Our scheme thus described is not easily solvable. The equations of motions are nonlinear, as the term describing the bimolecular diffusion into the encounter complex (namely, the transitions between state 1 and states 2 and 8 in Figure 5) depends on the concentration of both the free receptor L (bound to A), and the free ligand B. We employ a pseudo-first-order approximation with respect to the concentration of the linker (AL), so that the assumption is that the concentration of the receptor is constant in time. The equations of motion of AL and B thus modeled are symmetric, and specifically AL=B, so we may write
\(\left[AL\right]\left(t\right)=\left[B\right]\left(t\right)+\left[AL\right]\left(0\right)-\left[B\right]\left(0\right)\)
The pseudo-first-order approximation would thus be a valid simplification when: \(\frac{\left[B\right]\left(t\right)}{\left(\left[AL\right]\left(0\right)-\left[B\right]\left(0\right)\right)}<<1\)
That is, when the concentration of the free ligand is very small throughout the reaction with respect to the initial concentration of free receptor. With this assumption, the equations of motion (for occupation of states) become linear.It remains a question, whether we should include the encounter complexes in our analysis. As mentioned in an earlier section (3.3.4. Steady-state approximation of an irreversible mechanism), using a steady-state approximation one can reduce the reaction-diffusion mechanism to its effective rate. One may thus describe the system without its encounter complexes, using effective rates for each reaction-diffusion process (Figure 4c). However, in the context of our pseudo-first-order approximation, the system is completely solvable, and using a steady-state approximation we may actually over-complicate the solution, or obscure important features of it.
Thus our model is constructed and we may now examine its behavior. In the next section, I define the ratio of interest formally, and discuss further simplifications that are assumed in order to solve the system. Then, I describe the rates of transition between states using rates obtained via SSS theory. Finally, interesting features of the model are highlighted.
Steady-state
Kinetic scheme
We find the steady-state population of the 'active' states in two systems - with and without the linker (see Figure 5). More formally, we are interested in the ratio:
\({\epsilon}{\equiv}\frac{{v}_{3}+{v}_{4}+{v}_{5}}{\sum_{i}^{}{v}_{i}}/\frac{{v}_{0,3}}{\sum_{i}^{}{v}_{0,i}}\) (19)
Where \(v_0\) is the steady-state population of states of the system without the linker, and v the corresponding vector for the system with the linker; the elements of the vectors correspond to the different states of the reactants. States 3-5 are considered active states, in which the ligands A and B interact directly. Both vectors are normalized so that the sum of all state populations is unity. State 3 is considered as the active state of the system without the linker (as shown in the periphery of the scheme in Figure 5). In general, the steady-state described by a linear kinetic scheme, such as in our case, is the null space of the matrix in the master equation. For a kinetic scheme describing a cycle, this matrix is:
The null space is generally one dimensional, unless the rates have special relations between them. An expression for this space may be written most generally as:
\(\left\{ \left.\left(\alpha\sum_{p=1}^{n}\prod_{i=p\to j-1}k_{i+1,i}\prod_{i=j\to p-1}k_{i,i+1}\right)_{j=1}^{n}\right|\alpha\in\mathbb{R}^{+}\right\} \)(20)
Where the following notation is used:
\(\prod_{i=s\to r}a_{i}\equiv\begin{cases} \prod_{i=s}^{r}a_{i} & s\le r\\ \prod_{i=s}^{n}a_{i}\prod_{i=1}^{r}a_{i} & s>r \end{cases}\)(21)
The relaxation modes (or transient solutions) of the system are given by other eigenvectors of the master equation matrix, and the eigenvalues of the system are the roots of the 8th-degree characteristic polynomial of the master equation matrix. We do not discuss them in this work, and they remain the subject of future study.From expression (20) above, we can see that the solution for the model with the linker, made up of 8 states, is not a short expression. The sum of active states may be written formally (cf. Figure 5):
For the model without the linker, we have 3 states, and the expression for the population of the active state is simpler:The required quantity is the ratio of the former quantity with the latter (see equation (19)). Exploring this 12-parameter space is complicated, even though all terms appear linearly in the numerator and denominator. One may look for solutions that reach equilibrium; for these solutions detailed balance holds, and the concentration ratios become functions of the equilibrium constants (as opposed to individual rates), whereby the number of free parameters is cut down in half. However we do not approach the exploration of the general parameters space here. Instead, we are interested in the specific underlying physical system of a flexible multivalent receptor, and its rates as determined by the properties of the linker and its binding sites.
Results
Model of a multivalent flexible polymer receptor
We substitute the generic rate constants with the expressions derived from SSS theory, which include appropriate physical parameters, as depicted in Figure 5. Following \cite{Szabo_1980} and \cite{Lapidus_2000}, we assign (cf. 3.3.3. SSS theory, equations (12), (13), (16), (18)):
Where S is the Jacobson-Stockmayer factor (see equation (10)). The intrinsic reaction-limited on- and off-rate are labeled qAB and AB (respectively, see Figure 5) - the underlying short-range physical interactions are beyond the focus of this work.The quantities for the BX and ABX complexes are defined similarly. The expression is the reaction radius (the sum of reactant radii) between the two species X and Y. Specifically, the subscript ABX refers to the encounter complex between the AB complex (B bound to A) and the other binding site X. The expression for the ratio obtained in this manner is too lengthy to quote in its entirety here, and is instead given in Appendix C. However, the expression is greatly simplified, as we mentioned, if we assume detailed balance, which occurs when:
\(\frac{\prod_{i=1}^{n}k_{i,i+1}}{\prod_{i=1}^{n}k_{i+1,i}}=\left[\frac{r_{\text{ABX}}}{r_{\text{BX}}}\right]^3=1 \) (22)
This assumption is valid is we assume that after binding to A, the reactive radius of B with respect to X remains unchanged. As mentioned before, in the case of detailed balance, the expression becomes a function of the equilibrium constants, namely the ratios of the on/off rates. The ratio between occupancies of adjacent states may be expressed using the previously defined equilibrium constants. For brevity, we define:
\({K}_{1}={q}_{1}/{{\lambda}}_{1};\ \ \ \ {K}_{2}={q}_{2}/{{\lambda}}_{2}\)
\({x}_{1}={{\frac{4{\pi}}{3}r}_{AB}}^{3}{c}_{AL};\ \ \ \ {x}_{2}={{\frac{4{\pi}}{3}r}_{BX}}^{3}{c}_{AL};\ \ \ \ \tilde{c}={S}^{-1}/{c}_{AL}\) (23)
Using these definitions, the equilibrium constants are:
\(\frac{{v}_{2}}{{v}_{1}}={x}_{1};\ \frac{{v}_{3}}{{v}_{2}}={K}_{1};\ \frac{{v}_{4}}{{v}_{3}}={x}_{2}\tilde{c};\ \frac{{v}_{5}}{{v}_{4}}={K}_{2}\)
\(\frac{{v}_{8}}{{v}_{1}}={x}_{2};\ \frac{{v}_{7}}{{v}_{8}}={K}_{2};\ \frac{{v}_{6}}{{v}_{7}}={x}_{1}\tilde{c};\ \frac{{v}_{5}}{{v}_{6}}={K}_{1}\)
Note that all of these are association constants, so the direction of the ratios have flipped between the first and second half of the cycle. We divide the numerator and denominator in equation (19) with v1and obtain:The expression is simplified further if we define:
\({{\eta}}_{1}={\left({K}_{1}+1\right)}^{-1}={{\lambda}}_{AB}/\left({q}_{AB}+{{\lambda}}_{AB}\right)\)
\({{\eta}}_{2}={\left({K}_{2}+1\right)}^{-1}={{\lambda}}_{BX}/\left({q}_{BX}+{{\lambda}}_{BX}\right)\)
It then becomes:
(3.0.2)
\(\epsilon=\frac{\left(\eta_1+x_1\right)\left(\eta_2+x_2\tilde{c}\right)}{\eta_1x_2+\eta_2x_1+\eta_1\eta_2+\left(1-\eta_1\eta_2\right)x_1x_2\tilde{c}}\) (24)
This is a monotonically increasing function of c. x1 and x2 are the equilibrium constants (for the first and second binding sites) of the diffusion-limited reaction (equations (15) and (17)). They are small as long as the reaction radius for the appropriate binding site is small compared to the interparticle distance. 1 and 2 are the fraction of unreacted population in the reaction-limited subprocess. Their values range between 0 (completely reactive) and 1 (unreactive). c is the ratio of the effective concentration to the receptor concentration. We recall that the effective concentration is a measure of the probability of the intramolecular distance between the two ends of the polymer. It is large when the polymer ends tend to be proximal, as would be the case for short, flexible and/or otherwise constrained polymers (see equation (10)).The ratio (24) is larger than unity as long as the following equivalent conditions hold:
(3.0.1)
\(\tilde{c}>\frac{1}{1+\eta_2x_1}\Rightarrow\frac{1}{\left[2\pi L^2/3\right]^{\frac{3}{2}}c_{\text{AL}}}>\frac{1}{1+\frac{4\pi r_{\text{AB}}^3}{3}\frac{\lambda_{\text{BX}}}{\lambda_{\text{BX}}+q_{\text{BX}}}c_{\text{AL}}}\) (25)
Inequality (25) determines whether the presence of the linker contributes to formation of the active complex (positive cooperativity), or whether it impedes on this process (negative cooperativity) (Figure 6). This condition is independent of x2 and 1. c is determined by the characteristics of the linker, x1 by the geometry of the AB encounter complex, and 2 by the reactivity of the BX complex. Note that the inequality is scaled by cAL, the receptor concentration.