Growing-Dividing Autocatalytic Systems

• Growing-dividing autocatalytic systems are self-organizing reaction networks that combine autocatalytic growth with regulated division events, forming the basis of self-reproducing dynamics.
• The mathematical framework employs deterministic ODEs, stochastic dynamics, and geometric phase space analysis to elucidate transitions from exponential growth to instability or system death.
• Applications in synthetic biology and origin-of-life research guide the engineering of robust self-replicating systems while explaining cell size homeostasis and division control strategies.

Growing-dividing autocatalytic systems are self-organizing chemical or biochemical reaction networks that couple autocatalytic (self-amplifying) growth with explicit division or splitting events. These systems are fundamental to the paper of the origin of life, the construction and regulation of synthetic cells, and the design of robust self-reproducing synthetic biological circuits. Such systems display a range of collective behaviors—including balanced exponential growth, nonexponential growth, homeostasis, and various forms of instability or system death—depending on the interplay between internal autocatalytic kinetics and the external division (or fission) strategy.

1. Mathematical Foundations and Representations

Growing-dividing autocatalytic systems are formalized as deterministic or stochastic dynamical systems evolving in a chemical state space. The state at time $t$ is $X(t) = (X_1(t), X_2(t), ..., X_N(t))$, the populations of $N$ chemical species within a compartment. The deterministic growth phase is governed by a system of ordinary differential equations,

$\frac{dX_i}{dt} = f_i(X), \quad i = 1,\ldots,N,$

where $f_i(X)$ encodes the net rate of production (or degradation) of species $i$ via autocatalytic and other reactions.

A canonical example is the linear Hinshelwood cycle:

$$\frac{dX}{dt} = k_1 Y, \qquad \frac{dY}{dt} = k_2 X,$$

whose solutions generically converge to an asymptotic growth trajectory (AGT), e.g., $Y = m_A X$ with $m_A = \sqrt{k_2/k_1}$, along which all species grow exponentially at rate $\lambda = \sqrt{k_1 k_2}$.

Division is modeled as:

• A division control function $D(X)$ (homogeneous polynomial or monomial, e.g., total cell volume $V = \sum_i X_i$ or more generally, $$D(X) = X_1^{\alpha_1} X_2^{\alpha_2} \cdots X_N^{\alpha_N}$$).
• A division threshold $d > 0$ such that division is triggered when $D(X) = d$.
• A birth map $B: \mathbb{R}^N \to \mathbb{R}^N$ specifying the distribution of chemicals in the daughter cell. The standard map is $B(X) = X/2$ (equal partitioning), while nonstandard maps can reset individual species or impose fixed numbers.

The resulting evolution is a hybrid system: deterministic ODE flow punctuated by discrete stochastic (or deterministic) division events.

2. Conditions for Emergent Self-Reproduction and Growth Patterns

Self-reproduction (balanced growth and division cycles) arises only if the autocatalytic growth law and division strategy are mutually compatible.

• If $D(X)$ is homogeneous of positive degree (nonintensive, i.e., scales with system size), and the birth map properly maps the division surface $S_D = \{ X : D(X) = d \}$ into a region traversed by the AGT, the system approaches a growth-division steady state (GDSS):
  • Trajectories in phase space spiral toward the AGT and undergo repeated division at regular intervals; statistical properties (concentration ratios, interdivision time) become invariant from generation to generation.
  • For the Hinshelwood cycle and standard map, every division halves the chemical populations but preserves the trajectory on the AGT, ensuring exponential recovery.
• If incompatibilities exist—e.g., an intensive $D(X)$ (homogeneous of degree $0$) with the standard birth map—there is no net growth: birth maps do not move the system off the division surface, leading to stalling or system death.

Growth pattern outcomes fall into three categories: | Growth Pattern | Mechanistic Cause | System Behavior | |-------------------------------|------------------------------------|-------------------------| | Balanced Exponential Growth | Compatible growth/division (nonintensive $D(X)$, standard map) | All species grow exponentially at identical rate $\lambda$; ratios fixed; robust reproduction cycle | | Balanced Nonexponential Growth| Nonstandard birth maps (e.g., resetting one variable) | System converges to periodic orbits but departures from strict exponential growth can occur | | System Death | Incompatible division (e.g., intense $D(X)$, poor resetting) | Populations explode or collapse to zero; cell cycle cannot sustain itself |

The explicit criteria can be diagnosed geometrically: the intersection structure of the AGT, the division surface $S_D$, and the image of $S_D$ under the birth map $S_B$ determines the global attractor.

3. Division Strategies and Their Consequences

Division mechanisms dictate the stability and architecture of the growing-dividing system:

• Standard partitioning ($B(X) = X/2$) preserves concentration ratios; necessary for robustness when division control is based on extensive quantities.
• Non-standard resets (e.g., $B(X, Y) = (r, Y/2)$) model explicit biochemical control (e.g., initiation proteins in cell-cycle regulation), or non-equimolar partitioning. Such resets can stabilize systems with intensive $D(X)$ (e.g., concentration-based triggers) or induce nonexponential periodic behavior where a reinitialized molecule restarts growth cycles.
• Death regime: If division resets a critical variable to zero or fails to move the system to the AGT (for instance, if division is triggered by the concentration of a component, but the corresponding birth reset is not applied), the cell cannot balance growth and division, leading to extinction or uncontrolled growth.

The geometric language makes explicit when a chosen birth map $B$ and division surface $S_D$ are compatible with the AGT: precisely when $B(S_D) \cap \mathrm{AGT}$ is nonempty and transverse.

4. Geometric Structures and Phase Space Analysis

The authors develop a geometric framework in which the state space $\Gamma$ is partitioned as follows:

• The growth region is the subset $D(X) < d$ where the system evolves continuously according to the ODE flow.
• Upon reaching the division surface $S_D$ ($D(X) = d$), the state is reset to the birth surface $S_B = B(S_D)$.
• The AGT is a one-dimensional invariant manifold toward which all growth trajectories relax.

The full system trajectory alternates between deterministic growth along the AGT and discrete jumps dictated by $B$. If the reset preserves the AGT structure, a closed growth-division cycle forms: the GDSS.

This approach generalizes to both linear (e.g., Hinshelwood cycle) and nonlinear (coarse-grained protocell or bacterial cell) autocatalytic models. Visualizations in the $(X, Y)$ plane with AGT and division/birth surfaces clarify the necessary conditions for robust homeostasis.

5. Broader Implications and Applications

This geometric and dynamical formalism for growing-dividing autocatalytic systems has several implications:

• Synthetic biology and origin-of-life research: The analysis informs protocols for engineering robustly self-replicating protocells or minimal cell-like systems, guiding the design of division algorithms (e.g., which quantity to monitor for division, which state variables to reset upon division).
• Cell size control in bacteria: The results rationalize biologically observed strategies, such as division by absolute abundance or volume (nonintensive triggers), instead of concentration-based cues, to maintain homeostasis.
• Experimental design: Variations in the division control mechanism and resetting rule lead to experimentally testable predictions—such as transitions from exponential to nonexponential growth, onset of instability, or cycles with increasing cell-to-cell variability.
• Analysis of homeostasis loss: The geometric criteria explain transitions to nonhomeostatic regimes, such as cells that fail to divide, become progressively larger or smaller, or undergo catastrophic collapse, as observed empirically in certain mutants or stressed bacterial populations.

The theory further opens avenues for the analysis of stochasticity in growing-dividing systems, hybrid deterministic-stochastic descriptions, and the extension to nonhomogeneous or spatially structured chemical systems.

6. Prospects for Stabilization and Control

The theoretical framework identifies which division processes can stabilize or destabilize growth, offering explicit strategies for engineering robust self-reproducing systems:

• Robust stabilization: Choose nonintensive division variables (e.g., extensive quantities like molecule number or volume) with standard halving maps for error correction.
• Controlled destabilization or functional switching: Employ intensive division variables with nonstandard resetting (e.g., forced reinitialization of master regulators) to deliberately alter growth patterns.
• The impact of division policies extends to the possible evolutionary selection of cell cycle control mechanisms in both simple protocells and complex bacteria.

7. Summary Table: Outcomes of Growth-Division Compatibility

Division Variable $D(X)$	Birth Map $B$	Emergent Behavior
Nonintensive (e.g., total volume)	Standard (halving)	Exponential balanced growth; robust self-reproduction; steady GDSS
Intensive (e.g., concentration)	Nonstandard (reset)	Nonexponential periodic growth; possible functional cycles
Mismatched (e.g., intensive + standard map)	—	Homeostasis lost; growth halts or explodes (system death)

Compatibility between autocatalytic chemical dynamics, choice of division control variable, and division reset map is thus both necessary and sufficient for balanced self-reproduction in growing-dividing systems (Pandey et al., 7 Oct 2025).

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In conclusion, the geometry of growing-dividing autocatalytic dynamical systems reveals that self-reproduction mandates precise compatibility between internal autocatalytic growth mechanisms and explicit division strategies. The theoretical structure provided enables both a deep understanding of homeostatic growth-division cycles and the rational engineering of synthetic self-replicating cellular systems.