To understand the repercussions and limitations of this model, we examine the limits of the parameter space for the inequality (see Table 1). Strikingly, the modeled system becomes cooperative for effective concentrations that are smaller (and not larger) than the receptor concentration. However, the conditions under which the term on the right hand side (RHS) of inequality (25) deviates from unity, or equivalently for the term 2x1 to be large, are extreme. 2 is capped at unity, and is only large if the reaction between B and X is unfavorable. x1 is determined by the reaction radius for a protein-protein interaction, which may range between the radius of a pair of amino acids (roughly 9~10Å) to a pair of domains (roughly up to 40Å) \cite{Lobanov_2008}, and the concentration of the linker, cAL. x1 is generally small - to exceed 10-1, the concentration of the linker cAL must be as high as 0.6-40mM. We keep in mind that x1 expresses the fraction of ligands B located at a distance of the reaction radius or closer to their partners A. For this quantity to approach unity means that nearly every B molecule present forms an encounter complex. Deviation from this assumption also means that the population of the state forming more than one encounter complex is not negligible (see discussion in 5.1.2. The kinetic scheme). Even for the realization of such a system, the value of the right-hand side will remain in the order of unity, which means the condition will hold only when effective concentration is in the order of the ligand concentration. Experimental work shows that short (2- to 5-mers) ethylene glycol polymers exceed effective concentrations of 10mM \cite{Krishnamurthy_2007}, though it may be more difficult for polypeptides to obtain the same values for the same number of polymer units, as they are less flexible (the polymer mean square end-to-end tends to be larger for stiffer chains; cf. equation (10)). Thus, the term in the RHS of inequality (25) is assumed to be close to unity. We keep in mind that both x1 and x2 depend on the chemical properties of the interaction between the specific reacting groups. On the other hand, c is governed by the effective concentration, which is determined by polymer properties, such as its length, stiffness and its interactions with the solvent. While evolution may find ways to modify chemical interactions (e.g. by mutation or post-translational modification), rational design of such modifications is more difficult than modifying the properties of the polymer. We therefore adopt a view where our variable of interest is the effective concentration, and inequality (25) as a condition on that variable. Whether it exceeds the concentration of the ligand determines if the system achieves positive or negative cooperativity (respectively). One may ask not only when the system becomes cooperative (or anti-cooperative), but to what extent does this occur. As we have established that it makes sense to analyze the ratio (24) using the effective concentration variable, we examine in Figure 7 the ratio response to different polymer lengths, which correspond to different effective concentrations. Cooperativity switching occurs between the system with and without the linker, but also, albeit less accentuated in this example, between the system with different surroundings. Specifically, we demonstrate the effect of two different solvent conditions by the right arrow in Figure 7 and the gap between the two curves which follows). It is convenient to describe the graph through four numbers different than its parameters: the limits at c0 (infinite chain length), c (zero chain length), the point where the ratio becomes unity, and the slope of the curve at that intersection. The first two limits determine how cooperative or anti-cooperative the model can get.
(26)
    The limit at c0 is always smaller than unity, and the limit at c is always greater than unity. Notably, the infinity-limit is independent of x2. The slope of the curve as it intersects unity is given by the linear coefficient in the series expansion around that point where the inequality (25) becomes equality:
(27) 
The steeper the slope is, the more susceptible the system is to switching of the second sort (solvent-dependent) as demonstrated in Figure 7.
There is at least one important reservation to be made regarding this model: in the limit 21, where the second binding site is completely neutralized, one can still achieve positive/negative cooperativity. The expression for the ratio in this case is non-constant, and achieves its maximal value, due to monotony, at c: 
\(\lim(0\le x\le1)\epsilon=\frac{1}{1-\eta_1}\left(1+\frac{\eta_1}{x_1}\right)\)
Which is strictly larger than unity. This result is due to miscounting of states in phase space:
when 21 holds, states 5-7 are inaccessible (see Figure 5) and the system should behave as a system without the linker (and therefore should tend towards unity). However, since both states 4 and 8 remain accessible, and since state 4 is counted towards occupancy of the active state, one can still balance probability in/out these states. To remedy this feature, the rates must be adjusted to account for the fact that some of the phase space which is represented by state 3 in the system without the linker, is represented by states 3 and 4 in the system with the linker. This problem lies outside the scope of this work.

Discussion

In this work we set out to construct and characterize a model system which exhibits cooperativity via the introduction of another binding site at the edge of a flexible linker. For this system, we define positive (negative) cooperativity as an increase (decrease) in the population of the active states of the system due to the introduction of the linker. We found specific criteria for obtaining positive or negative cooperativity. The system may shift from negative cooperativity to positive cooperativity as the linker length shortens (equation XX). Generally, positive cooperativity occurs when the effective concentration introduced by the flexible linker surpasses the concentration of the linker in the solution. The switch between these two regimes of cooperativity is continuous, and the threshold is defined in terms of the equilibrium constants of the system (equation XX). This establishes a framework for evaluating cooperativity in real-world biological systems. We have examined a specific set of parameters and demonstrated ways of modulating the system’s cooperativity (Figure \ref{795695}). We have also obtained expressions describing the magnitude of these effects (equations XX and XX). Finally, we provided a brief overview of some of the places where flexible proteins and their interactions may be applied in research problems we have been facing. Specific features of our modeled should\ref{907536} be noted. First, the effective concentration \(S^{-1}\) (equation XX) need not be greater than the linker concentration \(c_{AL}\), but of a fraction lower than unity. However, while one may formulate a system in which this fraction is large, it is typically small under normal biological conditions. Following this conclusion, we may focus on the effective concentration, or more specifically, the polymer mean square end-to-end-distance, as the variable which governs the switch in behavior. Modulation of the polymer may be affected by many biological processes: through alternative splicing, production of species with variable linker lengths may be regulated; post-translational modifications and other chemical morphogenesis may modify the stiffness, accessibility and reactivity of partners; external fields such as temperature, acidity or an electrical field, which can often be manipulated in experiment, may change the interaction balance between the receptor, ligands and their environment, affecting their flexibility and binding affinity; protein intracellular targeting may increase the local concentration in specific compartments. All of these parameters can modify the ratio between the effective concentration and the free species concentration c, and therefore determine the type and magnitude of cooperativity the multivalent receptor provides (i.e. the tuning of the system in regard to  the curve in Figure 7).