Biophysics Tools and Techniques for the Physics of Life (Mark C. Leake) (1)

Chapter 8 – Theoretical Biophysics 351

Szilard model, which allows reversible transitions to occur between droplets of different size

as they exchange material through diffusion, both through droplets growing (or “ripening”)

and shrinking.

For a simple theoretical treatment of this effect, if N is the number of biomolecules phase

separated into a droplet, then the free energy ΔG in the super-saturated regime can be mod

eled as –AN + BN^2/3 where A and B are positive constants that depend upon the enthalpic of

interaction and energy per unit area at the interface between the droplet and the surrounding

water solvent due to surface tension (see Worked Case Example 8.3). This results in a max

imum value ΔGmax as a function of N, which thus serves as a nucleation activation barrier;

from ΔG in the range 0–ΔGmax, the effect from surface tension slows down the rate of droplet

growth, whereas above ΔGmax, droplet growth is less impaired by surface tension and more

dominated by the net gain in enthalpy, at the expense of depleting the population of smaller

droplets (this process is known as Ostwald ripening (also known as coarsening) and is a sig

nature of liquid–liquid phase transitions).

Sviliard modeling can explain the qualitative appearance of size distributions of droplets,

but it does not explain what drives the fine-tuning of the A and B parameters, which is down

to molecular scale interaction forces. A valuable modeling approach which has emerged to

address these questions has involved a stickers-and-spacers framework that has been adapted

from the field of interacting polymers (Choi et al., 2020). This approach models interacting

polymers as strings which contain several sticker regions separated by noninteracting spacer

sequences, such that the spacers can undergo spatial fluctuations to enable interactions

between stickers either from the same molecule or with neighbors. Stickers are defined

at specific locations of the molecule due to the likelihood of electrostatic or hydrophobic

interactions (which are dependent on the nature of the sequence of the associated polymer,

typically either RNA or a peptide) to enable insight into how multivalent proteins and RNA

molecules can drive phase transitions that give rise to biomolecular condensates. The reduc

tion in complexity in modeling a polymer as a string with sticky regions avails the approach

to coarse-graining computation and so has been very successful in simulating how droplets

form over relatively extended durations of several microseconds even for systems containing

many thousands of molecules.

8.4 REACTION, DIFFUSION, AND FLOW

Reaction–diffusion continuum mathematical models can be applied to characterize systems

that involve a combination of chemical reaction kinetics and mobility through diffusional

processes. This covers a wide range of phenomena in the life sciences. These mathematical

descriptions can sometimes be made more tractable by first solving in the limits of being

either diffusion limited (i.e., fast reaction kinetics, slow diffusion) or reaction limited (fast

diffusion, slow reaction kinetics), though several processes occur in an intermediate regime in

which both reaction and diffusion effects need to be considered. For example, the movement

of molecular motors on tracks in general comprises both a random 1D diffusional element

and a chemical reaction element that results in bias of the direction of the motor motion on

the track. Other important continuum approaches include methods that characterize fluid

flow in and around the biological structures.

8.4.1 MARKOV MODELS

The simplest general reaction–diffusion equation that considers the spatial distribution of

the localization probability P of a biomolecule as a function of time t at a given point in space

is as follows:

(8.64)

∂

∂⁼

∇

+ ( )

v P