Unraveling biochemical spatial patterns: machine learning approaches to the inverse problem of Turing patterns

TL;DR

The findings reveal the significant promise of machine learning in steering the creation of synthetic patterns in bioengineering, thereby advancing the grasp of morphological intricacies within biological systems while acknowledging existing limitations.

Abstract

The diffusion-driven Turing instability is a potential mechanism for spatial pattern formation in numerous biological and chemical systems. However, engineering these patterns and demonstrating that they are produced by this mechanism is challenging. To address this, we aim to solve the inverse problem in artificial and experimental Turing patterns. This task is challenging since high levels of noise corrupt the patterns and slight changes in initial conditions can lead to different patterns. We used both least squares to explore the problem and physics-informed neural networks to build a noise-robust method. We elucidate the functionality of our network in scenarios mimicking biological noise levels and showcase its application through a prototype involving an experimentally obtained chemical pattern. The findings reveal the significant promise of machine learning in steering the creation of synthetic patterns in bioengineering, thereby advancing our grasp of morphological intricacies within biological systems while acknowledging existing limitations.

Figures (5)

• Figure 1: Turing patterns and methods for the inverse problem.(A) Network of canonical Turing pattern with a short-range activator and a long-range inhibitor. (B) Turing patterns from different models. The top-left pattern showing spots is produced with the Schnakenberg model, the top-right pattern showing labyrinths with FitzHugh-Nagumo and the bottom row, also showing spots and labyrinths, with the Brusselator and two different parameter sets. (C) Methodology of the paper. Starting from a Turing patterns, we build a PDE loss which is minimized with respect to the parameters $\beta$. This is done using two different methods, LS and RBF-PINNs.
• Figure 2: Least squares for parameter inference.(A) In LS, noise changes the loss function that we are minimizing and the optimal value shifts, leading to some error in the inferred parameters. (B) Resulting patterns obtained from LS with different noise levels. After corrupting the original pattern with relative noise of different levels, parameters are obtained using LS and the model is solved with the inferred parameters to obtain the new patterns shown. The upper row corresponds to the FitzHugh-Nagumo model, the lower row to the Schnakenberg model. Enlarged patterns on the left are the original ones. It can be observed that at 1% noise, the predicted patterns are noticeably different from the original ones across the models, and that we can obtain several patterns at the same level of noise. (C) Radially averaged power spectra (RAPS) obtained for different patterns recovered at different noise levels (shown in (B)) from the Schnakenberg model. Red corresponds to 1% noise, green to 0.6%, orange to 0.3% and blue is the original pattern. Inset shows the same plot but the $y$ axis in log-scale. (D) Scatter plot of RAPS differences for different relative noise levels with noticeable sudden increases due to discrete changes in the predicted Turing patterns. (E) Average relative error of inferred parameters for different relative noise levels in the Schnakenberg and FitzHugh-Nagumo models, with a sketch of the parameter space explaining the difference in spread. Green point represents original set of parameters values and orange points are the different optimal sets resulting from the minimization. The bigger standard deviation in relative error occurs when the optimal sets are close to the original so the scale of the error changes drastically (can be very close or far), while the smaller standard deviation occurs when the optimal points are further away and the error is always on a similar scale.
• Figure 3: Effect of the number of pixels in the least squares method.(A) Pattern from the FitzHugh-Nagumo model with three different cropped regions given by small squares of different size. It can be seen that a region of $3\times3$ pixels is sufficient to recover accurate enough parameters such that the predicted and original patterns (right and left respectively) are indistinguishable. (B) Schematic of choosing $N$ randomly selected points (black) on the Turing pattern. (C) Effect of increasing the number of randomly selected pixels on the average relative error in the inferred parameters with (orange) and without (blue) added noise to the original pattern for the FitzHugh-Nagumo model. We used $N$ in the range $5$ to $2000$ ($50\times50=2500$ being the maximum possible) and sampled $10$ different sets of pixels for each $N$, and we measured the relative error for each of the inferred parameters. There is almost no effect without noise, but with noise there is a steady reduction in the relative error. Also shown is the slope of the line of best fit to the data (orange).
• Figure 4: Physics-informed neural networks for parameter inference.(A) Architecture of RBF-PINNs, where the input is space coordinates ($x,y$) and the output is the pattern. Input is shown in green, variables which are trained in blue and input to the losses in yellow. Red arrows show denote usage in the loss and yellow arrows backpropagation. From the network the partial derivatives can be efficiently computed using automatic differentiation and used in the PDE loss, where the PDE parameters are also network parameters. (B) Illustration of the three parameters of the Gaussian kernel and their interpretation. (C) Results from RBF-PINNs with different levels of added noise. After adding noise to the Turing pattern, the network is used to obtain a parameter set, which is subsequently used to predict the pattern. Patterns to the left are the original ones. (D) Relative error in the parameters and the RAPS difference for the parameters and patterns used for (C).
• Figure 5: Application to chemical patterns.(A) Explanation of scaling procedure, showing numerical and experimental patterns and how the scaling to the free-scale variable and the rescaling using the shift and scale parameters are performed. (B) Architecture of RBF-PINNs for the experimental case, with the division into the $u$ and $v$ approximation, the rescaling and the different losses. (C) Results from RBF-PINNs to the numerical pattern (first column) and the experimental pattern (second column). The top images show the original patterns, while the bottom images show the predicted patterns using the inferred parameters and our numerical solver. (D) Time evolution of simulated experimental pattern showing how labyrinths are present at the initial time points and plot of convergence of scaling parameters.