350 8.3  Mechanics of Biopolymers

Here, the polymer is typically modeled as either an FJC or WLC, but then a nontrivial

potential energy function U is constructed corresponding to the summed effects from each

segment, including contributions from a segment-​bonding potential, chain bending, excluded

volume effects due to the physical presence of the polymer itself not permitting certain spa­

tial conformations, electrostatic contributions, and vdW effects between the biopolymer and

nanopore wall. Then, the Langevin equation is applied that equates the force experienced by

any given segment to the sum of the grad of U with an additional contribution due to random

stochastic coupling to the water solvent thermal bath. By then, solving this force equation for

each chain segment in ~10⁻¹⁵ s time step predictions can be made as to position and orienta­

tion of each given segment as a function of time, to simulate the translocation process

through the nanopore.

The biophysics of a matrix of biopolymers adds a further layer of complication to the the­

oretical analysis. This is seen, for example, in hydrogels, which consist of an aqueous network

of concentrated biopolymers held together through a combination of solvation and electro­

static forces and long range vdW interactions. However, some simple power-​law analysis

makes useful predications here as to the variation of the elastic modulus G of a gel that varies

as ~C^2.25, where C is the biopolymer concentration, which illustrates a very sensitive depend­

ence to gel stiffness for comparatively small changes in C (much of the original analytical

work in this area, including this, was done by one of the pioneers of biopolymer mechanics

modeling, Pierre Gilles de Gennes).

8.3.6  MODELING BIOMOLECULAR LIQUID–​LIQUID PHASE SEPARATION

A recent emergence of myriad experimental studies of biomolecular liquid–​liquid phase sep­

aration (LLPS, see Chapter 2) has catalyzed the development of new models to investigate

how LLPS droplets form and are regulated. The traditional method to investigate liquid–​

liquid phase separation is Flory–​Huggins solution theory, which models the dissimilarity in

molecular sizes in polymer solutions taking into account the entropic and enthalpic changes

that drive the free energy of mixing. It uses a random walk approach for polymer molecules

on a lattice. To obtain the free energy, you need calculate the interaction energies for a given

lattice square with its nearest neighbors, which can therefore involve “like” interactions such

as polymer–​polymer and solvent–​solvent, or “unlike” interactions of polymer–​solvent.

During phase separation, demixing results in an increase in order in reducing the number

of available thermodynamic microstates in the system, thus a decrease in entropy. Therefore,

the driving force in phase separation is the net gain in enthalpy that can occur in allowing like

polymer and solvent molecules to interact, and much of modern theory research into LLPS

lies in trying to understand specifically what causes this imbalance between entropic loss

and enthalpic gain. For computational simplicity, a mean-​field treatment is often used, which

generates an average forcefield across the lattice to account for all neighbor effects. Flory–​

Huggins theory and its variants can be used to fit data from a range of experiments and

construct phase diagrams, for example to predict temperature boundaries at which phase

separation can occur and providing semi-​quantitative explanations for the effects of ionic

strength and sequence dependence of RNA and proteins have on the shape of these phase

diagrams.

One weakness with traditional Flory–​Huggins theory is that it fails to predict an interesting

feature of real biomolecular LLPS in live cells, that droplets have a preferred length scale—​

Flory–​Huggins theory predicts that either side of a boundary on the phase transition dia­

gram a phase transition ultimately either completely mixes or, in the super-​saturation side

of the boundary, demixes completely after a sufficiently long time such that ultimately all of

the polymer material will phase separate—​in essence resulting in two physically separated

states of just polymer and solvent; in other words, this would be manifest as one very large

droplet inside a cell were it to occur. However, what is observed in general is a range of

droplet diameters typically from a few tens of nanometers up to several hundred nanometers.

One approach to account for this distribution and preferment of length scale involves mod­

eling the effects of surface tension in droplet growth embodied in the classical nucleation

KEY BIOLOGICAL

APPLICATIONS:

BIOPOLYMER

MECHANICS

ANALYSIS TOOLS

Modeling molecular elasti­

city and force dependence of

unfolding transitions.