Population dynamics, information transfer, and spatial organization in a chemical reaction network under spatial confinement and crowding conditions

We investigate, via Brownian dynamics simulations, the reaction dynamics of a generic, nonlinear chemical network under spatial confinement and crowding conditions. In detail, the Willamowski-Rossler chemical reaction system has been “extended” and considered as a prototype reaction-diffusion system. Our results are potentially relevant to a number of open problems in biophysics and biochemistry, such as the synthesis of primitive cellular units (protocells) and the definition of their role in the chemical origin of life and the characterization of vesicle-mediated drug delivery processes. More generally, the computational approach presented in this work makes the case for the use of spatial stochastic simulation methods for the study of biochemical networks in vivo where the “well-mixed” approximation is invalid and both thermal and intrinsic fluctuations linked to the possible presence of molecular species in low number copies cannot be averaged out.

Figure 1

Schematic view of the Willamowski-Rossler chemical network. (a) The full version. (b) The minimal version analyzed in this study obtained by setting $k_{-2}=k_{-3}=k_{-4}=0$. In red we highlight the Lotka-Volterra component of the network [19, 25]. The overbar on species $E_1$, $E_2$, and $E_3$ indicates that the concentrations of those species are constants and the diagonal segment crossing $P_1$ and $P_2$ indicates that those species are instantaneously eliminated from the system (see Sec. 2 for details).

Figure 2

Population time series (partial time windows) obtained each from a single trajectory of our Brownian dynamics simulations of the MWR network with kset3 parametrization. (a) Population time series for systems with varying volume (radii varying from 0.4 to 0.65 $\mu \text{m}$). (b) Population time series for systems with varying number of crowders and with constant container volume. The radius of the spherical container is $R=0.4\mu \text{m}$.

Figure 3

Average population values for species $A$ [(a) and (b)], $B$ [(c) and (d)], and $C$ [(e) and (f)] with parametrizations kset1, kset2, and kset3. (a), (c), and (e) The average populations are plotted against the volume of the spherical container, no inert crowders are present. The average population of the reactive species grows linearly with the volume of the spherical container. (b), (d), and (f) The average populations are plotted against the number of inert crowders. A slight decrease in the average population is observed when the number of inert crowders within a spherical container of radius $R=0.4\mu \text{m}$ is increased from 0 to 8000. The presence of the inert crowders affects differently the three chemical species in the MWR network.

Figure 4

Transfer entropy as a function of the number of inert crowders (constant container volume). Data refer to kset3 parametrization.

Figure 5

(a) Statistical complexity measure as a function of the container volume with no inert crowders present. (b) Statistical complexity measure as a function of the number of inert crowders at constant container volume. Data refer to kset3 parametrization.

Figure 6

Average number of clusters (a) and average maximum cluster size (b) as a function of the number of inert clusters. Data refer to species $A$, $B$, and $C$ under kset3 parametrization. The average number of clusters shows a weak tendency to increase for all three chemical species. The average maximum cluster size decreases with denser crowding conditions. The maximum cluster size in species $C$ displays the largest decrease rate.

Figure 7

Time evolution for the population size (blue) and the maximum cluster size (yellow). The temporal pattern in the maximum cluster size accurately mirrors the population size. Data refer to species $A$ under kset3 parametrization. The mutual information between the two sets of temporal data decreases with increasing number of crowders (see Table 1).