Introduction
Macromolecular interactions drive cell function. Such interactions include protein-protein interactions in signaling and regulatory networks, the interactions of receptors with their ligands, enzymatic reactions, DNA binding of transcription factors and more. Macromolecular interactions in the cell often utilize cooperativity for function enhancement and regulation. For instance, local enrichment of molecular concentration could be an efficient way to increase the effective affinity of bimolecular interactions. An important example of cooperativity in molecular binding can be mediated by intrinsically disordered regions (IDRs) in proteins. While proteins typically adopt a stable globular structure, many protein regions are flexible, sampling multiple states in conformation space [Dunker et al. (2000)]. Such intrinsically disordered regions (IDRs) are prevalent and allow for a sophisticated behavior of biological systems \cite{van_der_Lee_2014}. Interestingly, it is common to observe two (or more) adjacent binding sites that are coupled in the sense of function, either within a flexible region, or connected by a flexible linker. Upon binding of one motif, it may be more favorable, or less so, for the ligand to bind the other motif. This behavior is the result of a modification of the energy landscape from the first binding event, so that other structural motifs bind ligands more strongly, resulting in positive cooperativity (or more weakly - negative cooperativity). Alternatively, when bound, the ligand may be spatially constrained, increasing (decreasing) the effective concentration of motifs connected to it, a phenomenon known as avidity \cite{Mammen_1998}. Here, we present a simple biophysical model of such a system and define the conditions for manifesting cooperativity effects. We rely on polymer theory and solutions to the Smoluchowski equation in reaction-diffusion context (3.3. Diffusion-reaction), to describe the model system (5.1. Construction of a model). We then analyze quantitatively how output quantities of biological relevance are affected by the system physical properties and their consequences on cooperativity (5.2. Steady-state).
Model
General considerations
Chemical reaction network theory
There in a correspondence between the discrete set of complexes and their concentrations, to the phase space of the underlying physical system and its probability density distribution. The complexes in a system correspond to energy equivalence classes of the phase space, or phase “cells” (see [
van Kampen (2007) p. 108]). Whether such a description is appropriate for biological cellular processes is not a simple question to answer. Such processes often involve many actors, with an overflow of chemical detail, and it is not always clear how to divide the phase space into energetically-equivalent “cells”. Many cellular processes are however compartmentalized, macroscopically confined in time and space[
Mitrea et al. (2016)]. We may theorize that these compartments are small enough to present a uniform (well mixed) environment in which chemical reactions occur, so that the complexes are well-defined energetically, but large enough to use the tools of statistical mechanics. We then will assume the localized steady-states are perturbed, e.g. by changing the boundary conditions, followed by return to equilibration. Whether the system reaches equilibrium or a nonequilibrium steady state depends on the properties of the system and the reactions (see
3.3.4. Steady state and equilibrium).
Detailed balance
where we use the fact that the ratio of occupancies are given by the ratio of transition rates, to define the equilibrium constant Ki (with Kn=k1,nkn,1). The condition on the rates is evident by taking the product of all n occupancy ratios:
i=1nki+1,ii=1nki,i+1=i=1nKi=1
Rates must fulfill this detailed balance condition for the system to reach equilibrium. For nonequilibrium steady-states, the magnitude of the currents Ii may be used to quantify the entropy production rate [
Zia & Schmittmann (2007)].
The Random Walk chain
y combining the three components and integrating over all directions R for a given distance R, we obtain the distribution for the end-to-end distance:
p(R)=4R232nl232e-3R22nl2(4) |
The result we obtained for the polymer mean square end-to-end distance holds for a simplified freely-jointed chain model, and does not take any interactions into account. Real polymers have complex energy landscapes, leading to - sometimes considerably - different statistics, specifically the mean square end-to-end distance.
Depending on whether the interactions of the polymer are favorable over those with the solvent or not, one may have the polymer mean square end-to-end distance (equation
(3)) scale differently with n - with an exponent typically marked . When excluded volume effects become negligible due to a balance between the interaction of the polymer with the solvent (neutral solvent), long polymers behave as a freely-jointed polymer, and thus =1/2. In good solvents, interaction with the solvent is favored and the polymer expands, so >1/2. In poor solvents the opposite is true. A polymer will behave as a rigid rod for =1. Typically will fall somewhere between 1/2 and unity for relevant biological solvents, resulting in a larger end-to-end distance estimate. It was shown that for self-avoiding chains, the value of this exponent should approach =3/5[
Flory (1949)].
Other measures of chain stiffness, such as the persistence length or the Kuhn length, may be used to consider the polymer as being a freely jointed polymer using an effective unit length. Thus, one may write a generalized formula for the mean square end-to-end distance of a real polymer: