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Controlling the stochastic dynamics of biological populations is a challenge that arises across various biological contexts. However, these dynamics are inherently nonlinear and involve a discrete state space, i.e., the number of molecules, cells, or organisms. Additionally, the possibility of extinction has a significant impact on both the dynamics and control strategies, particularly when the population size is small. These factors hamper the direct application of conventional control theories to biological systems. To address these challenges, we formulate the optimal control problem for stochastic population dynamics by utilizing a control cost function based on the Kullback-Leibler divergence. This approach naturally accounts for population-specific factors and simplifies the complex nonlinear Hamilton-Jacobi-Bellman equation into a linear form, facilitating efficient computation of optimal solutions. We demonstrate the effectiveness of our approach by applying it to the control of interacting random walkers, Moran processes, and SIR models, and observe the mode-switching phenomena in the control strategies. Our approach provides new opportunities for applying control theory to a wide range of biological problems., Comment: 12 pages, 4 figures

Understanding deaths and life-death boundaries of cells is a fundamental challenge in biological sciences. In this study, we present a theoretical framework for investigating cell death. We conceptualize cell death as a controllability problem within dynamical systems, and compute the life-death boundary through the development of "stoichiometric rays". This method utilizes enzyme activity as control parameters, exploiting the inherent property of enzymes to enhance reaction rates without shifting equilibrium states. This approach facilitates the efficient evaluation of the global controllability of models. We demonstrate the utility of our framework using its application to a toy metabolic model, where we delineate the life-death boundary. The formulation of cell death through mathematical principles provides a foundation for the theoretical study of cellular mortality.

We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient and mirror descent. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz-Hodge decomposition and the Otto structure in $L^{2}$ Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can extend the applicability of information geometry to various problems of linear and nonlinear dynamics., Comment: 92 pages, 9 figures

The optimization of high-dimensional black-box functions is a challenging problem. When a low-dimensional linear embedding structure can be assumed, existing Bayesian optimization (BO) methods often transform the original problem into optimization in a low-dimensional space. They exploit the low-dimensional structure and reduce the computational burden. However, we reveal that this approach could be limited or inefficient in exploring the high-dimensional space mainly due to the biased reconstruction of the high-dimensional queries from the low-dimensional queries. In this paper, we investigate a simple alternative approach: tackling the problem in the original high-dimensional space using the information from the learned low-dimensional structure. We provide a theoretical analysis of the exploration ability. Furthermore, we show that our method is applicable to batch optimization problems with thousands of dimensions without any computational difficulty. We demonstrate the effectiveness of our method on high-dimensional benchmarks and a real-world function.

In multicellular systems, the single-cell behaviors should be coordinated consistently with the overall population dynamics and functions. However, the interrelation between single-cell rules and the population-level goal is still elusive. In this work, we reveal that these two levels are naturally connected via a gradient flow structure of the heterogeneous cellular population and that biologically prevalent single-cell rules such as unidirectional type-switching and hierarchical order in types emerge from this structure. We also demonstrate the gradient flow structure in a standard model of the T-cell immune response. This theoretical framework works as a basis for understanding multicellular dynamics and functions.