeing it spread out. And in those quantitative measurements, you can sometimes then see differences perhaps between different cells, or different conditions for the same type of cell, that may have biological relevance but that you wouldn&#8217;t have necessarily identified without the quantitative analysis,” England says.

⌇  Fakhri et al. (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science, 344(1687), 1031-5
See also:⌇  Levine and MacKintosh (2009) The mechanics and fluctuation spectrum of active gels. J Phys Chem B, 113, 3820–3830⌇  MacKintosh and Levine (2008) Nonequilibrium mechanics and dynamics of motor-activated gels. Phys Rev Lett, 100, 018104⌇  Lau et al. (2003) Microrheology, stress fluctuations, and active behavior of living cells. Phys Rev Lett, 91, 198101⇢  Related post : water&#8217;s SED failure in molecular orientational diffusion
Molecular motions inside the cell
A paper in Science this week describes the use of carbon nanotubes to pinpoint the movements of the living cell in fine detail, making for a really nice study in quantitative/mathematical biology.

Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random “stirring” that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.

The &#8220;mesoscopic" scale is more often seen in the context of pure and applied physics (microelectronics, nanofabrication and nanotechnology), though journals such as Soft Matter present research articles giving the same &#8216;condensed matter&#8217; treatment to biological systems (“Where physics meets chemistry meets biology”).
From ancient Greek μέσος it refers simply to a &#8216;middle&#8217;/intermediate between the molecular and macroscopic scale, where neither atomistic/quantum nor classical physics/bulk models best describe observed behaviour, and novel effects may be described — from interference effects, to quantum confinement (giving rise to band gaps) and charging effects (such as the Coulomb blockade/staircase).
Although often presented as a water-based solvent, the cytosol is more accurately described as a “highly dynamic composite material ” with mechanical properties dominated by microtubules (MTs), F-actin and intermediate filaments; all driven by metabolism-energised polymerisation of actin and tubulin and from motor proteins (specifically nucleotide triphosphate hydrolysis).
The traditional technique to observe cells in motion is fluorescence microscopy, though long-term tracking of single molecules has been hindered by fluorophores&#8217; instabilities and the fluorescence background in cells.
Though biological networks have been termed ‘scale-free’ or ‘-invariant’, and metabolic rate for example is well known to follow a power law, the internal structure of the cell itself is far from self-similar across scales. 

At short times (microseconds to milliseconds), thermal motions should dominate. Between milliseconds and seconds, thermal diffusion might still be relevant, but there is mounting evidence, both in vitro and in vivo, that the motion of larger objects couples to myosin-driven stress fluctuations in the cytoskeleton.

»  Mizuno (2007) Nonequilibrium mechanics of active cytoskeletal networks.
»  Brangwynne (2008) Cytoplasmic diffusion: Molecular motors mix it up.

Here, temporal fluctuations, reminiscent of thermal diffusion in liquids, can arise from nonequilibrium dynamics in the viscoelastic cytoskeleton. On longer time scales, from minutes to hours, directed transport and larger-scale collective motions typically dominate. The motion of probe particles tracked inside cells has been classified as subdiffusive, diffusive, or superdiffusive. Such classifications, however, obscure the distinction between thermally driven and nonequilibrium fluctuations and are inadequate to identify intracellular material properties.

Motor proteins direct a whole host of molecular motions, kinesins and myosins being among the most heavily studied in vitro. Using fluorescence microscopy to track individual motor proteins is not only limited by instability of fluorophores, but the quality of the images taken (&#8220;signal to noise&#8221;) and efficiency of targetting probes to specific molecules.
Modern optical equipment and carefully designed fluorescent dyes have enabled experiments tracking single molecules at a time, though in living cells the authors note these experiments&#8217; timeframes have been limited to around a second.

Their solution was to use single-walled carbon nanotubes (SWNTs), “stiff quasi–one-dimensional tubular all-carbon nanostructures with diameters of ~1 nm and persistence lengths above 10 μm” — which have the convenient property of luminescence in the near-infrared, a region ‘virtually free of autofluorescence in biological tissues’. Not only this, but the excitation time is ~100ps, such that high excitation can give ~1&#160;ms resolution (1&#160;ms = 109 ps).
The nanotubes were dispersed throughout the cell wrapped in short DNA oligonucleotides, with HaloTag protein fusion tags covalently attaching them specifically onto kinesin motor proteins (see Fig. 1, above).

Besides observing directed kinesin-driven transport on MTs, it is possible to directly observe fluctuations of the MT network because a moving kinesin must be bound to a MT. The MT tracks are embedded in the viscoelastic actin cytoskeleton, which in turn fluctuates as a result of stresses generated by cytoplasmic myosins.


With just 100 per cell, the group could track kinesin for up to an hour and a half, observing ~30% of them moving with some sense of direction; the rest locally constrained and moving in a random (stochastic) manner.
Some of the kinesins moved the whole length of the cell, suggesting they had cargo vesicles [along with other motor proteins] attached. Calculating mean squared displacement, MSD, of the molecules&#8217; trajectories showed it grew over time following a power law which could be used to characterise the motion, 〈Δr2(τ)〉∝ τα(where r is distance travelled in the focal plane and τ the lag time). The exponent α shifted from ¼ to 1 between 5&#160;ms and 2.5&#160;s, indicating clear scale variance to the motion.
After this, the group acquired the nanotubes&#8217; fluorescent signal at a rate of four frames per second ‒ using this 250&#160;ms window to observe an intermediate between the thermal diffusion seen on the short timescales and directed motor activity.
With a well-designed control or two, they showed that the transverse motion of the nanotube-marked microtubules was not due to kinesin motors, but reflecting intrinsic dynamics of the cytoskeleton.

The way the relatively rigid MTs report these dynamics depends on two restoring forces: the elastic force of bent MTs and the force exerted by the strained cytoskeletal matrix in which the MTs are embedded. Because it is hard to bend an elastic rod on short length scales, the surrounding matrix yields to the MT when it is deformed on short length scales. By contrast, the MT yields to matrix forces for deflections of wavelength larger than ~1 μm. The shorter-wavelength MT deflections relax faster than our 5-ms frame rate. Therefore, we assume that the transverse MT motion we observe reflects the (active or passive) strain fluctuations of the surrounding matrix.
The MSD power-law exponent α generally reflects the randomness of motion. More precisely, in any medium, the MSD of an embedded probe particle is governed both by the material properties of the medium and the temporal characteristics of the forces driving the particle. For thermally driven Brownian motion in simple liquids, the MSD exponent α = 1. For thermal motion in viscoelastic media, which exhibit time- and frequency-dependent viscosity and elasticity, α &lt; 1 strictly holds. For viscoelastic materials, the stiffness G(ω) typically increases with a power of frequency ω: G(ω) ∝ ωβ. This is observed in polymer solutions, where the viscoelastic exponent β ≈ 0.5 to 0.8, as well as in cells, where β ≈ 0.1 to 0.2 on time scales on the order of seconds. This value of the exponent is close to what is expected for purely elastic materials, where β = 0. 
The nearly elastic behavior of cells can be understood as a consequence of strong cross-linking in the cytoskeleton.
 Knowing the driving forces, it is possible to construct a relation between MSD exponent α and viscoelastic exponent β. For thermal driving forces, the MSD exponent α = β. Thermal fluctuations can therefore never appear as “superdiffusive” motion with α &gt; 1. Nonthermal driving, by contrast, can result in superdiffusive motion. Theory provides a specific prediction for motion in nearly elastic solids driven by random stress fluctuations with long correlation times and sudden transitions: α = 1 + 2β. This prediction is expected to apply for cytoskeletal stress fluctuations caused by randomly distributed cytoplasmic myosin minifilaments. Myosin locally contracts the actin network with an attachment time of several seconds, followed by sudden release. Some hints of this predicted scaling have been reported for cells and reconstituted acto-myosin model systems. When β = 0 (i.e., in the elastic limit), the resulting MSDs can look deceptively like Brownian motion in a simple liquid, although the physical reason is entirely different. For observation times τ longer than the correlation time of the driving forces, the MSD is predicted to level off, as we observed. In our experiments, the stress correlation time should correspond to typical cytoplasmic myosin motor engagement times, which are indeed reported to be ~10&#160;s in cells.

Still attached to microtubules, the kinesin molecules exhibit vigorous random (Brownian-like) motion as they are buffeted by myosins as described ‒ likely thrusting MTs into the path of other cellular particles. Tubulin forms strong tubular filaments embedded in a more flexible actin network. Nonmuscle myosin II exerts mechanical stress on it, which is released ‘suddenly’ as random stirring of the whole filament network, including the microtubules.

We observed a transition between thermal dynamics in the dominantly elastic cytoskeleton at short times to strongly nonequilibrium power-law dynamics, likely driven by myosin activity, at intermediate times. When the time exceeded the correlation time of the random stress generators, the intermediate regime was followed by a saturation to a maximum MSD, nearly constant over time. Note that in this regime, the MSD amplitude corresponds to a root mean square displacement of ~500 nm, which is larger than the estimated mesh size of the actin network, and thus larger than the expected spacing of obstacles in the crowded cytoplasm.

The authors lastly used myosin inhibitor blebbistatin to block myosin from the actin network, confirming their hypothesis with a dose-dependent reduction in what they call the amplitude of active stirring, and exponent α, “establishing nonmuscle myosin II as the dominant driving factor for random cytoskeletal stirring”.

We can explain the regimes we observe by a quantitative model of cytoskeletal fluctuations and directed motor motion that describes the transition from thermal motion to nonequilibrium stirring dynamics driven by myosin, as well as the transition from stirring dynamics to directed transport driven by kinesin. Our observations were made possible by the use of SWNT labels for broadband molecular tracking in cells. Many questions concerning motor transport in cells will now be addressable using this approach. We have focused here on the stirring dynamics, which constitute an important mode of active intracellular transport between the limits of random thermal diffusion and directed transport, accelerating nonspecific transport through the nanoporous cytoskeleton.

Lead author Nikta Fakhri will soon leave the Göttingen Institute for Biophysics to join the faculty at MIT as assistant professor of physics. Fakhri gave a talk in Massachussets last year on the topic, to the Chemical Engineering department in which some of the details of this paper made their debut:

The discovery of fullerenes provided exciting insights into how highly symmetrical structures of pure carbon can have remarkable physical properties. Single-walled carbon nanotubes (SWNTs) are the vanguard of such architectures. The organization of the hexagonal honeycomb carbon lattice into high-aspect-ratio cylinders with a variety of helical symmetries creates very unusual macromolecular structures representing an emerging research area in condensed matter physics and materials science: traditionally hard materials appearing in new soft matter applications and environments.
… the dynamics of SWNTs in liquids are essentially polymer-like. By exploiting the intrinsic near-infrared fluorescence of semiconducting SWNTs, we have imaged the Brownian motion of individual nanotubes in water and have measured directly the bending stiffness of SWNTs. The semiflexible chain model represents accurately the configurational dynamics of SWNTs suspended in water. Interestingly, the persistence length of SWNTs is comparable to that of biopolymers. This finding paves the way for using SWNTs as a model system for semiflexible polymers to answer long-standing fundamental questions in polymer physics.
… the confined dynamics of stiff macromolecules in crowded environments [are] a common feature of polymer composites and the cell cytoskeleton. In fixed porous networks, we find that even a small bending flexibility strongly enhances SWNTs&#8217; motion. This ends a 30-year-old debate in polymer physics: the rotational diffusion constant is proportional to the filament bending compliance and counter-intuitively, independent of the network porosity. The dynamics of SWNTs in equilibrium and non-equilibrium biopolymer networks is more complex.
At long times, SWNTs reptate in the networks. At short times SWNTs can sample the spectrum of local stresses in equilibrium networks. In the non-equilibrium networks we observe strong local shape fluctuations driven by force generating molecular motors. I will discuss a newly developed microrheology technique in which we use nanotubes as “stealth probes” to measure viscoelastic properties of the host media. Finally, I will introduce a new single-molecule technique based on ultra-stable near-infrared fluorescence of short SWNTs, to study intracellular transport dynamics in living cells and in whole organisms. The combination of long-time stability and high signal-to-noise ratio enables the accurate long-term tracking of single motor proteins tagged with SWNTs traversing the entire cell. Remarkably, we can distinguish the motor protein’s motion along its microtubule track from the track’s underlying random non-thermal fluctuations.

She envisions the technology as applicable beyond probing biophysical questions, in the design of 'active' technical materials.
“Imagine a microscopic biomedical device that mixes tiny samples of blood with reagents to detect disease or smart filters that separate squishy from rigid materials.”
Fakhri will join the Physics of Living Systems group, seemingly on such a bio-materials science project. MIT lab colleague Jeremy England, known for work showing that E. coli reproduction is close to thermodynamic limits of efficiency, spoke of common interest in the cytosol and diffusive processes.

“We&#8217;re interested in the non-equilibrium thermodynamics of biological organization, so that could be construed to be about evolution and the origins of life or just about how you make or design self-replicators with desired properties.”
“Increasingly there are now instruments where you can make quantitative measurements on fluorescently labeled proteins in live cells,” England explains. “The cell biologists have their language and their frame of analysis that they&#8217;re most comfortable with for describing the phenomenon, but if there are interesting phenomena that are only going to be identifiable if you do the right quantitative analysis on all these numbers that you can now measure in the cell, then it&#8217;s useful to have people who are a bit more theoretically minded or physics minded who are there, when rubber meets road, when the data is being generated and helping to influence what kind of experiments get done.”
“We&#8217;re looking, for example, at diffusion of proteins in cells. Diffusion as a qualitative phenomenon is just things spreading out over space, but as a quantitative phenomenon, you can look at things like how rapidly a protein that&#8217;s labeled over here in the cell will wander over to another region of the cell that&#8217;s a certain distance away, and if you can make measurements of that, then you can start to say things that are more specific about characteristics of the diffusion that you are observing than simply seeing it spread out. And in those quantitative measurements, you can sometimes then see differences perhaps between different cells, or different conditions for the same type of cell, that may have biological relevance but that you wouldn&#8217;t have necessarily identified without the quantitative analysis,” England says.

⌇  Fakhri et al. (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science, 344(1687), 1031-5
See also:⌇  Levine and MacKintosh (2009) The mechanics and fluctuation spectrum of active gels. J Phys Chem B, 113, 3820–3830⌇  MacKintosh and Levine (2008) Nonequilibrium mechanics and dynamics of motor-activated gels. Phys Rev Lett, 100, 018104⌇  Lau et al. (2003) Microrheology, stress fluctuations, and active behavior of living cells. Phys Rev Lett, 91, 198101⇢  Related post : water&#8217;s SED failure in molecular orientational diffusion

Molecular motions inside the cell

A paper in Science this week describes the use of carbon nanotubes to pinpoint the movements of the living cell in fine detail, making for a really nice study in quantitative/mathematical biology.

Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random “stirring” that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.

The “mesoscopic" scale is more often seen in the context of pure and applied physics (microelectronics, nanofabrication and nanotechnology), though journals such as Soft Matter present research articles giving the same ‘condensed matter’ treatment to biological systems (“Where physics meets chemistry meets biology”).

From ancient Greek μέσος it refers simply to a ‘middle’/intermediate between the molecular and macroscopic scale, where neither atomistic/quantum nor classical physics/bulk models best describe observed behaviour, and novel effects may be described — from interference effects, to quantum confinement (giving rise to band gaps) and charging effects (such as the Coulomb blockade/staircase).

Although often presented as a water-based solvent, the cytosol is more accurately described as a “highly dynamic composite material ” with mechanical properties dominated by microtubules (MTs), F-actin and intermediate filaments; all driven by metabolism-energised polymerisation of actin and tubulin and from motor proteins (specifically nucleotide triphosphate hydrolysis).

The traditional technique to observe cells in motion is fluorescence microscopy, though long-term tracking of single molecules has been hindered by fluorophores’ instabilities and the fluorescence background in cells.

Though biological networks have been termed ‘scale-free’ or ‘-invariant’, and metabolic rate for example is well known to follow a power law, the internal structure of the cell itself is far from self-similar across scales. 

At short times (microseconds to milliseconds), thermal motions should dominate. Between milliseconds and seconds, thermal diffusion might still be relevant, but there is mounting evidence, both in vitro and in vivo, that the motion of larger objects couples to myosin-driven stress fluctuations in the cytoskeleton.

»  Mizuno (2007) Nonequilibrium mechanics of active cytoskeletal networks.

»  Brangwynne (2008) Cytoplasmic diffusion: Molecular motors mix it up.

Here, temporal fluctuations, reminiscent of thermal diffusion in liquids, can arise from nonequilibrium dynamics in the viscoelastic cytoskeleton. On longer time scales, from minutes to hours, directed transport and larger-scale collective motions typically dominate. The motion of probe particles tracked inside cells has been classified as subdiffusive, diffusive, or superdiffusive. Such classifications, however, obscure the distinction between thermally driven and nonequilibrium fluctuations and are inadequate to identify intracellular material properties.

Motor proteins direct a whole host of molecular motions, kinesins and myosins being among the most heavily studied in vitro. Using fluorescence microscopy to track individual motor proteins is not only limited by instability of fluorophores, but the quality of the images taken (“signal to noise”) and efficiency of targetting probes to specific molecules.

Modern optical equipment and carefully designed fluorescent dyes have enabled experiments tracking single molecules at a time, though in living cells the authors note these experiments’ timeframes have been limited to around a second.

Their solution was to use single-walled carbon nanotubes (SWNTs), “stiff quasi–one-dimensional tubular all-carbon nanostructures with diameters of ~1 nm and persistence lengths above 10 μm” — which have the convenient property of luminescence in the near-infrared, a region ‘virtually free of autofluorescence in biological tissues’. Not only this, but the excitation time is ~100ps, such that high excitation can give ~1 ms resolution (1 ms = 109 ps).

The nanotubes were dispersed throughout the cell wrapped in short DNA oligonucleotides, with HaloTag protein fusion tags covalently attaching them specifically onto kinesin motor proteins (see Fig. 1, above).

Besides observing directed kinesin-driven transport on MTs, it is possible to directly observe fluctuations of the MT network because a moving kinesin must be bound to a MT. The MT tracks are embedded in the viscoelastic actin cytoskeleton, which in turn fluctuates as a result of stresses generated by cytoplasmic myosins.

image

With just 100 per cell, the group could track kinesin for up to an hour and a half, observing ~30% of them moving with some sense of direction; the rest locally constrained and moving in a random (stochastic) manner.

Some of the kinesins moved the whole length of the cell, suggesting they had cargo vesicles [along with other motor proteins] attached. Calculating mean squared displacement, MSD, of the molecules’ trajectories showed it grew over time following a power law which could be used to characterise the motion, 〈Δr2(τ)〉∝ τα(where r is distance travelled in the focal plane and τ the lag time). The exponent α shifted from ¼ to 1 between 5 ms and 2.5 s, indicating clear scale variance to the motion.

After this, the group acquired the nanotubes’ fluorescent signal at a rate of four frames per second ‒ using this 250 ms window to observe an intermediate between the thermal diffusion seen on the short timescales and directed motor activity.

With a well-designed control or two, they showed that the transverse motion of the nanotube-marked microtubules was not due to kinesin motors, but reflecting intrinsic dynamics of the cytoskeleton.

The way the relatively rigid MTs report these dynamics depends on two restoring forces: the elastic force of bent MTs and the force exerted by the strained cytoskeletal matrix in which the MTs are embedded. Because it is hard to bend an elastic rod on short length scales, the surrounding matrix yields to the MT when it is deformed on short length scales. By contrast, the MT yields to matrix forces for deflections of wavelength larger than ~1 μm. The shorter-wavelength MT deflections relax faster than our 5-ms frame rate. Therefore, we assume that the transverse MT motion we observe reflects the (active or passive) strain fluctuations of the surrounding matrix.

The MSD power-law exponent α generally reflects the randomness of motion. More precisely, in any medium, the MSD of an embedded probe particle is governed both by the material properties of the medium and the temporal characteristics of the forces driving the particle. For thermally driven Brownian motion in simple liquids, the MSD exponent α = 1. For thermal motion in viscoelastic media, which exhibit time- and frequency-dependent viscosity and elasticity, α < 1 strictly holds. For viscoelastic materials, the stiffness G(ω) typically increases with a power of frequency ω: G(ω) ∝ ωβ. This is observed in polymer solutions, where the viscoelastic exponent β ≈ 0.5 to 0.8, as well as in cells, where β ≈ 0.1 to 0.2 on time scales on the order of seconds. This value of the exponent is close to what is expected for purely elastic materials, where β = 0.

The nearly elastic behavior of cells can be understood as a consequence of strong cross-linking in the cytoskeleton.

Knowing the driving forces, it is possible to construct a relation between MSD exponent α and viscoelastic exponent β. For thermal driving forces, the MSD exponent α = β. Thermal fluctuations can therefore never appear as “superdiffusive” motion with α > 1. Nonthermal driving, by contrast, can result in superdiffusive motion. Theory provides a specific prediction for motion in nearly elastic solids driven by random stress fluctuations with long correlation times and sudden transitions: α = 1 + 2β. This prediction is expected to apply for cytoskeletal stress fluctuations caused by randomly distributed cytoplasmic myosin minifilaments. Myosin locally contracts the actin network with an attachment time of several seconds, followed by sudden release. Some hints of this predicted scaling have been reported for cells and reconstituted acto-myosin model systems. When β = 0 (i.e., in the elastic limit), the resulting MSDs can look deceptively like Brownian motion in a simple liquid, although the physical reason is entirely different. For observation times τ longer than the correlation time of the driving forces, the MSD is predicted to level off, as we observed. In our experiments, the stress correlation time should correspond to typical cytoplasmic myosin motor engagement times, which are indeed reported to be ~10 s in cells.

Still attached to microtubules, the kinesin molecules exhibit vigorous random (Brownian-like) motion as they are buffeted by myosins as described ‒ likely thrusting MTs into the path of other cellular particles. Tubulin forms strong tubular filaments embedded in a more flexible actin network. Nonmuscle myosin II exerts mechanical stress on it, which is released ‘suddenly’ as random stirring of the whole filament network, including the microtubules.

We observed a transition between thermal dynamics in the dominantly elastic cytoskeleton at short times to strongly nonequilibrium power-law dynamics, likely driven by myosin activity, at intermediate times. When the time exceeded the correlation time of the random stress generators, the intermediate regime was followed by a saturation to a maximum MSD, nearly constant over time. Note that in this regime, the MSD amplitude corresponds to a root mean square displacement of ~500 nm, which is larger than the estimated mesh size of the actin network, and thus larger than the expected spacing of obstacles in the crowded cytoplasm.

The authors lastly used myosin inhibitor blebbistatin to block myosin from the actin network, confirming their hypothesis with a dose-dependent reduction in what they call the amplitude of active stirring, and exponent α, “establishing nonmuscle myosin II as the dominant driving factor for random cytoskeletal stirring”.

We can explain the regimes we observe by a quantitative model of cytoskeletal fluctuations and directed motor motion that describes the transition from thermal motion to nonequilibrium stirring dynamics driven by myosin, as well as the transition from stirring dynamics to directed transport driven by kinesin. Our observations were made possible by the use of SWNT labels for broadband molecular tracking in cells. Many questions concerning motor transport in cells will now be addressable using this approach. We have focused here on the stirring dynamics, which constitute an important mode of active intracellular transport between the limits of random thermal diffusion and directed transport, accelerating nonspecific transport through the nanoporous cytoskeleton.

Lead author Nikta Fakhri will soon leave the Göttingen Institute for Biophysics to join the faculty at MIT as assistant professor of physics. Fakhri gave a talk in Massachussets last year on the topic, to the Chemical Engineering department in which some of the details of this paper made their debut:

The discovery of fullerenes provided exciting insights into how highly symmetrical structures of pure carbon can have remarkable physical properties. Single-walled carbon nanotubes (SWNTs) are the vanguard of such architectures. The organization of the hexagonal honeycomb carbon lattice into high-aspect-ratio cylinders with a variety of helical symmetries creates very unusual macromolecular structures representing an emerging research area in condensed matter physics and materials science: traditionally hard materials appearing in new soft matter applications and environments.

… the dynamics of SWNTs in liquids are essentially polymer-like. By exploiting the intrinsic near-infrared fluorescence of semiconducting SWNTs, we have imaged the Brownian motion of individual nanotubes in water and have measured directly the bending stiffness of SWNTs. The semiflexible chain model represents accurately the configurational dynamics of SWNTs suspended in water. Interestingly, the persistence length of SWNTs is comparable to that of biopolymers. This finding paves the way for using SWNTs as a model system for semiflexible polymers to answer long-standing fundamental questions in polymer physics.

… the confined dynamics of stiff macromolecules in crowded environments [are] a common feature of polymer composites and the cell cytoskeleton. In fixed porous networks, we find that even a small bending flexibility strongly enhances SWNTs’ motion. This ends a 30-year-old debate in polymer physics: the rotational diffusion constant is proportional to the filament bending compliance and counter-intuitively, independent of the network porosity. The dynamics of SWNTs in equilibrium and non-equilibrium biopolymer networks is more complex.

At long times, SWNTs reptate in the networks. At short times SWNTs can sample the spectrum of local stresses in equilibrium networks. In the non-equilibrium networks we observe strong local shape fluctuations driven by force generating molecular motors. I will discuss a newly developed microrheology technique in which we use nanotubes as “stealth probes” to measure viscoelastic properties of the host media. Finally, I will introduce a new single-molecule technique based on ultra-stable near-infrared fluorescence of short SWNTs, to study intracellular transport dynamics in living cells and in whole organisms. The combination of long-time stability and high signal-to-noise ratio enables the accurate long-term tracking of single motor proteins tagged with SWNTs traversing the entire cell. Remarkably, we can distinguish the motor protein’s motion along its microtubule track from the track’s underlying random non-thermal fluctuations.

She envisions the technology as applicable beyond probing biophysical questions, in the design of 'active' technical materials.

“Imagine a microscopic biomedical device that mixes tiny samples of blood with reagents to detect disease or smart filters that separate squishy from rigid materials.”

Fakhri will join the Physics of Living Systems group, seemingly on such a bio-materials science project. MIT lab colleague Jeremy England, known for work showing that E. coli reproduction is close to thermodynamic limits of efficiency, spoke of common interest in the cytosol and diffusive processes.

“We’re interested in the non-equilibrium thermodynamics of biological organization, so that could be construed to be about evolution and the origins of life or just about how you make or design self-replicators with desired properties.”

“Increasingly there are now instruments where you can make quantitative measurements on fluorescently labeled proteins in live cells,” England explains. “The cell biologists have their language and their frame of analysis that they’re most comfortable with for describing the phenomenon, but if there are interesting phenomena that are only going to be identifiable if you do the right quantitative analysis on all these numbers that you can now measure in the cell, then it’s useful to have people who are a bit more theoretically minded or physics minded who are there, when rubber meets road, when the data is being generated and helping to influence what kind of experiments get done.”

“We’re looking, for example, at diffusion of proteins in cells. Diffusion as a qualitative phenomenon is just things spreading out over space, but as a quantitative phenomenon, you can look at things like how rapidly a protein that’s labeled over here in the cell will wander over to another region of the cell that’s a certain distance away, and if you can make measurements of that, then you can start to say things that are more specific about characteristics of the diffusion that you are observing than simply seeing it spread out. And in those quantitative measurements, you can sometimes then see differences perhaps between different cells, or different conditions for the same type of cell, that may have biological relevance but that you wouldn’t have necessarily identified without the quantitative analysis,” England says.

⌇  Fakhri et al. (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubesScience344(1687), 1031-5

See also:
⌇  Levine and MacKintosh (2009) The mechanics and fluctuation spectrum of active gels. J Phys Chem B, 113, 3820–3830
⌇  MacKintosh and Levine (2008) Nonequilibrium mechanics and dynamics of motor-activated gels. Phys Rev Lett, 100, 018104
⌇  Lau et al. (2003) Microrheology, stress fluctuations, and active behavior of living cells. Phys Rev Lett91, 198101

⇢  Related post : water’s SED failure in molecular orientational diffusion

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I know that it is a hopeless undertaking to debate about fundamental value judgments. For instance if someone approves, as a goal, the extirpation of the human race from the earth, one cannot refute such a viewpoint on rational grounds. But if there is agreement on certain goals and values, one can argue rationally about the means by which these objectives may be attained. Let us, then, indicate two goals which may well be agreed upon by nearly all who read these lines.

  1. Those instrumental goods which should serve to maintain the life and health of all human beings should be produced by the least possible labor of all.

  2. The satisfaction of physical needs is indeed the indispensable precondition of a satisfactory existence, but in itself it is not enough. In order to be content men must also have the possibility of developing their intellectual and artistic powers to whatever extent accord with their personal characteristics and abilities.

The first of these two goals requires the promotion of all knowledge relating to the laws of nature and the laws of social processes, that is, the promotion of all scientific endeavor. For scientific endeavor is a natural whole the parts of which mutually support one another in a way which, to be sure, no one can anticipate. However, the progress of science presupposes the possibility of unrestricted communication of all results and judgments—freedom of expression and instruction in all realms of intellectual endeavor. By freedom I understand social conditions of such a kind that the expression of opinions and assertions about general and particular matters of knowledge will not involve dangers or serious disadvantages for him who expresses them. This freedom of communication is indispensable for the development and extension of scientific knowledge, a consideration of much practical import. In the first instance it must be guaranteed by law. But laws alone cannot secure freedom of expression; in order that every man may present his views without penalty there must be a spirit of tolerance in the entire population. Such an ideal of external liberty can never be fully attained but must be sought unremittingly if scientific thought, and philosophical and creative thinking in general, are to be advanced as far as possible.

If the second goal, that is, the possibility of the spiritual development of all individuals, is to be secured, a second kind of outward freedom is necessary. Man should not have to work for the achievement of the necessities of life to such an extent that he has neither time nor strength for personal activities. Without this second kind of outward liberty, freedom of expression is useless for him. Advances in technology would provide the possibility of this kind of freedom if the problem of a reasonable division of labor were solved.

The development of science and of the creative activities of the spirit in general requires still another kind of freedom, which may be characterized as inward freedom. It is this freedom of the spirit which consists in the independence of thought from the restrictions of authoritarian and social prejudices as well as from unphilosophical routinizing and habit in general. This inward freedom is an infrequent gift of nature and a worthy objective for the individual. Yet the community can do much to further this achievement, too, at least by not interfering with its development. Thus schools may interfere with the development of inward freedom through authoritarian influences and through imposing on young people excessive spiritual burdens; on the other hand schools may favor such freedom by encouraging independent thought. Only if outward and inner freedom are constantly and consciously pursued is there a possibility of spiritual development and perfection and thus of improving man’s outward and inner life.

“On Freedom” by Albert Einstein, in the anthology of his essays Out of My Later Years.

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Overlay journals and supplement review

An overlay journal (or repository journal) is a specific type of open access journal (generally online). While not producing content itself, it selects from texts that are freely available ‒ deriving content from preprint servers such as arXiv, or from commercial publishers using self-archived, pre-, or post-prints.

Editors locate and evaluate suitable material, anywhere from the decision of a single editor up to a full peer review process. Brown (2010) writes that

An overlay journal performs all the activities of a scholarly journal and relies on structural links with one or more archives or repositories to perform its activities.

while Peter Suber (2003) defines one as “an open access journal that takes submissions from the preprints deposited at an archive… and subjects them to peer-review”. Referring to them as repository journals, he also wrote that they

use the institutional repository as the journal infrastructure. Submissions are deposited in the journal as preprints, and accepted articles are redeposited as postprints, labelled to show that they have been peer-reviewed.

In a footnote, Harley and Acord (2011) detail how this new phenomenon is “still fairly speculative at present

An overlay journal would mine self-archived “raw” author manuscripts from repositories and carry out certain publishing functions like peer-review management, editing, and perhaps branding (Swan 2010). The actual published content would continue to reside in the repository, perhaps with an updated postprint incorporating any revisions and updated metadata reflecting the journal/society brand that carried out the peer review. The overlay journal would then link to the repository content via a traditional table of contents.

…minimalist journals that provide peer review but not a publishing platform.

The emphasis therefore is on creating the canon of ‘prized’ literature, seemingly in response to the insidious practice of valuation according to use (citation) and location (quantified as ‘impact’ in the Impact Factor). Whether any editing would take place is an interesting question.

In a formal remarks section of this discussion paper, librarian (and Dean of Libraries, Professor of Public Policy, Economics, and Information, Former Provost and Executive Vice President for Academic Affairs!) Paul Courant at the University of Michigan adds his thoughts.

Why change the peer-review publishing system if mandates and Green OA can solve the problem of open access?

I would be surprised if mandates for open access would solve the problem; that would imply significant change in the relationship between peer review and publication. I think what is imagined by this question is that the institutional repositories to which people are making deposits serve as a bin from which to create overlay journals.

I do wonder whether having heavily populated institutional repositories, which were open to the world and for non-commercial republishing of various kinds, would solve a lot of the problem. I think that an as likely route would be if the federal government mandated an open access version of all scholarship produced with federal funds within six months or a year of publication, and that that version is allowed to be read and recombined in various ways. That would certainly have a salutary effect on the way the commercial publishing industry works. Additionally, universities should reserve a slice of copyright so that they can always make that happen.

Lior Pachter, a computational biologist who notably published the Cufflinks RNA-seq transcript assembly software, has spoken of science’s lapsed obligation to its ‘supplementary’ data (those research data that are moved out of the main article, and as such often neglected during peer review).

These limitations arise from the traditional scholarly publishing system, but with literature now both read and submitted electronically it seems counterproductive to still be doing so.

To replace this facet of the research process, Lior writes his own blog, and draws regular comment from others in the field. This type of activitity is usually pitched as post-publication peer review, though I’m yet to hear of any other avenues through which supplementary materials could be subjected to scrutiny and held to adequate account.

It seems quite important to me for previous research to be discussed and posts of worth to be brought to the attention of others in a way that steps outside of the publication-PR cycle. The push for (and increasing presence of) open access in academia certainly makes the ability to change the canon of ‘top literature’ more feasible — something both overlay journals and post-publication peer review aim to achieve.

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It is difficult to obtain conclusive proof of absence of transcription because transcription may only occur under particular environmental conditions that do not match those assayed in the laboratory.

Wilson & Masel (2011), in a paper also detailing the potential for “a simple product of chance or phylogenetic confounding” to undermine conclusions drawn from lab assays.


Professor of Biology and Biochemistry at the University of Houston Dan Graur, who writes as judgestarling, pointed out this neat review from 2011, when prompted for an example of transcribed and translated “junk” DNA.

Another paper has been published by Manolis Kellis et al, members of the ENCODE project which set out to define, describe and quantify function in the genome (the dispute over which I covered here last year).

The developments in this story aside, this paper details how putatively noncoding transcripts show extensive association with ribosomes. Whole genes arising de novo from non-coding DNA amounts to a sort of virgin birth in evolutionary terms — at odds with gradual change and adaptation.

There have been recent surprising reports that whole genes can evolve de novo from noncoding sequences. This would be extraordinary if the noncoding sequences were random with respect to amino acid identity. However, if the noncoding sequences were previously translated at low rates, with the most strongly deleterious cryptic polypeptides purged by selection, then de novo gene origination would be more plausible.

I’ll write more on this soon, in the mean time the paper is open access at the link below, and for more on de novo gene formation see my previous post and Carl Zimmer’s recent piece in the New York times in which he speaks to Christian Schlötterer, the author of the eLife paper I mentioned midway through.

 Wilson, B. & Masel, J. (2011) Putatively noncoding transcripts show extensive association with ribosomes. Genome Biology & Evolution, 3, 1245-1252

Masel had previously written a more mathematical paper on fitness, and a “local solution [that] facilitates the genetic assimilation of cryptic genetic variation”:
  Rajon and Masel (2011) Evolution of molecular error rates and the consequences for evolvability. PNAS, 108(3) 1082‒1087

⇢  The Masel group’s web page has further details on this and their other lines of research, including the connection between de novo gene birth and the need for preventing aggregation as a constraint on sequence evolution therein

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