Modeling the Intercelular Exchange of Signalling Molecules depending on Intra- and Inter-Cellular Environmental Parameters

Exchange of biochemical substances is essential way in establishing communication between bacterial cells. It is noticeable that all phases of the process are heavily influenced by perturbations of either internal or external parameters. Therefore, instead to develop an accurate quantitative model of substances exchange between bacterial cells, we are interested in formalization of the basic shape of the process, and creating the appropriate strategy that allows further investigation of synchronization. Using a form of coupled difference logistic equations we investigated synchronization of substances exchange between abstract cells and its sensitivity to fluctuations of environmental parameters using methods of nonlinear dynamics.

💡 Research Summary

The manuscript by Balaz and Mihailovi tackles the problem of inter‑cellular communication by focusing on the exchange of signaling molecules between two bacterial cells. Rather than constructing a detailed biochemical network, the authors aim to capture the essential dynamical structure of the process and to explore how internal (protein disorder, intrinsic noise) and external (environmental fluctuations, diffusion distance, fluid properties) perturbations affect the ability of the two cells to synchronize their signaling activities.

In the introductory sections the authors review the ubiquity of cell‑to‑cell communication, from quorum sensing in bacteria to hormonal regulation in vertebrates, and emphasize that despite the diversity of molecular players the underlying scheme—release, diffusion, uptake, and regulatory response—remains remarkably conserved. They argue that robustness, i.e., the capacity to preserve functional output in the face of strong stochastic disturbances, is a central yet incompletely understood property of biological signaling systems. Consequently, they propose a minimalist mathematical representation that isolates the key parameters governing exchange:

• p – the affinity of cellular receptors/transporters for the signaling molecule, which encapsulates protein‑level disorder and intrinsic noise;
• c – a coupling coefficient that merges the concentration of the molecule in the surrounding medium with the effectiveness of the cellular response;
• r – the logistic growth parameter that reflects the overall suitability of the environment for molecule production and accumulation.

Using these quantities, the authors derive a pair of coupled logistic maps:

 xₜ₊₁ = r xₜ(1 − xₜ) + c p yₜ
 yₜ₊₁ = r yₜ(1 − yₜ) + c p xₜ

where xₜ and yₜ denote the intracellular concentrations of the signaling molecule in cell 1 and cell 2, respectively. The maps are constrained to the unit interval (0, 1) and the exponent p is chosen to avoid the trivial zero‑attractor.

The core of the study is a systematic numerical exploration of the three‑parameter space (c, r, p). The authors fix r = 3.95—close to the chaotic regime of the classic logistic map—and vary c from 0 to 1 in fine steps. For each value of c they compute the largest Lyapunov exponent for several discrete values of p (0, 0.1, 0.2, 0.3, 0.4). Positive Lyapunov exponents indicate divergent, chaotic trajectories (non‑synchronised), whereas negative values signal convergence to a synchronized manifold. The results reveal a clear threshold around c ≈ 0.4: below this concentration the system exhibits predominantly chaotic, desynchronised behaviour with occasional narrow windows of synchrony; above the threshold the trajectories become uniformly synchronised, as evidenced by consistently negative Lyapunov exponents. Increasing p shifts the threshold to lower c values and broadens the synchronised region, reflecting the intuitive notion that higher receptor affinity strengthens coupling and promotes robustness against environmental noise.

To complement the Lyapunov analysis, the authors employ Cross‑Sample Entropy (Cross‑SampEn) as a statistical measure of asynchrony between the two time series. Cross‑SampEn quantifies the regularity of the joint dynamics; lower values correspond to higher synchrony. The entropy curves corroborate the Lyapunov findings: a sharp decline in Cross‑SampEn occurs near c = 0.4, and for p ≥ 0.6 the entropy remains low across the entire c range, indicating that strong affinity can essentially eliminate the desynchronised regime.

The discussion highlights both strengths and limitations of the approach. The model’s elegance lies in its reduction of a complex biochemical exchange to three dimensionless parameters while retaining the essential nonlinear feedback and stochastic coupling. This enables a clear analytical and computational investigation of robustness and synchronisation. However, the authors acknowledge several simplifications: (i) real bacterial communities involve many interacting signals and multiple cell types, which cannot be captured by a two‑dimensional map; (ii) the affinity parameter p is treated as static, whereas in vivo receptor affinity can be modulated by post‑translational modifications, expression levels, and ligand‑induced conformational changes; (iii) the logistic growth term approximates saturation and feedback but may underestimate higher‑order inhibitory mechanisms that become important at high molecule concentrations.

Future work is suggested along three lines: expanding the framework to larger networks to study cluster formation and wave propagation; incorporating stochastic differential equations to model continuous noise sources rather than discrete map perturbations; and validating the theoretical predictions with experimental data, such as fluorescence‑based measurements of intracellular signaling molecule concentrations under controlled perturbations.

In summary, the paper provides a concise yet insightful mathematical treatment of inter‑cellular signaling molecule exchange, demonstrates how environmental coupling strength and receptor affinity govern the transition between chaotic and synchronized dynamics, and offers a foundation for more elaborate models that could bridge the gap between abstract nonlinear dynamics and concrete biological observations.