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The stochastic evolution of a protocell. The Gillespie algorithm in a dynamically varying volume
Timoteo Carletti, Alessandro Filisetti
TL;DR
The paper addresses how to simulate protocell reactions when the reaction volume changes with molecule production, a setting inadequately handled by standard Gillespie modeling. It introduces a stochastic integration algorithm with self-consistent volume variation, compares protocell models with deterministic counterparts, and emphasizes fluctuation-dependent dynamics while noting important computational and biological boundary cases.
Problem
Existing protocell models often use deterministic concentrations, although small molecule numbers and thermal noise require stochastic treatment; varying-volume Gillespie methods have rarely modeled volume growth as a consequence of membrane-molecule production.
Method
The paper extends Gillespie simulation by making reaction propensities depend on a protocell volume that changes with the amount of contained molecules.
Results
The paper presents several stochastic protocell models, compares them with analogous deterministic ODE models, and finds richer dynamics with fluctuations that depend on the initial molecule amount.
Takeaways & Limitations
Stochastic protocell dynamics merit study because reducing molecule numbers increases fluctuations and can make deterministic descriptions incomplete.
Takeaways & Limitations
The time-dependent propensities make next-reaction calculations less straightforward and require suitable integral approximations; a missing next reaction time can represent protocell death by dilution.
Abstract
from arXiv · showhide
In the present paper we propose an improvement of the Gillespie algorithm allowing us to study the time evolution of an ensemble of chemical reactions occurring in a varying volume, whose growth is directly related to the amount of some specific molecules, belonging to the reactions set. This allows us to study the stochastic evolution of a protocell, whose volume increases because of the production of container molecules. Several protocells models are considered and compared with the deterministic models.
1. Introduction
Protocell research seeks abstract models of primitive life that capture growth, duplication, limited genetics, and stochastic molecular behavior. This paper adapts Gillespie simulation to protocells whose reaction volume changes with growth.
- Motivation: Primitive protocells were likely simpler than modern cells, with rudimentary embodiment, metabolism, and genetics.
- Motivation: Artificial protocells capable of growth, duplication, and limited genetics have not yet been reproduced in the laboratory.The paper therefore motivates high-level reference models that capture essential protocell behaviors without adding unnecessary complexity.
- Stochastic modeling: Deterministic concentration-based models are limited because cellular molecule numbers are small integers and thermal noise makes evolution stochastic.
- Stochastic modeling: The Gillespie algorithm numerically determines the most probable reaction and its occurrence time from the system’s current state.
- Contribution: The paper adapts Gillespie simulation for protocell reactions in a varying volume whose growth follows the production of membrane lipids.
2. Surface Reaction Models
The Surface Reaction Model represents a protocell with container and genetic-memory molecules whose surface reactions drive membrane growth and division. Its deterministic dynamics provide the reference models for the paper’s stochastic simulations.
- Model assumptions: The Surface Reaction Model contains container molecules C and Genetic Memory Molecules X that synthesize the membrane and replicators.
- Model assumptions: Its reactions occur on the protocell surface, where produced container molecules enlarge the membrane until physical instability causes division.
- Deterministic formulation: Container and genetic-memory molecule numbers follow deterministic equations parameterized by reaction constants, molecular amounts, geometry, and lipid density.
- Division cycle: The protocell divides when container molecules double from C(t0) = C0 to C(t0 + ∆T1) = 2C0.
- Division cycle: Each offspring receives half of the mother’s Genetic Memory Molecules, and suitable conditions can produce synchronized growth and information production.
3. The method
The method extends Gillespie simulation to reaction propensities that vary with a protocell’s dynamically growing volume. Volume is linked to container-molecule count, so reactions and growth evolve self-consistently.
- System representation: The stochastic model treats N Genetic Memory Molecules inside a well-stirred vesicle undergoing m reaction channels within volume V(t).
- Dynamic volume: Protocell volume increases with container-molecule count according to V = C/ρ, where C is membrane-molecule number and ρ is lipid density.
- Reaction propensities: For each reaction, propensity equals the number of possible molecular combinations multiplied by the reaction rate: aµ(Y) = hµ(Y)cµ.
- Reaction propensities: Binary reaction rates scale as cµ = kµ/V, while reactions involving n species scale as cµ = kµ/V^(n−1); single-molecule decays are volume-independent.
- Stochastic simulation: Because propensities depend on time-varying volume, the next-reaction calculation requires an approximation for an otherwise nontrivial integral.
- Stochastic simulation: The algorithm samples the next reaction channel and time from a cumulative distribution, then updates the state by the reaction’s stoichiometric vector.
4. The stochastic simulation algorithm in a growing volume
The paper adapts Gillespie simulation to protocells whose reaction propensities vary with a dynamically growing volume. It derives approximations for the next reaction time, identifies when that time does not exist, and determines the next reaction channel using a second random draw.
- Algorithm: The algorithm samples the next reaction time and channel from the time-dependent probability distribution, then updates the molecular state by the reaction stoichiometry.It uses the cumulative distribution and inversion method while accounting for time-dependent propensities.
- Adiabatic assumption: Under the fast-reaction assumption, the volume is treated as constant during the interval, yielding an explicit approximation for the next reaction time.This assumes chemical reactions occur much faster than container-molecule production and volume growth.
- Adiabatic assumption: As the volume increases, the time-dependent propensity contribution can decrease because AQ ∼ 1/V.The resulting expression has the standard Gillespie form but includes the time dependence induced by volume growth.
- Next-order correction: A higher-order estimate approximates the integrated propensity for comparable chemical and container-growth time scales and computes the next reaction time with corrections of order τ^3.The derivative of the propensity is obtained by expressing volume through the container-molecule amount and approximating its short-time growth deterministically.
- Next-order correction: If volume growth is sufficiently fast, the next reaction time may not exist because the defining equation has no real solution; in a protocell, this can correspond to death by dilution.Geometrically, the horizontal level fails to intersect the concave parabola defining τGill.
- Next reaction channel: When a next reaction time exists, the channel is selected by the classical Gillespie rule; if every reaction involves p chemicals, this channel choice is independent of volume.The volume factorizes out of the channel-selection expression when all reactions have the same molecular order.
5. Some applications
Applications compare the stochastic protocell dynamics with deterministic models across one- and two-GMM systems, showing that fluctuations grow as molecule numbers decrease and can alter long-term outcomes.
- Model applications: The applications compare stochastic and deterministic dynamics for protocells containing one or two Genetic Memory Molecules.The one-GMM model activates two chemical channels, while the two-GMM model considers two linear non-interacting GMMs.
- One single Genetic Memory Molecule: Large molecule populations produce small stochastic fluctuations whose averages remain close to deterministic dynamics.This behavior is reported for the time evolution of GMM and container molecules and for division-cycle observables.
- One single Genetic Memory Molecule: 20 times fewer initial molecules produce larger stochastic fluctuations than the corresponding larger-population model.The comparison concerns both molecular time evolution and GMM amounts and division times at cycle beginnings.
- Fluctuation dependence: The standard deviation of division time decreases with initial molecule number and follows a power law with exponent −0.54 ± 0.03.The deterministic division time remains constant when kinetic constants are fixed, whereas stochastic fluctuations increase as initial X0 and C0 decrease.
- Two non–interacting Genetic Memory Molecules: In the two-GMM model, fluctuation strength follows a power law with exponent −0.51 ± 0.05, while deterministic dynamics can still approximate large-population behavior.Division times nevertheless exhibit large fluctuations around the deterministic value, and reducing molecule numbers dramatically increases stochastic variability.
- Two non–interacting Genetic Memory Molecules: With few initial molecules, stochastic fluctuations can break symmetry between identical GMMs, causing one replicator to dilute and acquire a different fate.The deterministic model would keep identical replicators synchronized, whereas stochastic fluctuations can eventually remove one species.
6. Conclusion
The paper introduces a Gillespie-based stochastic algorithm for protocells whose volume changes with container-molecule production, and applies it to compare stochastic and deterministic models. The study highlights fluctuation-dependent dynamics and identifies richer behavior for future investigation.
- The algorithm explicitly relates volume variation to the amount of contained molecules, allowing volume to evolve self-consistently.
- The method models protocells whose volume increases through production of container molecules during chemical reactions.
- Several stochastic protocell models are compared with analogous deterministic models solved using ordinary differential equations.
- The stochastic dynamics are richer than the deterministic dynamics, with fluctuations depending on the initial amount of molecules.
- Future work includes cross catalysis, nonlinear interactions, and time-periodic patterns generated by fluctuations.