Geometric Path Integrals. A Language for Multiscale Biology and Systems Robustness

In this paper we suggest that, under suitable conditions, supervised learning can provide the basis to formulate at the microscopic level quantitative questions on the phenotype structure of multicellular organisms. The problem of explaining the robustness of the phenotype structure is rephrased as a real geometrical problem on a fixed domain. We further suggest a generalization of path integrals that reduces the problem of deciding whether a given molecular network can generate specific phenotypes to a numerical property of a robustness function with complex output, for which we give heuristic justification. Finally, we use our formalism to interpret a pointedly quantitative developmental biology problem on the allowed number of pairs of legs in centipedes.

💡 Research Summary

The paper proposes a novel mathematical framework for linking microscopic molecular states to macroscopic phenotypic traits in multicellular organisms, and for assessing the robustness of this link against variations in underlying parameters. The authors begin by assuming that at a given embryonic time point one can measure protein activity levels (X = (X_1,\dots,X_N)) and their time derivatives (\dot X). Using supervised learning, they train a single‑output neural network (F(X,\dot X)) that acts as an “interval classifier”: if a sample belongs to phenotypic class (C_m) then the output satisfies (m-1 < F < m). This construction forces each class to occupy a distinct real interval, which later enables a compact analytic formulation.

The molecular dynamics are modeled as a polynomial (or analytic) ordinary differential equation system (\dot x = f(x,a)), where (a) denotes a vector of environmental or genetic parameters. A “network classification” is defined by the existence of a state (X) and a scalar (Y) with (m-1<Y<m) such that (F(X,f(X,a)) - Y = 0). The central notion of “class robustness” (Definition 3) requires that for every parameter vector (a) in a prescribed neighbourhood (A) of a nominal value (a_0) there exists at least one such pair ((X,Y)). In other words, the zeros of the function \