Stochastic sequestration dynamics: a minimal model with extrinsic noise for bimodal distributions and competitors correlation

Many biological processes are known to be based on molecular sequestration. This kind of dynamics involves two types of molecular species, namely targets and sequestrants, that bind to form a complex. In the simple framework of mass-action law, key features of these systems appear to be threshold-like profiles of the amounts of free molecules as a function of the parameters determining their possible maximum abundance. However, biochemical processes are probabilistic and take place in stochastically fluctuating environments. How these different sources of noise affect the final outcome of the network is not completely characterised yet. In this paper we specifically investigate the effects induced by a source of extrinsic noise onto a minimal stochastic model of molecular sequestration. We analytically show how bimodal distributions of the targets can appear and characterise them as a result of noise filtering mediated by the threshold response. We then address the correlations between target species induced by the sequestrant and discuss how extrinsic noise can turn the negative correlation caused by competition into a positive one. Finally, we consider the more complex scenario of competitive inhibition for enzymatic kinetics and discuss the relevance of our findings with respect to applications.

Mean and variance of the discrete uniform distribution are defined as: As described in the main text for an S T with a Gaussian distribution, we derive the free target probability distribution as a weighted superposition of conditional probabilities that are solution of the master equation (eq. (8) in the main text): In Fig. S1 we present some examples of P (T ) originated from sequestrant probability distributions with mean and variance comparable to the ones of Fig. 1d in the main text. A uniform extrinsic noise is not able to induce bimodal distributions of the target free amount. Systems that pick a value of S T below threshold are concentrated by the threshold response into a repressed peak with value of T close to 0. Differently than the Gaussian case, the expressed peak, which corresponded to the peak of the distribution of S T , cannot be obtained in the uniform case. Indeed, each value of S T has the same probability and the threshold response does not have any effect in the expressed regime. As a result, the free target probability distribution presents a flat plateau in correspondence to the expressed regime.
In order to obtain bimodal distributions of the free target amount, the extrinsic noise must have a peaked distribution, sufficiently broad to sample both below and above threshold.

Correlation dependence on the total amount of sequestrant
In the main text we kept fixed S T and studied the correlation by varying the total amounts of the targets and the level of extrinsic noise, i.e. the variance of the distribution P (S T ). Following this approach, we here investigate the dependence of the correlation on the mean of the extrinsic noise distribution. In the following analysis, the extrinsic noise level is evaluated in terms of the coefficient of variations (CV = σ S T / S T ). Since this quantity depends on the mean of the distribution, when S T is varied, the standard deviation is accordingly tuned in order to keep the CV constant.   Figure  The contour plot of the correlation in presence of extrinsic noise as a function of S T and T 2T , for a fixed value of T 1T , is presented in Fig. S2a. As in the analysis of the main text, we observe a region of positive correlation located in proximity to the threshold (the equimolarity point). This behaviour can be better observed in Fig. S2d, where the value of S T , for which the maximum correlation is attained ( S T * ), is plotted as a function of the total amount of the second target T 2T . With T 1T kept fixed, the position of the correlation maximum increases linearly with T 2T , closely following the threshold point. Fig. S2b,c show the behaviour of the correlation, as a function of S T , for different levels of extrinsic noise. As in the main text, also in this case we observe that the extrinsic fluctuations oppose the negative correlation induced by competition and eventually lead to a peak of positive correlation which, as described above, is located in proximity to the threshold. The competition-induced negative correlation tends to dominate when the total amount of target molecules is much greater than the sequestrant one, while the global correlation tends to 0 when the sequestrant molecules saturate the system.

Correlation dependence on the dissociation constants
Let us now explore the dependence of the correlation on the two dissociation constants by keeping fixed the total amounts of molecules T 1T and T 2T . We focus on the case in which the total amount of molecules of one target and its dissociation constant are fixed (e.g. K d 1 and T 1T fixed). By varying the dissociation constant of the other target (in this case K d 2 ), the correlation profile can be non monotonic, displaying a minimum, see figure S3e. We shall refer to the value of the dissociation constant that minimises the Pearson correlation as K 2 d * . As shown in figure S3f, the value of K 2 d * depends linearly on the fixed dissociation constant of the competing target (K d 1 ) and the slope of the dependence is set by the total amount of target molecules T 2T . Keeping S T fixed, the slope decreases as T 2T increases and vanishes when T 2T = S T . In the regime in which the target outnumbers the sequestrant (T 2T ≥ S T ), the correlation profile is monotonic as a function of the dissociation constant and its lowest value is reached for vanishing K d 2 , regardless of the value of K d 1 . This means that the minimum of the correlation is obtained for the highest affinity between the target and the sequestrant.
Besides the slope of the linear dependence on K d 1 of the minimum position K 2 d * , the total amount of target molecules T 2T influences also the offset. When the total number of targets exceeds that of the sequestrant (T 1T + T 2T ≤ S T ) at K d 1 = 0, the correlation is minimised by a finite value of K 2 d * . Conversely, for T 1T + T 2T > S T the offset disappears and a vanishing K d 1 corresponds to a vanishing K 2 d * . In this region of the parameters, the minimum of the correlation is reached for the highest affinity of both the targets. The existence of the offset in the position of K 2 d * as a function of K d 1 indicates the presence of two sub-regimes characterised by a different joint dependence on the two dissociation constants K d 1 and K d 2 . What determines these regimes is the total number of target molecules in the system T 1T + T 2T , compared to the total number of sequestrant molecules S T . When the sequestrant is more abundant than the two targets (T 1T + T 2T ≤ S T ) there are always some free molecules of the sequestrant and both the targets are in the repressed state. Nonetheless, these two targets can be correlated and their correlation presents a global minimum as a function of the two dissociation constants, see figure S3 (a, b). The presence of a global minimum for finite values of the dissociation constants justifies the existence of the offset in the 1-dimensional plots. For systems in which the number of target molecules is small, the global minimum can be located at relatively large values of the dissociation constants, corresponding to a condition of weak interaction between the targets and the sequestrant.
As an example, let us consider again figure S3. In the first regime, for the case in which T 1T = T 2T = 15 and S T = 30, we see that the minimum of the correlation is present and is obtained for values of the dissociation constants for which the average number of free target is low T 1 = T 2 2 and S 4. Moving into the second regime, where T 1T + T 2T ≥ S T , we see that there is no global minimum and correlation is lower for lower values of the dissociation constants. Nonetheless, even when the global minimum is absent, considering slices of the contour plot (e.g. for K d 1 fixed) the correlation still presents a minimum if T 2T < S T . Finally, when T 2T > S T , the local minimum in the 1-D plot of the correlation as a function of K d 2 is lost and the minimal value of correlation is reached for vanishing K d 2 .

Correlation in presence of higher levels of extrinsic noise
We here report a plot analogous to Fig. 3 in the main text but for higher levels of extrinsic noise (a standard deviation of 6 instead of 4) and higher dissociation constants. The main qualitative difference is the behaviour is contained in Fig. S4e where the abundance of target 1, for which the maximum correlation is attained (T * 1T ), is plotted as a function of the abundance of the second target T 2T . When the dissociation constants of the two targets are different the linear decrease of the maximum position may have a slope larger than −1. The profile after the overall number of target molecules exceeds that of the sequestrant depends more markedly on the specific dissociation constants and is, in general, non monotonic. The other features are qualitatively unchanged.

Mutual Information
In this section we provide additional plots for parameters that are not show in the main text. As presented in the main text and shown in Fig. S5, the mutual information profile starts at zero and sharply increases in the vicinity to the theoretical threshold. As for the Pearson correlation, when plotted as a function of T 1T , the dissociation constant K d 1 governs the steepness of the profile, while K d 2 mainly controls the maximum value of mutual information that can be achieved, see Fig.  S5. In presence of extrinsic noise, the mutual information conveys qualitatively the same message as the Pearson correlation. To compare directly the two quantities it is useful to express correlation in units of − 1 2 log[1 − ρ 2 ] where ρ is Pearson correlation coefficient (i.e. the mutual information that two jointly Gaussian variables of correlation ρ would have). Fig. S6 shows how the profiles of the two quantities display a qualitatively similar behaviour.

Competitive inhibition
We here present in detail the model of competitive inhibition kinetics that is discussed in the main text. This system is a model of enzymatic kinetics based on the interaction between an enzyme and its inhibitor. The inhibitor plays the role of the sequestrant and the enzyme is the target.
A free molecule of target, T F , can become active by binding to a substrate which is assumed to be at fixed concentration. The activation of the target occurs with a rate that depends on the intrinsic activation rate k f and on the concentration of substrate c S . The active target molecule, T A , can be deactivated with rate k r becoming T F . The free molecule of target  Figure S6. Comparison between the mutual information as a function of T 1T and the correlation in units of (−1/2 log(1 − ρ 2 )) for different levels of extrinsic noise: no extrinsic noise (a), σ S T = 4 (b), σ S T = 6 (c).
can also be bound by a free molecule of sequestrant, S, forming the complex T S with rate k + . The complex can in turn dissociate with rate k − , returning a free molecule of target and one of sequestrant. The reaction network is then: We assume that the concentrations of substrate and product are large, we then neglect their fluctuations and study the stochastic dynamics of target and sequestrant. In addition to the reactions above, we assume that the total amounts of target and sequestrant molecules are conserved, defining then the following conservation laws: As a consequence of the conservation laws, the number of free variables for this system is reduced to 2. We here focus on T A and T S. The master equation describing the time evolution of their joint probability distribution reads: We focus on the case of quasi-equilibrium dynamics in which the reaction of product formation is much slower than the others, which means neglecting the product forming reaction. The master equation for the general case has a very similar structure, with the main difference that there is now an additional reaction converting an active enzyme (bound to the substrate) T A to an inactive and free one T F . For the general case, then, the rate k r should be replaced by k r + k p .
The steady-state solution of the master equation can be written recalling the grand canonical distribution for ideal particle mixtures [2] and reads: Since we are interested in the effects of extrinsic noise on this model of competitive inhibition, we again assume that the total amount of sequestrant molecules is a fluctuating quantity described by a discretised Gaussian distribution P (S T ). The analytic equilibrium solution in presence of extrinsic noise can be obtained as a weighted superposition of conditional probabilities: