# Emergent Chemical Behavior in Variable-Volume Protocells

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reactor Models

**Figure 1.**Chemical Reactor Models. (

**a**) A chemical reaction held far-from-equilibrium by use of two constant concentration reservoirs (reservoir conditions). (

**b**) Continuous-flow stirred tank reactor (CSTR) model with non-limiting outflow, constant solvent volume and constant reaction temperature. (

**c**) Unilamellar lipid vesicle, with variable internal solvent volume determined by osmotic water flow equalizing the total solute concentration difference across the semi-permeable membrane. (

**d**) Lipid vesicle morphology space, with vesicle viability space drawn as the sub-region of all possible morphologies (grayscale region). Axes express vesicle volume Ω and surface area S

_{µ}as the nm diameter of a sphere having volume Ω or surface area S

_{µ}, respectively. Spherical vesicles (Φ = 1) are located when Ω (in nm)= S

_{µ}(in nm). Circles represent vesicle cross-sections, drawn to scale (8.5% of the axes’ scale). Blue crosses indicate how the morphology of a spherical d = 800 nm giant unilamellar vesicle (GUV) changes, when (i) gaining surface or (ii) losing volume. Both conditions bring the vesicle toward more filamentous or prolate states. Prolate states are depicted as strings of smaller spherical vesicles, only to give an idea of the volume and surface ratio at the {Ω, S

_{µ}} point; the meaning is not that the vesicle has necessarily divided into identical daughters at this stage. Red crosses indicate that the spherical d = 800 nm vesicle will burst by (iii) losing too much surface or, conversely, (iv) gaining too much volume.

_{1}, · · · , s

_{N}} is the vector of all species concentrations inside the reactor, function ${{r}}_{i}(\overrightarrow{s})$ contains all reaction MAK terms producing or consuming the i-th species, ${s}_{i}^{f}$ is the constant concentration of species i in the reactor feed pipe and θ is the mean residence time of the CSTR. The mean residence time is defined as the ratio of reactor solvent volume (constant) divided by the flow rate of solvent into the reactor θ = Ω

_{CSTR}/Q

_{f}and represents that solutes are “washed out” faster from a reactor having a higher inflow rate or a smaller volume.

_{A}$\mathcal{V}$ (Avogadro’s constant multiplied by the liter volume of the vesicle), B

_{T}> 0 is the number of non-reacting impermeable buffer molecules trapped inside the vesicle [33], C

_{ε}is the total external concentration, that is the sum of all the external solute concentrations ${C}_{\epsilon}={b}_{\epsilon}+{\displaystyle {\sum}_{j=1}^{N}}{s}_{j}^{\epsilon}$, and ${\sum}_{j=1}^{N}}{s}_{j$ is the total internal concentration of chemically-reacting solute species inside the vesicle. The assumption of instantaneous vesicle volume change is based on the observation that water permeates a fatty acid membrane on a time scale that is orders of magnitude faster than the passive diffusion of solutes [34] and simplifies the model, because it does not require treating Ω as an extra state variable.

_{ε}being very high with respect to the possible range of external and internal solute concentrations (b

_{ε}$\gg {s}_{j}^{\epsilon}$ and ${b}_{\epsilon}\gg {s}_{j}$). Under such conditions, there is little net water movement across the membrane, and the vesicle volume is approximately constant at ${\Omega}^{\varnothing}\approx {B}_{T}/{b}_{\epsilon}$. Assuming well-stirred kinetics within the vesicle water pool [35], the dynamic behavior of the concentration of the i-th solute species inside the vesicle would be described by:

_{i}is the diffusion constant for species s

_{i}calculated as:

_{ribose}= 2.65 × 10

^{8}dm

^{2}s

^{−}

^{1}mol

^{−}

^{1}[11], and the vesicle bilayer is considered to have constant thickness λ = 4 × 10

^{−}

^{8}dm. Fick’s law provides the basis for the membrane diffusion term.

_{µ}is now variable and given by Equation (3). The above expression is formed by substituting Equation (2) into Equation (4) and then adding the dilution term [14] to properly account for changing volume in a concentration ODE. Considering that vesicle volume is a function of internal solute concentrations Equation (2), the dilution term is given by:

_{j }/dt = 0, j = 1, . . . , N. In this state, the vesicle volume and surface are stationary.

#### 2.1. Solution of Vesicle Reactor Model: Graphical Method

_{µ}term in Equation (3) still makes the remaining equations difficult to solve for zeros, even numerically.

**Step 1:**The fixed points for a variable volume vesicle reactor with constant surface ${S}_{\mu}^{\varnothing}$ are solved. With constant surface, the concentration derivatives at fixed points simplify to a set of multivariate polynomials in the species concentrations:

**Step 2:**The solute concentrations at each fixed point are converted to the corresponding fixed point vesicle volume Ω

^{∗}, using Equation (2). Then, all of the fixed point volumes are plotted on a two-dimensional graph, which we call the vesicle morphology space, where the x-axis represents vesicle volume and the y-axis represents vesicle surface (Figure 1d). The fixed point volumes are plotted along horizontal line y = ${S}_{\mu}^{\varnothing}$in this space.

**Step 3:**Vesicle surface ${S}_{\mu}^{\varnothing}$ is incremented, and the process repeated from Step 1.

_{µ}.

_{µ}} that allow the encapsulated reaction network to reach steady state, for a given model parameter set. The fixed point solutions to the full reactor model Equation (7) are precisely where the branches of the bifurcation curve intersect the line, giving spherical vesicle morphologies (Φ = 1 line; see the next subsection). Local stabilities of fixed points can also be calculated at Step 1. However, at best, these are “quasi-stabilities” or predictors of stability in the full model, because the dilution term and the fact that surface area is actually variable are both not taken into account.

#### 2.2. Vesicle Viability Space within Vesicle Morphology Space

## 3. Results

#### 3.1. Case Study 1: Compartment as a Bottleneck to the Internal Reaction System

^{−4}, 6.32×10

^{−4},5.25×10

^{4},2.85×10

^{−3},9.15×10

^{−2},7.15×10

^{−3}}, where concentrations are in molar, first order reaction rate constants in s

^{−1}and third order in M

^{−2}s

^{−1}. The following molar concentrations represent the low stable state, the unstable state and the high stable state of species Y respectively: $\{{y}_{1}^{\ast},{y}_{u}^{\ast},{y}_{2}^{\ast}\}$ = {5.03×10

^{−}

^{5}, 4.01×10

^{−}

^{3},7.86×10

^{−}

^{3}}.

_{ε}= x and z

_{ε}= z) and using the same reaction rate parameters. The remaining parameters were assigned at random in 5000 different combinations, in ranges given in the Methods section. The vesicle surface area ${S}_{\mu}^{\varnothing}$ was fixed at that of a 400-nm diameter sphere.

_{1}, k

_{2}, k

_{3}, k

_{4}}= {1.17×10

^{−3}, any, 5.86×10

^{2}, 9.26×10

^{2}, 5.75×10

^{2}, 9.98×10

^{−2}} with units the same as before and second order reaction rates in M

^{−1}s

^{−1}. The following three solution pairs (in molar concentrations) for fixed points of the intermediate species are obtained: $\{{y}_{1}^{\ast},{z}_{1}^{\ast}\}$ = {0, 0}, $\{{y}_{u}^{\ast},{z}_{u}^{\ast}\}$ = {1.20×10

^{−4}, 1.94×10

^{−5}}, $\{{y}_{2}^{\ast},{z}_{2}^{\ast}\}$ = {1.07×10

^{−}

^{3}, 1.55×10

^{−}

^{3}}.

#### 3.2. Case Study 2: Compartment as Enabling New Steady States for Single Reaction Sets

**Figure 2.**Main results: vesicle bistability in Case Studies 1–3. The reaction sets of (

**a**) Case Study 1, (

**b**) Case Study 2 and (

**c**) Case Study 3 can yield bistability in the vesicle model under suitable parameter conditions (given in Supplementary Material). Circles (

**middle**) depict spherical vesicle shapes (drawn to scale) at which steady states can occur, corresponding to where stable branches (solid lines) and unstable branches (dashed lines) on the bifurcation diagrams (

**right**) cross the the Φ = 1 line. Labels A–F on the bifurcation diagram (c) refer to non-spherical morphologies drawn in Figure S2 of the Supplementary Material.

**Figure 3.**Steady State 1 (SS1) and SS2 internal solute concentrations. Solute concentrations inside the vesicle reactor, at spherical steady states SS1 and SS2, for each reaction scheme reported in Figure 2. Comparing SS1 and SS2 for each scheme, it can be observed that there are quantifiable differences in solute concentrations between steady states, but these differences are fairly small. Owing to the osmotic water balance, the total concentration of solutes inside the vesicle (height of the stacked bars) is always constrained to be equal to the total external solute concentration of the environment, C

_{ε}. Thus, in the vesicle reactor model, the main feature distinguishing steady states is vesicle size (see Section 4.2 for details). Symbol b

^{∗}denotes the steady-state concentration of the B

_{T}buffer molecules trapped inside the vesicle. At SS2, b

^{∗}→ 0, due to large vesicle sizes, and the diffusing solutes constitute the majority of total internal concentration. The Supplementary Material supplies data supporting the figure.

_{µ}≈ 80 nm diameter) and another stable state at a much larger GUV vesicle size (Ω and S

_{µ}≈ 1200 nm diameter). To re-iterate, the stable steady state means that the vesicle sphere is providing the correct diffusion surface and inner volume for all of the solute concentrations in the reaction network to be stationary, and at the same time, the total concentration of solutes and buffer inside the vesicle is equal to the total external concentration; and so, there is no net movement of water across the bilayer membrane. The steady-state sizes are separated by an unstable saddle point at an intermediate vesicle size.

#### 3.3. Case Study 3: Compartment as Osmotically Coupling Two Chemically-Independent Reaction Sets

**Figure 4.**Switching dynamics: bistability in two unimolecular reactions. Encapsulating two unimolecular reactions X → Y and P → Q in the variable-volume vesicle reactor model gives a bistable system under the correct parameter regime (Figure 2c(i)). Below, switching dynamics between steady states SS1 and SS2 are demonstrated by injecting molecules into the reactor by a simulated syringe. Following four different two-minute injections of molecules, changes in (

**a**) spherical vesicle diameter, (

**b**) vesicle internal species numbers and (

**c**) vesicle internal species concentrations are monitored. Injection

**I1**releases both X and Q into the vesicle at a linear rate of 1000 molecules per second. This perturbation is not sufficiently strong to switch the reactor into SS2, but injection

**I2**, releasing X and Q at 3500 molecules per second, is able to prompt the transition. Once in the larger vesicle SS2 state, the switch back to SS1 is achieved by injecting a new species U into the reactor. This species undergoes reaction $U+Q\stackrel{k}{\to}W$, which depletes Q inside the vesicle by quickly transforming it into waste W (k = 60.0) that rapidly diffuses out of the compartment (${D}_{W}^{\times}\text{}=\text{}100.0,\text{}{D}_{U}^{\times}\text{}=\text{}1.0$). Injection

**I3**releases U into the vesicle at a rate of 8000 molecules per second, but cannot initiate the switch back to SS1. Injection

**I4**successfully completes the transition, releasing U at a rate of 10, 000 molecules per second. Time is divided into windows to accommodate different timescales (from minutes to days).

_{2}to the number of impermeable buffer molecules B

_{T}trapped inside the vesicle. Conversely, from the view point of the P → Q reaction (Figure 5a(iii)), species X and Y appear as inert, and they add B

_{1}extra buffer molecules to B

_{T }. Therefore, we have the situation that the total number of buffer molecules “experienced” by one reaction depends on the instantaneous species concentrations of the other reaction.

_{1}= f

_{R}

_{1}(B

_{2}). This function returns the total effective number of molecules that reaction X → Y (Reaction 1) has at steady state, B

_{1}, given that B

_{T}+ B

_{2}buffer molecules exist inside the vesicle. The green line (plotted normally: x-axis independent) is function B

_{2}= f

_{R}

_{2}(B

_{1}), which returns the total effective number of molecules that reaction P → Q (Reaction 2) has at steady state, B

_{2}, given that there are B

_{T}+ B

_{1}buffer molecules inside the vesicle. The whole two-reaction system has a steady state only when the following cyclic condition is fulfilled: Reaction 1, “seeing” B

_{2}extra buffer molecules inside the vesicle, has a steady state equivalent to B

_{1}extra buffer molecules, and Reaction 2, “seeing” B

_{1}extra buffer molecules inside the vesicle, has a steady state equivalent to B

_{2}extra buffer molecules. The cyclic condition is fulfilled at three points, marked by circles in Figure 5b.

**Figure 5.**Graphical intuition into emergent bistability through osmotic coupling. （a）Emergent bistability in the vesicle reactor model: (i) two chemically-independent unimolecular reactions can be understood by taking a “reactions-eye view” from the perspective of each reaction; (ii) from the perspective of the X → Y reaction (Reaction 1), all molecules associated with the P → Q reaction simply act as extra inert buffer (B

_{2}) in addition to the trapped impermeable buffer molecules B

_{T}inside the compartment; (iii) conversely, from the perspective of the P → Q reaction (Reaction 2), all molecules associated with the P → Q reaction act as extra inert buffer (B

_{1}). (

**b**) Graph showing how the total steady-state particle number of each reaction responds to the extra number of buffer molecules that the other reaction is providing, where Reaction 1 has the y-axis as independent and Reaction 2 has the x-axis as independent. The three cross points represent fulfillment of the cyclic condition referred to in the text. The dotted line shows the relation B1 = B2. Two chemically-independent reaction sets with identical stoichiometry and identical kinetic constants would give curves that are reflections in this line. (

**c**) When the chemical transformation between P and Q is removed, the latter solutes simply diffuse across the membrane until their respective concentration gradients are equalized. A unimolecular reaction sharing the vesicle compartment with such inert diffusing species cannot be bistable (see the text).

_{R}

_{1}and f

_{R}

_{2}, i.e., from the non-linear response that the steady state of a reaction has to a modification in the total number of buffer molecules inside the vesicle [46]. This non-linearity allows multiple cross points of the red and green curves in Figure 5b. Indeed, writing function f

_{R}

_{1}in explicit form (derivation and constants K in the Supplementary Material):

_{1}is a highly non-linear function of B

_{2}. Likewise, B

_{2}will be a similar non-linear function of B

_{1}.

_{ε}− p

_{ε}− q

_{ε}. Nevertheless, even if inert diffusing solutes cannot expand steady-state possibilities already present, they do change dynamic trajectories toward existing steady states.

## 4. Discussion

#### 4.1. Significance of the Results Obtained

#### 4.2. Notions of Bistability

_{ε}(fixed). It follows, then, that all reactions inside the vesicle are running at around the same speed (chemical transformations per unit time) in each steady state, which is in contrast to bistable chemical systems in bulk, where, usually, some/all reactions in the “high” state are running at a much accelerated rate. In any case, the absolute number of solute molecules inside the vesicle in SS1 (small vesicle size) is several orders of magnitude less than the number of molecules at SS2 (large vesicle size).

#### 4.3. Comments on Graphical Solution Method

_{µ}} pairs in vesicle morphology space) supporting steady-state concentrations of the internal reaction network. In this work, the kinetics of the vesicle surface area were modeled, keeping the restriction of spherical shape: all {Ω, S

_{µ}} pairs on the Φ = 1 line. However, different membrane kinetics schemes may allow wider movement within vesicle viability space, for example if the vesicle surface area is an extra state variable not directly determined by the volume. If this is the case, information on non-spherical {Ω, S

_{µ}}pairs giving a steady state is useful. A visual idea of non-spherical vesicle morphologies supporting a steady state for encapsulated reaction scheme X → Y, P → Q is given in Figure S2 of the Supplementary Material, which represents Points A–F on the bifurcation diagram of Figure 2c(i) as prolate spheroids.

#### 4.4. Limitations of the Current Approach

_{µ}). Besides, we assumed a well-mixed homogeneous medium inside and outside the vesicle reactor, and we followed a deterministic treatment.

#### 4.5. Future Challenges

## 5. Methods

#### 5.1. Vesicle Model Parameter Space: Search Methodology

_{(}

_{→}

_{)}+ 2R

_{（}

_{⇌}

_{）}+ 2) dimensions [55], where N is the number of membrane permeable solute species that are involved in a total of R

_{(}

_{→}

_{)}irreversible and R

_{（}

_{⇌}

_{）}reversible reactions inside the vesicle.

^{∗}< 2000−nm diameter sphere. It should be noted that fixed points could have existed outside this maximum volume limit, and they would have been screened out.

**d**

_{i}($\overrightarrow{s}$) is the diffusion function for the i-th species across the vesicle membrane, and K

_{i }is the constant:

_{µ}

_{,}D

_{i }and B

_{T}are changed in such a way that each Ki remains the same as previously, then we are solving the same set of multivariate polynomials as before. Hence, the fixed point concentration solutions remain unchanged. However, now, depending on the parameter changes made, the fixed points will happen at shifted {Ω, S

_{µ}} points in morphology space.

_{i }as:

- 1. If each diffusion constant D
_{i }is multiplied by a factor of a, then the fixed point volumes stay constant (because B_{T}remains unchanged and the steady-state concentrations are the same as before), but the fixed point surface areas have to change by a factor of 1/a to compensate the scaling of the diffusion constants. The bifurcation curve is stretched vertically up (decreasing a) and down (increasing a) in vesicle morphology space. - If each diffusion constant Di and the number of trapped buffer molecules BT are both multiplied by a common factor c, then the fixed point surface areas stay constant (because ratio $\frac{{D}_{i}}{{B}_{T}}$ remains the same), but the volume changes by factor c, because B
_{T}has been scaled. The bifurcation curve is translated and stretched left (decreasing c) and right (increasing c) in vesicle morphology space. - If just B
_{T}is multiplied by a factor b, then the surface area and volume of all fixed points are multiplied by b. The bifurcation curve is translated and stretched diagonally in vesicle morphology space.

#### 5.2. Rescaling Vesicle Model Parameters for Different Concentration Ranges

_{µ}}positions in morphology space. The procedure outlined below leaves the time scale of the dynamics unaffected.

_{1}) remain unchanged; second order reaction rates (e.g., k

_{2}) require multiplying by n and third order reaction rates (e.g., k

_{3}) multiplying by n

^{2}. Dividing the number of trapped internal buffer particles B

_{T}by n ensures that the scaled concentrations create the same steady-state aqueous volume of the system Ω as before, i.e.:

_{X }.

#### 5.3. Vesicle Parameter Set Ranges For Schlögl and Wilhelm Models: Case Study 1

_{T}∈ (2, 2000).

## Acknowledgments

## Author Contributions

## Supplementary Materials

## Conflicts of Interest

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**MDPI and ACS Style**

Shirt-Ediss, B.; Solé, R.V.; Ruiz-Mirazo, K. Emergent Chemical Behavior in Variable-Volume Protocells. *Life* **2015**, *5*, 181-211.
https://doi.org/10.3390/life5010181

**AMA Style**

Shirt-Ediss B, Solé RV, Ruiz-Mirazo K. Emergent Chemical Behavior in Variable-Volume Protocells. *Life*. 2015; 5(1):181-211.
https://doi.org/10.3390/life5010181

**Chicago/Turabian Style**

Shirt-Ediss, Ben, Ricard V. Solé, and Kepa Ruiz-Mirazo. 2015. "Emergent Chemical Behavior in Variable-Volume Protocells" *Life* 5, no. 1: 181-211.
https://doi.org/10.3390/life5010181