Identification of the governing equation of stimulus-response data for run-and-tumble dynamics

Abstract

The run-and-tumble behavior is a simple yet powerful mechanism that enables microorganisms to efficiently navigate and adapt to their environment. These organisms run and tumble alternately, with transition rates modulated by intracellular chemical concentration. We introduce a neural network-based model capable of identifying the governing equations underlying run-and-tumble dynamics. This model accommodates the nonlinear functions describing movement responses to intracellular biochemical reactions by integrating the general structure of ODEs that represent these reactions, without requiring explicit reconstruction of the reaction mechanisms. It is trained on datasets of measured responses to simple, controllable signals. The resulting model is capable of predicting movement responses in more realistic, complex, temporally varying environments. Moreover, the model can be used to deduce the underlying structure of hidden intracellular biochemical dynamics. We have successfully tested the validity of the identified equations based on various models of Escherichia coli chemotaxis, demonstrating efficacy even in the presence of noisy measurements. Moreover, we have identified the governing equation of the photo-response of Euglena gracilis cells using experimental data, which was previously unknown, and predicted the potential architecture of the intracellular photo-response pathways for these cells.

Fig 1. Schematic of SIVM and DIVM.

The input dataset for SIVM and DIVM are respectively discrete pairs $(f_i,s_i)$ and quadruplets $(f_i,s_i,f_i^{\prime },s_i^{\prime })$ ($i=1,2,\cdots ,N^{\text{train}}$). The neural networks $G_{{\theta }_1}^{NN}(f,s)$ and $G_{{\theta }_2}^{NN}(f,s)$ are trained through SIVM loss ${ℒ}_{\mathrm{S}\mathrm{I}\mathrm{V}\mathrm{M}}$, while $G_{{\theta }_3}^{NN}(f,s,f^{\prime },s^{\prime })$ and $G_{{\theta }_4}^{NN}(f,s,f^{\prime },s^{\prime })$ are constrained by DIVM loss ${ℒ}_{\mathrm{D}\mathrm{I}\mathrm{V}\mathrm{M}}$. During the training of neural networks, we assess the performance of the validation set through $E_{\text{vali}}$ between ${\hat{f}}_i$ and f_i to avoid overfitting.

Fig 2. Performance of SIVM and DIVM on E. coli models I-III under ELCC signals.

(A) Model I; (B) Model II; (C) Model III. Top row: the simplified topological relationships among external stimuli, internal variables, and tumbling fractions in the three different E. coli models. $x\to y$ denotes that x influences y, which can be both excitation or inhibition. Second row: The outside signal s(t) of the form ELCC. The third row: the predictive performance of SIVM. The fourth row: the predictive performance of DIVM. Here, the response data (gray circles) are obtained by adding Gaussian noise to the numerical solutions of the corresponding models with $\mathrm{S}\mathrm{N}\mathrm{R}$ following the uniform distribution U(20,22). These noisy data, after smoothing, serve as the reference values (black solid line).

Fig 3. Performance of SIVM and DIVM on E. gracilis experimental data.

(A) Typical cell trajectory with uniform light intensity, where the red dots highlight tumble events. (B-C) Measured tumbling fraction with PWC and PWCL stimuli. (D-E) Performance of SIVM and DIVM for different LCC stimuli (yellow dotted line). Gray circles are the experimental data. (D) is for $s_{max}^{\prime }=4.18\times {10}^{-3}$ $\mathrm{W}/({\mathrm{m}}^2\mathrm{·}\mathrm{s})$ and (E) is for $s_{max}^{\prime }=10.441\times {10}^{-3}$ $\mathrm{W}/({\mathrm{m}}^2\mathrm{·}\mathrm{s})$. Top row: the LCC stimuli; middle row: predictive performance of SIVM; bottom row: predictive performance of DIVM.

Fig 4. Identify the s gradient threshold leading to model II degeneration from SIVM and DIVM test errors.

External stimuli are compressed in time, as shown in (A). We gradually compress the T = 200 s stimulus signal to T₁ = {180,160,140,120,100} s. The absolute maximum value of the gradient $s_{max}^{\prime }$ is correlated with SIVM and DIVM test errors in (B). The error points from left to right correspond to T₁ = {200,180,160,140,120,100} s respectively. (C) illustrates the SIVM (blue dashed line) and DIVM (red dashed line) prediction outcomes of three experiments with different T₁. Here, ELCC stimuli are used in the test set; the subplots from left to right correspond to the three error points for T₁ = 140 s, T₁ = 120 s, and T₁ = 100 s in (B).