Fig 1.
Sketch describing the concept of memory within a reaction network.
A network is divided into bulk and subnetwork, x axis represents time. Concentration changes in the subnetwork act as signals that leave the subnetwork and travel into the bulk. There they interact with other bulk species and return at a later timepoint via bulk-to-subnetwork interactions. The net effect of such interactions is thus that the subnetwork reacts to its own past. The precise influence of past subnetwork states is governed by memory functions that depend on the time difference, i.e. on how long ago the relevant signal has left the subnetwork.
Fig 2.
Example of application of Zwanzig-Mori projection.
(A) Illustration of the methodology, for the example of a cross repressive motif. First the nonlinear thermodynamic reactions (left) are expanded into mass action reactions with an appropriate timescale separation (centre). This generates additional nodes that represent the possible DNA conformations for both proteins, with e.g. g1DNA/g2Prot indicating DNA for gene 1 with protein 2 bound to it. To the expanded network we can apply the projection approach, retaining only the concentration of protein 1 in the subnetwork. The effect of the rest of the network—the bulk—is captured via memory terms (right). (B) Comparison of the cross repressive motif described via the original thermodynamic equations and the expanded mass action equations with and without timescale separation. Already for a moderate fast rate factor of γ = 10 the mass action and thermodynamic time evolutions are visually indistinguishable. Two time courses are plotted in Fig. (B-D), for g1Prot & g2Prot. (C-D) Demonstration of the projection approach with g1Prot in subnetwork and g2Prot in bulk. The projected equations track the dynamics of the original thermodynamic equation with a reduced system that contains memory functions. However, one can observe in (C) that accuracy can be lost in the transient if the system is initiated too far away from the fixed steady state. (D) Starting nearer to the steady state (at t = 1) substantially increases the accuracy of the projected description. (E) Example of memory functions of protein 1 to itself, decaying with time difference. The linear memory function is positive as expected from the network (positive feedback loop), while the nonlinear term is negative to correct for range-limiting nonlinearities that the linear terms cannot capture. Parameters used are α1 = 1, α2 = 1, wp1 = 1, wp2 = 2, w1 = 2, w2 = 2, β1 = 1/2, β2 = 1/2.
Fig 3.
Patterning of the vertebrate neural tube.
(A) Antibody staining of Wild type (WT) mouse stained for three of the main bands in dorso-ventral patterning. (Image provided by Katherine Exelby). (B) Illustration of neural tube patterning: ventral Shh secreted from the notocord and floor plate (termed “Source”) generates patterned domains along the dorso-ventral axis. Each domain is defined by the expression of a characteristic set of genes. (C) GRN that patterns the three most ventral domains of the neural tube. The chosen separation of bulk (purple) and subnetwork (green) in the application of the Zwanzig-Mori projection is also shown. (D) Simulations of steady state pattern along the dorsoventral axis using a set of thermodynamic equations of the form of eqs (1 and 2). These equations were taken—along with appropriate initial conditions of 0 for all species—from [24]. For all plots where x axis represents neural tube position, zero corresponds to the most ventral point. (E) Full bifurcation diagram illustrating the multistable nature of the network. Shown are steady state concentrations of the four molecular species against neural tube position, with unstable steady states marked dashed. Colours in (D, E) identify genes/proteins in the same way as in the labelling of the illustration (B) and of the network nodes in (C). This colour code is used throughout the paper unless otherwise noted.
Fig 4.
Memory amplitude and temporal dynamics.
(A) Amplitude of memory (memory function at Δt = 0) of Nkx2.2 to itself along the neural tube. There are multiple lines as the analysis was performed at all possible stable steady states. The vertical axis is logarithmic to make the range of amplitudes easier to appreciate. Colours identify the memory amplitude contribution from each of the two possible bulk channels, via Irx3 and Pax6, respectively. Thick lines indicate physiological states, while thin lines indicate states that are not usually observed in vivo. (B) Linear memory amplitude of (past) Olig2 on Nkx2.2 along the neural tube. The memory via Pax6 is for the most part below the memory via Irx3 in each pair of corresponding curves. (C,D) Memory amplitudes of Olig2 to Nkx2.2 (C) and to itself (D). No channel decomposition is performed as Olig2 receives memory only via the Irx3 channel. (E) Nonlinear memory of (past) Olig2 squared on Nkx2.2 in the p3 domain, where dynamics are dominated by Irx3; memory via Pax6 is negligible by comparison. (F) Nonlinear memory function of (past) Olig2 squared on Nkx2.2 from position 0.2 in (E), plotted to show that the relative contribution of the Pax6 channel is small also for all time differences. (G,H) Nonlinear memory functions of Olig2 to Nkx2.2 (G) and itself (H) in the p2 domain at position 0.7, exemplifying the potential for nontrivial time dependences (including non-monotonicity and sign changes) in the nonlinear memory functions.
Fig 5.
Olig2 repression of Pax6 increases robustness to initial conditions.
(A) Patterning without the repressive link from Olig2 to Pax6, showing steady states reached from standard initial conditions. Compared to the full network, the qualitative domain structure is conserved. (B) Patterning for the full network for initial conditions with high levels of Pax6 and Irx3 is qualitatively identical to low initial levels of Pax6 and Irx3. Initial conditions [Pax6] = [Irx3] = 1. (C) Patterning without the repressive link from Olig2 to Pax6 for initial conditions with high levels of Pax6 and Irx3. The p3 domain is lost as a consequence of the different initial conditions. (D) Bifurcation diagram of the network without Olig2-Pax6 repression. The region in which the pMN state (Olig2 high) is stable has expanded ventrally, thus making the ventralmost region bistable.