Two diffusion processes are relevant to our work. One is of the freely diffusing particle. It is described by a uniform probability density distribution at equilibrium. The leading term of the solution of the above, in terms of a/R, reproduces the result obtained in 1901 by Smoluchowski for the effective reaction rate (when ):

Where C=4R3/3-1 is the concentration of the particle; so the bimolecular rate is kf/C.

The second process of interest is the diffusion of two ends of a polymer, or of two particles pinned to the edges of a polymer. It is described by the polymer end-to-end distribution. By taking the limit R, a solution was obtained in orders of a/L, where L is the root mean square end-to-end distance of the polymer (equation (3)), with a leading term of:

Keeping in mind that this is a unimolecular binding rate, and by comparing this expression to that of the freely diffusing particle (equation (8)), it is evident why the factor,

is often termed the effective concentration of the polymer ends in relation to one another. The inverse of this quantity appears as the probability of the two ends of the polymer to meet, relative to the volume defined by the encounter distance, in [Jacobson & Stockmayer (1950)]. S is termed the Jacobson-Stockmayer factor.

We study a minimal system that can result in an enhanced (reduced) avidity. The system contains a linker with two binding sites, which may facilitate increased binding between its ligands. Such a protein-ligand interaction network may be constructed in several manners. In the most general case, there are three bivalent reactants - two ligands, A and B, and a flexible polymer receptor, dubbed the linker, L, which together enable three distinct complexes to form: AB, AL and BL (Figure 3). Here, we focus on a reduced system, with one of the ligands (A) being constitutively bound to a linker (L), at the other edge of which is a binding site (X) for another ligand (B). Thus the system consists of a multivalent receptor with two distinct sites, both of which bind a specific ligand - but in independent, non-competing modes. To assess cooperativity in such a system, we compare the extent of formation of the complex AB (ligand binding to activating site), which we call the active complex or product, to the formation of this complex in a similar system without a linker. When the ratio between the former and the latter is greater than (or less than) unity, we state that the system is positively (or negatively) cooperative. Different approaches have been taken towards solving this kind of a problem. One may consider the statistical-mechanical problem of dynamics of the probability distribution in a continuous phase-space, and solve the continuity equation explicitly (Figure 3; see Section 3.4.3; e.g. \cite{Szabo_1980}, \cite{Sunagawa_1975}). Descriptions may be simplified by examining a discrete phase lattice (e.g. \cite{Rubin_1982} \cite{Szabo_1991}), or by dividing the phase space into a finite number of 'cells', and using rate equations to describe probability flow between these collections of states (cf. \cite{VAN_KAMPEN_2007}, e.g. \cite{Lapidus_2000}, \cite{Jacobson_1950}). Specifically, SSS theory \cite{Szabo_1980} provides a solution for the diffusion-reaction Smoluchowski (or Fokker-Planck) equation (equation (6)) in a setting where a ligand diffuses under an arbitrary field of force, with an absorber placed at the origin. Given arbitrary starting conditions, they obtain an effective rate for the overall, irreversible two-step process (see 3.4.3. SSS theory), and show it is closely described by single exponential decay for two cases of interest - a freely diffusing particle (equation (8)) and polymer end-to-end contact (equation (9)). The process described in the SSS theory may be decomposed into the process of binding between ligand and receptor into two subprocesses: diffusion and reaction, to construct a 3-state kinetic scheme \cite{Lapidus_2000}(see Figure 2). The first subprocess is governed by the diffusion-limited rates (on and off). The latter is determined by the efficiency of reaction on the surface of the receptor, and is thus termed the microscopic, intrinsic or reaction-limited rates. The time-dependent solution of such a system is bi-exponential, and it may be reduced to a single-exponential decay using a steady-state approximation (see Appendix A). This reduction produces the same results when comparing it to the leading terms of the SSS theory solutions. Thus, decomposing the full physical process into a kinetic scheme and the steady-state proves to be a simple tool for obtaining an effective rate for the irreversible diffusion-reaction mechanism. We embrace this method when approaching the problem of a reversible reaction. The scheme we build is thus made up of independent steps for the processes of diffusion into an encounter complex, and reaction into a bound complex. Similar to SSS theory, we treat our reactants as spheres, reacting isotropically. More sophisticated approaches to include localized binding sites or distance-dependent microscopic interaction have been suggested \cite{Shoup_1981}\cite{Agmon_1983}\cite{Zhou_1996}. Such results may be incorporated in our model, but their inclusion lie outside the scope of this study.