 Ordering My Thoughts (Summer 2013)

•  Cato

In this document I outline my ideas and goals for a handful of projects to work on over the summer (2013). The purpose is to maintain ordered thoughts and to be constructive/directed about tackling problems.

I have abandoned these questions (at least for the meantime) to pursue some more “useful” work.

Payam & Shura’s Papers

DNA electrophoresis in vicinity of a membrane pore: Rowghanian et al. (2013) and Rowghanian et al. (2013)

Some questions

1. What happens if an interior or bulk DNA piece approaches the pore?

2. In Rowghanian et al. (2013)a page 5, it is claimed that DNA motion will occur in a superposition of longitudinal and transverse orientiations (i.e. along the longest dimension, and perpendicular to it). Is this true? I think one of them will be unstable (though at high $$T$$ the orientation will be frequently randomised).

• Drag resistances (for elongated/prolate bodies) from Burgers (1995):
For a spherical particle: $$F_{\rm drag}^{\rm sph}=6\pi\eta a V$$.
For an end-on ellipsoid: $$F_{\rm drag}^{\rm ell-e}=\frac{4\pi\eta a V}{\ln(2a/b_0)-0.5}$$, where $$a$$ is half the length and $$b_0$$ is the equatorial radius.
For an end-on cylinder: $$F_{\rm drag}^{\rm cyl-e}=\frac{4\pi\eta a V}{\ln(2a/b)-0.72}$$ (slightly higher than ellipsoid).
For a face-on ellipsoid: $$F_{\rm drag}^{\rm ell-f}\simeq\frac{8\pi\eta a V}{\ln(2a/b)-0.5}$$.
For a face-on cylinder: similar to the face-on ellipsoid, but slightly higher.
As expected, the face-on bodies (where the motion is perpendicular to the long axis) encounter more resistance to motion than the end-on bodies – roughly twice as much, depending on the dimensions. Doesn’t this mean that, in steady-state, all the DNA will move in the longitudinal direction?

• Perhaps this is not a problem: we consider a low-Re regime, where viscous effects are much more important than inertia. Thus the instability or propensity to change orientation might not matter.

• Consider a stick moving through fluid at an angle; what is flow profile and where is the fluid pressure greatest? If the motivating force on the stick acts on the centre, there may still be a torque which aligns the stick in a preferred direction.

3. In Rowghanian et al. (2013) page 5, second column, they use dimensional analysis to justify a steady-state approximation used in the paper: the time taken to establish the fluid velocity profile around a section of DNA must be much shorter than the time taken the DNA to “move past” that fluid.

1. I want to try to do a better job than Payam’s scaling argument: what are the coefficients in the “$$\# \ll 1$$” relationship? This might be interesting because of a problem acknowledged in Payam’s other paper: his predictions do not fit the experimental data.

2. Dimensional analysis argument not fully understood: I guessed $$t_{\rm h}\sim \ell/c_{\rm s}$$ on physical grounds, but Payam says $$t_{\rm h}\sim \ell^2\rho/\eta$$ from dimensional analysis. What is the interpretation of Payam’s $$t_{\rm h}$$?

• The two $$t_{\rm h}$$s are measuring different things. Payam’s $$t_{\rm h}$$ is considering the diffusion of a particle’s momentum via vorticity in the fluid; or the characteristic time in the velocity autocorrelation of neighbouring particles. My $$t_{\rm h}$$ is about information flow via sonic waves: this is much smaller than the vorticity time-scale.
I want to think about which time scale is more appropriate in this context and why?

• So, what should the time-scale be from physical reasoning: what is the physics behind establishing the flow profile $$\vec{v}(\vec{x})$$ in the fluid around the DNA/jet?
Landau et al. (1959) argue $$\vec{v}(\vec{x})$$ purely from momentum conservation / Gauss’ Law, and symmetries of the momentum flux tensor. Time plays no role because the source of the momentum is static with respect to the (“Eulerian”) fluid.
Can we do the same analysis with a moving submerged jet? Say the jet moves through stationary fluid along the jet axis with constant velocity $$U\hat z$$: how do the fluid velocity equations change? (Could use perturbation theory?)

• Does Payam assume incompressible fluid? Yes – he follows Landau et al. (1959) pp. 86-88 in setting up the submerged jet; they assume zero divergence in the velocity field & use stress tensor for incompressible flow. (The derivation in Broman et al. (2010) also assumes incompressibility.) This means that the sound speed in his model will be infinite, and sonic information transfer will be irrelevant. This is why there is only one possible time-scale in the theory.

• Should I re-visit the submerged/elongated jet derivation with compressibility / sound included? Method 1: I would have to derive the formulae for the submerged jet in a compressible fluid, and perhaps solve the time-dependent Navier-Stokes equation to boot.
Method 2: tack some ad-hoc changes onto the existing work – e.g. treat the DNA/jet as an extended source of spherical waves. This will be inconsistent and crude, but probably easier than method 1.
In any case, a model with “sound” propagation might be a more accurate model (see references below). It may also be more illuminating as to how the velocity profile is established, and the consequences of interactions between neighbouring DNA molecules.

• Which process (sound waves or vorticity diffusion) carries more momentum? Read Landau et al. (1959) chapter 8 (p. 245), on sound. Specifically: how is a sound wave generated; how much momentum does it carry / how does it affect the surrounding medium; how does it compare with vorticity?
Sound waves are a higher-order effect than vorticity propagation. Read more.

• Another thing to consider: does it even matter? The time-scale they use in the paper is much larger than the sound-time, and places a more stringent test on the steady-state assumption. So my objection would only strengthen the conclusion.
The vorticity moves slowest, and the fluid velocity profile might only assume the anticipated form once this final contribution has had time to influence the entire profile.

Hinch (1975) application of Langevin equation to fluid suspensions; includes fluid inertia and interaction with suspension.
Ladd (1993) short-time motion of colloidal particles: theory.
Español (1995), Español et al. (1995) for discussion of how HD disturbances propagate, and evidence for faster-than-vorticity interaction. One considers incompressible and the other considers compressible; the vorticity time-scale is not the only important time-scale.
Clercx (1997) scaling of diffusion coefficient.
Kao et al. (1993) high-speed experiment finds interaction between colloids is faster than the vorticity timescale. Single-particle theory simply scales with parameters to account for interactions at various volume fractions.
Henderson et al. (2002): high-speed observations of two interacting colloids. Sonic effects.
Padding et al. (2006) coarse-graining approach; Langevin does a poor job.

3. Payam has (equation 12) $$v_{\rm DNA}=F_{\rm ext} \ln(\ell/d)/\eta\ell - \lambda E\ln(1+r_{\rm D}/d)/\eta$$, which we use to calculate $$t_{\rm D}$$. He finds $$t_{\rm D}\sim \ell\eta/E\lambda$$ – whence this simplification?

4. Is the elongated jet derivation valid? (Since integration along length of the DNA may break L&L’s “weak momentum injection” assumption.)
Yes it is valid – this is actually addressed on pp. 2-3: the criterion for the submerged jet formulae is the same as the L&L criterion for linear fluid velocity dependence.

5. Is there anything to be done about the discrepancy between experiment and observations in the DNA capture paper?

• The onset of the diffusion-limited regime occurs later than predicted when varying the DNA length (figure 4a). But when varying the potential difference, the crossover happens earlier (figure 4b).

• The approximations and order-of-magnitude guesses in the paper mean that we have no reason to expect high-fidelity predictions. They actually do surprisingly well.

• But there might be a question of internal consistency: does the fact that there are two discrepancies (figures 4a and 4b) in “different” directions point to some problem with the theory? I should read the paper more thoroughly and try to understand the significance of the different crossover regions.

• I could improve the work by putting error bars on the predictions – perhaps this will boost faith in the results, or point to some hitherto unaccounted-for issue.

6. Is there anything to learn from the inevitable energy dissipation as the DNA moves through the viscous fluid? Could predict this from hydrodynamic considerations. How much work is done by the electric field? Biological implications?

7. Where does the nonequilibrium stuff happen? There is no mention of linear response or anything like that.

Understanding Jeremie’s Swimmers

Some questions inspired by / about Jeremie’s experiment with active (diffusio-phoretic) colloids.

QUESTIONS

1. Why do they from a 2D layer? Answer: because they are heavy!

2. For now consider the simple case of a single inert particle in a (fixed) concentration gradient. Say it moves towards the source of the chemical. It will follow some mean trajectory, but with plenty of scatter. This is an out-of-equilibrium process.

1. Understand why it moves (diffusiophoresis): how fast and in what direction?

• Consider an uncharged particle for now.

• An osmotic pressure gradient the interfacial layer leads to drift velocity $$V_{\rm DP} = \mu\nabla c$$. I have been able to recover something which resembles this sort of.

• This is a complicated and difficult question – probably more productive if I just accept the established results and move on to other questions.

2. What are the statistics of this motion – how much variance is there?

• Look at notes. Imagine density/concentration/probability instead of a single particle.

3. Which part of this setup is out of equilibrium?

4. What does the scatter in trajectories tell us about the system? Anything useful? Use linear-response theory (is this justified? Why?).

3. Now consider active particle in concentration gradient. Do we get a simple superposition of two behaviours?

DIFFUSIOPHORESIS For charged interactions, the mobility $$\mu\propto 1/c$$. This leads to diffusio-phoretic velocity $$V = D\nabla\log c$$. This arises from the balance of osmotic and viscous forces in the Debye layer. The diffusion coefficient $$D \sim T/\eta\ell_{\rm B}$$, where $$\ell_{\rm B}$$ is the Bjerrum length (ratio of electrostatic to thermal energy). Derivation of fluid velocity done in Anderson (1989), page 14ish. One mechanism for diffusiophoresis of a charged colloid. Not so interesting to me now.

Jeremie’s Experiment: Clustering

\label{sec:cluster}

Note: need to restructure this section.

When there are many swimmers, they form clusters (read Theurkauff et al. (2012) and others). These clusters are very dynamic / “liquid”.

I want to understand:

1. how and why clusters form

2. why they are dynamic

3. what is the probability of breaking / re-forming (detailed balance / steady-state)?

4. what is the size distribution (for a given concentration, activity, …)?

I’m sure there are many tools from Matthieu’s class which would come in useful.

Thoughts:

Question 1:A first step is to solve the (coupled) diffusion equations of the medium and (mean field?). This gives coarse, deterministic behaviour. (Then need a way into the statistics.)

Can we use RG flow (see e.g. Bertin (2009), Matthieu)? Make a very simple Hamiltonian which captures some of the features of the swimmer clusters, then group them together in an increasingly course way. What happens to critical exponents, to couplings, etc? Will this actually be useful (given the Hamiltonian is necessarily very crude)? Instead of keeping track of all the particle positions $$\vec x_i$$, I could try putting them on a regular square lattice and just modifying the interaction strengths between them: $\sum_{\rm particles} J x_i \rightarrow \sum_{\rm grid} J_{ij}.$

Question 3: Random energy wells with some distribution of depths: what clustering statistics would this lead to? What distribution of wells gives a fractal distribution of clusters?

Master equation approach: $\frac{\mathrm{d}P}{\mathrm{d}t}(C,t) = -\sum_{C^\prime\neq C} W(C\rightarrow C^\prime) P(C,t) + \sum_{C^\prime\neq C} W(C^\prime \rightarrow C) P(C^\prime,t)$ The probability of taking a step depends on the number of neighbours.

Cooper Pairs

In superconductivity, electrons form Cooper Pairs. This is achieved not via direct interaction but through medium (lattice) mediation. A similar thing is happening in Jeremie’s experiment: swimmers deplete the medium around them, leading to attractive interactions and clustering behaviour (section \ref{sec:cluster}).

My idea Is the formalism for Cooper Pair formation applicable in any way to interactions of active particles? It could be an alternative way of thinking about things.

Define a “field” representing concentration. This will have creation and annihilation operators etc. Differences:

• All commutators are zero.

• All particles distingusihable.

• No Fermi / Bose statistics – perhaps this is essential to the method?

Investigate many-body interactions. Read Derek Lee’s notes on superconductivity.

Jarzynski Equality for Particle in a 1D Potential

Consider a particle in a 1D potential $$V(x)$$. Can I apply the JE to this system?

Some ideas:

Free energy

• What is temperature? Is there a way of turning mechanical energy into a free energy? How do I invoke a heat bath?

• Can I make a partition function (if I have an explicit $$V$$)?

• For a single particle in a monotonic potential, what counts as a macro/microstate. Is multiplicity always one?

• Free energy is work the can be done. But this is interpretation, not definition.

JE

• The JE shouldn’t depend on microscopic dynamics of the system, as long as they are microscopically reversible. Can I propose my own dynamics? E.g. the particle goes up or down the potential with some probability.

• Interpretation of the control parameter $$\lambda(t)$$ in this context: it is the “force” that a particle in a absolute zero vacuum would experience in the potential. Real experiment randomness.

• The work parameter provides the distinction between work and heat flow (at least in Crooks’ derivation (Crooks 1998)). (See also Crooks (1999) and Crooks (1999)a.)

• In the derivation of the JE, can we transform $$\lambda(t) \rightarrow \lambda(x)$$ to make relevance to fixed potential stronger? If so, then smooth $$V(x)$$ can be steps in $$H(t)$$.

Solving the problem

• Treat as absorbing boundary with drift, and solve the diffusion equation (see page 3 of Payam’s paper. Rowghanian et al. (2013)).

• If we can make free energy change zero, then $$\langle e^{-\beta W}\rangle = 1$$. With dissipative process.

Jeremie and Jarzynski

Once all the other stuff is understood, an eventual goal might be to apply JE to Jeremie’s experiment. This would be a nice test of the JE in a regime where it hasn’t been tested before (a local nonequilibrium process).

To do: formalise this.

To Do – short-term

• Look at Jeremie’s papers again.

• Review Matthieu’s notes with mind to a specific problem.

Miscellaneous Notes

• Renormalisation group good for things with structure, where correlations between sites are important. Could be good for CLUSTERING investigation, especially if I can write down a Hamiltonian for the whole system composed of other particles (will certainly have to simplify this a lot!).

• In Langevin analysis (e.g. Hinch (1975)) we define inertia and friction tensors $$\mathbf{m}$$ and $$\mathbf\zeta$$, which act on the derivative of the system’s state vector $$\mathbf{\dot x}$$. What does it mean when these are symmetrical? What does it mean when they are diagonal?

• Diagonalise the inertia tensor – reduce the motion to its modes.

• In regimes far from equilibrium, $$\mathbf\zeta$$ stops being symmetrical.

• Snowflake growth, renormalisation.