Temporal differentiation and the optimization of system output

We develop two simplified dynamical models with which to explore the conditions under which temporal differentiation leads to increased system output. By temporal differentiation, we mean a division of labor whereby different subtasks associated with performing a given task are done at different times. The idea is that, by focusing on one particular set of subtasks at a time, it is possible to increase the efficiency with which each subtask is performed, thereby allowing for faster completion of the overall task. In the first model, we consider the filling and emptying of a tank in the presence of a time-varying resource profile. If a given resource is available, the tank may be filled at some rate $r_f$. As long as the tank contains a resource, it may be emptied at a rate $r_e$, corresponding to processing into some product, which is either the final product of a process or an intermediate that is transported for further processing. Given a resource-availability profile over some time interval $T$, we develop an algorithm for determining the fill-empty profile that produces the maximum quantity of processed resource at the end of the time interval. We rigorously prove that the basic algorithm is one where the tank is filled when a resource is available and emptied when a resource is not available. In the second model, we consider a process whereby some resource is converted into some final product in a series of three agent-mediated steps. Temporal differentiation is incorporated by allowing the agents to oscillate between performing the first two steps and performing the last step. We find that temporal differentiation is favored when the number of agents is at intermediate values and when there are process intermediates that have long lifetimes compared to other characteristic time scales in the system. Based on these results, we speculate that temporal differentiation may provide an evolutionary basis for the emergence of phenomena such as sleep, distinct REM and non-REM sleep states, and circadian rhythms in general. The essential argument is that in sufficiently complex biological systems, a maximal amount of information and tasks can be processed and completed if the system follows a temporally differentiated “work plan,” whereby the system focuses on one or a few tasks at a time.

Figure 1

(Color online) Illustration of a tank that may be filled at a rate $r_f$ when an external resource is present or emptied at a rate $r_e$ when there is material in the tank to empty.

Figure 2

(Color online) Illustration of the two-gene resource processing model discussed in this paper. The dark blue rectangle denotes gene $G_1$ (black rectangle), and the dark blue circles denote protein $P_1$ (black circles), while the green rectangle (white rectangle) and circles (white circles) denote gene $G_2$ and protein $P_2$, respectively. The red circles (small black circles) denote the available external resource, which enters the system via $P_1$. The proteins $P_2$ process the resource. Processed resource is denoted by the small blue-green circles (small white circles).

Figure 3

Illustration of a periodic resource-availability profile.

Figure 4

Illustration of the “microsleep” solution profile.

Figure 5

Illustration of the time dependence of various enzyme levels in both the temporally differentiated and undifferentiated labor strategies.

Figure 6

(Color online) Comparison of the production rate of product $P$, as measured by the factor $1-\alpha$, for the temporally differentiated and undifferentiated pathways.