Matters of Life and Death in Computational Cell Biology

TL;DR

The paper addresses the lack of a principled life-death boundary in computational cell biology and argues that a formal theory of cellular viability is needed. It proposes a geometric framework built around viability regions and viability space decomposition to unify living dynamics with death across subcellular to multicellular scales, introducing mortality, ordering, and collapse manifolds as organizing principles. It contrasts extrinsic viability, imposed by models, with intrinsic viability, arising from self-maintaining networks, and demonstrates intrinsic viability through idealized models like Conway’s Game of Life. This framework aims to guide the development of more coherent whole-cell and multicellular models and to align theoretical viability with empirical manifold structures, ultimately supporting digital twins and predictive biology at multiple scales.

Abstract

Nearly all cell models explicitly or implicitly deal with the biophysical constraints that must be respected for life to persist. Despite this, there is almost no systematicity in how these constraints are implemented, and we lack a principled understanding of how cellular dynamics interact with them and how they originate in actual biology. Computational cell biology will only overcome these concerns once it treats the life-death boundary as a central concept, creating a theory of cellular viability. We lay the foundation for such a development by demonstrating how specific geometric structures can separate regions of qualitatively similar survival outcomes in our models, offering new global organizing principles for cell fate. We also argue that idealized models of emergent individuals offer a tractable way to begin understanding life's intrinsically generated limits.

Figures (6)

• Figure 1: Viability space as a geometric concept in a minimal 2D-3D visualization.A. In the space of essential variables, the viability boundary (light blue) marks the edge of the viability region where a cell can still be considered alive. B. The viability region is expanded across the range of unconstrained variables, such as location in the environment, although its structure is not modified across these dimensions.
• Figure 2: The geometry of multicellular viability constraints. To generalize the geometric idea of life-death boundaries to models of more than one cell, we need to begin thinking about multicellular dynamics as playing out in the intersection of all participating cells’ viability regions. This figure shows an example of three cells’ essential variables in a unified space. When the system’s dynamics result in the green cell violating its viability constraint, it dies, leaving the other two cells behind. This can be imagined as the intersection of viability regions collapsing in its dimensionality.
• Figure 3: Schematic of viability space decomposition for a single cell. A. A single attractor (dark blue) is located within the viability region such that all of the vectors on the boundary point inward, and the entire region is asymptotically viable (green). B. The attractor is outside of the viability region, such that every initial condition leads to a trajectory that will die in finite time and will never finish the path to the attractor (gray), making the whole region transiently viable (orange). C. The attractor is within the boundary of the viability region, but some of the initial conditions in its would-be basin die in the transient. To decompose the viability region into its asymptotically and transiently viable sets, we find where the vector field is tangent to the viability boundary (magenta point). Since this point leads to an asymptotically viable trajectory, it is a mortality point, and its backward time trajectory is a mortality manifold that separates the two sets.
• Figure 4: Multicellular viability space decomposition as a hybrid dynamical system.A. It is possible to visualize a multicellular model as a type of directed graph. Each node represents the cells currently alive with the corresponding continuous state space, and edges are the transitions that take place between nodes when one or more cells simultaneously die. Colored edges correspond to specific trajectories in the following subplots. B. Where both cells’ viability constraints meet (cyan square), the dynamics result in both cells perishing simultaneously. The backward time trajectory (dashed cyan trajectory) of this point forms an ordering manifold where either the blue or purple cell will have its viability constraints violated first. C. When the purple cell dies, the fate of the blue cell depends on its own state as it collapses into its independent one-dimensional space (double arrows). If the blue cell falls to the left of its unstable equilibrium point (red dot), it will head towards a terminal attractor, and if it falls to the right, it will be asymptotically viable. To know which initial conditions in the joint blue-purple cell space will result in either outcome, we can find the terminal point that collapses onto the unstable equilibrium (red square) and integrate it backward in time to get a collapse manifold (dashed red trajectory).
• Figure 5: A schematic of the intrinsic viability region of a glider in Conway’s Game of Life. The glider is defined by a pattern of on-cells (brown) and a surrounding layer of off-cells (yellow) that function as its physical boundary or membrane. As the glider moves through its environment, it encounters various environmental configurations that function as perturbations, which will either result in a viable (green) or nonviable transformation (orange) according to its conditions for self-maintenance. The states that lead to the green transitions are in the interior of the viability region (light gray), and the state that results in a terminal transition belongs to its boundary (light blue). The complement of the viable set are all the configurations that do not contain the glider.