Stochastic Responses May Allow Genetically Diverse Cell Populations to Optimize Performance with Simpler Signaling Networks

Two theories have emerged for the role that stochasticity plays in biological responses: first, that it degrades biological responses, so the performance of biological signaling machinery could be improved by increasing molecular copy numbers of key proteins; second, that it enhances biological performance, by enabling diversification of population-level responses. Using T cell biology as an example, we demonstrate that these roles for stochastic responses are not sufficient to understand experimental observations of stochastic response in complex biological systems that utilize environmental and genetic diversity to make cooperative responses. We propose a new role for stochastic responses in biology: they enable populations to make complex responses with simpler biochemical signaling machinery than would be required in the absence of stochasticity. Thus, the evolution of stochastic responses may be linked to the evolvability of different signaling machineries.


S1A. Variations of the simple T cell signaling model
In the main text, our simple model consisted of the cost function in Eq. 9, with c 1 =50, c 2 =0.2, and c 3 =40, and the probability model in Eq. 10 with distributions in Fig.   3A. (These parameters weight the cumulative mistakes against self and pathogenic pMHC roughly equally.) In this section, we show that a different cost function with different probability distributions leads to the same qualitative results as the example in the main text, suggesting the results do not particularly depend on the choices of these model inputs. The cost function and probability model presented here have the same qualitative properties motivated in the main text. The cost function is: The probability distributions for encounters with self and pathogenic pMHC are presented in Fig. S1A. As in the main text, we model only an intermediate range of stimulus, since it is assumed T cells will not activate at very weak stimulus.
As with the model in the main text, the best stochastic solution outperforms the best single sharp threshold ( Fig S1B). The percentage change is small, but suffices to confirm that stochastic decisions outperform single sharp thresholds. (Because of the simplifications, the model is not quantitative.)

S1A. Optimization of the simple T cell model
In the main text, we considered a simple model in which the host encounters a single infection. Which particular infection the host encounters is uncertain. The cost function in Eq. 9 and probability model in Eq. 10 set up an optimization problem for the decision rule. To simplify the calculation, we made the assumption in the main text that the number of encounters in each infection is large enough so that, within a particular infection, the distributions of stimuli from self and pathogenic pMHC are well-sampled.
In the main text, we introduced the notation f 0 and f 1 for the fractions of encounters with self and pathogenic pMHC that activate T cells. When the distributions are well sampled, the fractions f 0 and f 1 converge to probabilities: That is, the probability a T cell activates in an encounter with self pMHC is just the probability the T cell activates given the stimulus x (σ(x)) times the probability the stimulus is actually x in an encounter with self pMHC (P (x|s=0) The simple probability model we have chosen is constant over unit intervals of the stimulus (Fig. 3A), in order to simplify computation of the optimal decision rule. As a result, the optimization problem can be transformed from a functional optimization over all ! " to an optimization over vectors ! r v where: Each v i is constrained to be between 0 and 1, inclusive (because each interval is of unit length and the decision rule falls between 0 and 1, inclusive, for all x).
If more than one element of the optimal solution ! r v is not strictly 0 or 1, then a stochastic strategy is strictly better than a single sharp threshold, since sharp thresholds have v i equal to 0 or 1 (no or complete activation) on all intervals except the one the threshold falls in.
In general, the optimum decision rule ! " * corresponding to the optimal solution ! r v is degenerate, since Eq. S7 is not invertible. The stochastic decision rule plotted in Fig.   3B was obtained by letting σ(x) be constant over each interval. The deterministic decision rule plotted in Fig. 3D was obtained by taking σ(x)=1 over the first part of each interval, and then σ(x)=0 over the second part of each interval, such that the appropriate value for v i was obtained. Though slightly simpler deterministic decision rules can be found by varying the choice of σ(x), they are still more complicated than the stochastic decision rule or a single sharp threshold. The best single sharp threshold (Fig. 3B) was obtained by explicitly searching over all possible threshold locations.