# Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems

## Abstract

**:**

## 1. Introduction

**in**duplicating real artificial protocells through a specific oil-in-water droplet system with replicating information templates [31]. This result is certainly remarkable, but our approach does exclude the role of any information/replication dynamics. In doing so, we explore how far can we go by just taking into account general stability properties and energy imbalances to explain and characterize the duplication process. The work presented here runs in parallel to an interesting complementary approach taken in [32], where the kinetics involved in the duplication events of synthetic systems was studied in detail.

## 2. The System

#### 2.1. Gibbs Free Energy Landscape

#### 2.2. Helmholtz Free Energy

#### 2.3. Detailed Balance Condition in Duplication

#### 2.4. The Driving Protocol

#### 2.5. An Example: Ternary Emulsions

## 3. Duplication Thresholds

#### 3.1. Transition Probabilities between Macrostates

#### 3.2. Quasi-Static Approach

#### 3.3. Non Equilibrium Approach

#### 3.3.1. Free Energy Structure

**First relation**. Observe that we can decouple the general term ${\langle \delta G(\tau -\delta ,\tau )\rangle}_{\Sigma [\tau -\delta ,\tau ]}$ as follows: Let ${p}_{|{\tilde{\sigma}}^{t}}$ be the probability distribution of the actual macrostate ${\tilde{\sigma}}^{t}$, and ${p}_{|{\tilde{\sigma}}_{*}^{t}}$ be the equilibrium distribution associated to the equilibrium macrostate ${\tilde{\sigma}}^{t}$, under the conditions imposed by the protocol at time t. That is, the probability distribution that would correspond to ${\tilde{\sigma}}^{t}$ if it where in equilibrium, ${\tilde{\sigma}}^{t}={\tilde{\sigma}}_{*}^{t}$. In other words, we have an equilibrium distribution $p\left({\sigma}^{t}\right|{\tilde{\sigma}}_{*}^{t})\sim {e}^{-\beta G\left({\sigma}^{t}\right)}$, sharing the support (let p be a probability distribution defined over the set ${X}^{\prime}$ and let $X\subseteq {X}^{\prime}$ such that $X=\{{x}_{i}\in {X}^{\prime}:p\left({x}_{i}\right)>0\}$. X is the support set of the probability distribution p. In words, the support set is the set of elements whose probability is larger than 0) set with the actual, possibly non-equilibrium, distribution ${p}_{|{\tilde{\sigma}}^{t}}$. After rearrangements one finds that—see Appendix A for details:

**Second relation**. Recognizing that Equation (9) implies that:

#### 3.3.2. Non-Equilibrium Duplication Thresholds and Work Relations

#### 3.4. The Perpetuation of the Duplication Process

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation Details

## References

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**Figure 1.**Schematic characterization of the role of the protocol $\psi \left(t\right)$. (

**a**) At time $t=0$ the system is in contact to a thermal reservoir at inverse temperature $\beta $, and in an equilibrium state containing 1 aggregate. (

**b**,

**c**) The protocol starts by increasing the number of lipids and precursors and providing energy that may trigger chemical reactions. The action of the protocol is depicted by the red arrow. This process may change the energy landscape provided by Equation (4) and thereby destabilize the structure of the aggregates, eventually creating more and more frustration in the surface. (

**d**) At time $\tau $ the energy gradients favour the duplication and the protocol stops. (

**e**) The system relaxes towards an equilibrium state containing 2 aggregates.

**Figure 2.**Trajectories between macrostates. At time $\tau -\delta $ the macrostate containing 1 aggregate receives the action of the protocol and transits to a macrostate containing 2 aggregates. This transition can performed through any of the trajectories connecting the states that realize one or the other macrostate. The ensemble of trajectories connecting these two macrostates is called $\Sigma [\tau -\delta ,\tau ]$. Forward trajectories are depicted with solid lines. Dashed lines correspond to time reversal trajectories, i.e., trajectories obtained through the protocol running under time reversal.

**Figure 3.**Irreversible action of the Protocol $\psi $. (

**a**) At $t=0$ we have a macrostate ${\tilde{\sigma}}_{n}^{0}$ in equilibrium and the protocol induces a change in the boundary conditions that destabilizes the system eventually making it to jump to a non equilibrium state, producing an amount of entropy $\beta {Q}_{\psi}\left(t\right)$. Then the system relaxes—maybe not completely—until the next action of the protocol until there is a stable division event and the system relaxes completely. (

**b**) Details of the transition, with the thermodynamic quantities involved. The jump experienced by the system from its previous state is $D\left({p}_{|{\tilde{\sigma}}^{t}}\right|\left|{p}_{|{\tilde{\sigma}}_{*}^{t}}\right)$, and $\Delta \mathcal{F}$ is the energy gradient that leads to the new macrostate. The entropy produced through this, possibly partial, relaxation process is $D\left({p}_{\to}\right|\left|{p}_{\leftarrow}\right)$—see text.

**Figure 4.**Schematic picture of the conditions for the duplication process to be sustained in time. Duplication events are indicated by dashed circles. A system with n aggregates—gray line—in equilibrium receives the action of the protocol changing the energy landscape. Its Helmoltz free energy increases until a point in which the Helmholtz free energy of a macrostate containing $n+1$ aggregates—orange line—is lower than the one for the n aggregates, and a duplication event occurs. If we switch on the protocol again, the system increases its Helmholtz free energy until a point in which, eventually, the Helmholtz free energy differences trigger again a duplication event—blue line. If the protocol is able to destabilize the system from $n\left(t\right)$ to $n\left(t\right)+1$ aggregates for any t, the duplication process will continue unboundedly in time. In this figure we described a quasi-static approach, which makes use of equilibrium Helmholtz free energies for the sake of clarity. The non-equilibrium case is thoroughly discussed in the text.

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Corominas-Murtra, B.
Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems. *Life* **2019**, *9*, 9.
https://doi.org/10.3390/life9010009

**AMA Style**

Corominas-Murtra B.
Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems. *Life*. 2019; 9(1):9.
https://doi.org/10.3390/life9010009

**Chicago/Turabian Style**

Corominas-Murtra, Bernat.
2019. "Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems" *Life* 9, no. 1: 9.
https://doi.org/10.3390/life9010009