Modeling delayed processes in biological systems

Delayed processes are ubiquitous in biological systems and are often characterized by delay differential equations (DDEs) and their extension to include stochastic effects. DDEs do not explicitly incorporate intermediate states associated with a delayed process but instead use an estimated average delay time. In an effort to examine the validity of this approach, we study systems with significant delays by explicitly incorporating intermediate steps. We show that such explicit models often yield significantly different equilibrium distributions and transition times as compared to DDEs with deterministic delay values. Additionally, different explicit models with qualitatively different dynamics can give rise to the same DDEs revealing important ambiguities. We also show that DDE-based predictions of oscillatory behavior may fail for the corresponding explicit model.

Figure 1

(a) A schematic diagram of the self-activation circuit. (b) Mean residence time calculated with DSS and explicit model I. (c) Equilibrium distribution calculated with DSS. (d) Equilibrium distribution calculated by explicit model I. $p$ is the proportion occupied by the low number state. For all plots parameter values $\alpha =5,c=19,\gamma =ln\left(2\right)$, and $b=10$ are used. MRT and PDF stand for mean residence time and probability density function, respectively.

Figure 2

(a) Mean residence time calculated by the DSS and by explicit model II. (b) Equilibrium distribution calculated by explicit model II. $p$ is the proportion occupied by a low number state. All parameters are the same as in Fig. 1.

Figure 3

Probability density function (PDF) $p_n$. Here the mean delay is fixed to be 2.

Figure 4

(a) A schematic diagram of a toggle-switch circuit. (b) Mean residence time of $X<Y$ state calculated by DSSs and the explicit model. (c) Equilibrium distribution calculated by DSSs. (d) Equilibrium distribution calculated by the explicit model. $p$ is the proportion occupied by the $X<Y$ number state; $\beta =21.9,k=6.8$, and $\gamma =ln\left(2\right)$.

Figure 5

Equilibrium distribution of toggle-switch circuit for $\tau =2$. $p$ shows the probability of $X<Y$. Recall that in the DSS, $P(X<Y)\sim 0.5$.

Figure 6

(a) Time series of total number of molecules in DSS. (b) Time series of total number of molecules in explicit model. (c) Corresponding periodogram in DSS, where $\tau =10$. (d) Corresponding periodogram in explicit model. Here $A=100,B=4.1,C=1.0$, and $D=1/\tau =0.1$.

Figure 7

Time series of the total number of molecules in the explicit model with $n=4$. Parameters are the same as in Fig. 6.