aa r X i v : . [ phy s i c s . b i o - ph ] J a n January 25, 2021
THERMALLY DRIVEN FISSION OF PROTOCELLS
ROMAIN ATTAL
Abstract.
We propose a simple mechanism for the self-replication of protocells. Our main hypothesis is that theamphiphilic molecules composing the membrane bilayer are synthesised inside the protocell through globally exothermicchemical reactions. The slow increase of the inner temperature forces the hottest molecules to move from the innerleaflet to the outer leaflet of the bilayer. This asymmetric translocation process makes the outer leaflet grow fasterthan the inner leaflet. This differential growth increases the mean curvature and amplifies any local shrinking of theprotocell until it splits in two.
Contents Protocells and metabolism
Hypotheses of our model
Flows, forces and energy dissipation
Membrane geometry and growth equation
Thermal instability of cylindrical growth
Translocation between leaflets L mθ L mm L θθ Conclusions and perspectives
The mean curvature of the membrane
Solutions of the growth equation η = 1 and τ > τ + ( η ) or τ < τ − ( η ) 21B.2. Case 2 : η = 1 and τ ∈ { τ + ( η ) , τ − ( η ) } τ − ( η ) < τ < τ + ( η ) 22Appendix C. Smooth perturbation of cylindrical growth
Asymptotic expansion of F ( a ) 25References 271. Protocells and metabolism
The objects modeled in the present article are protocells, the putative ancestors of modern living cells [23, 34]. Inthe absence of fossils [38], we ignore their detailed properties. However, we can sketch a minimalist functional diagramof protocells (FIG. 1).
Key words and phrases. protocell, bilayer, translocation, self-replication, thermodynamical instability.
Food WasteMetabolismHeatBiomass
Figure 1.
Protocells initiate the fundamental process of life : Food → Biomass + Heat + Waste.
The protocell is a vesicle bounded by a bilayer made of amphiphilic molecules. Nutrient molecules (food) enter bymere diffusion, since they are consumed inside, where their concentration is lower than outside. Conversely, wastemolecules have a larger concentration inside and therefore diffuse passively to the outside. The metabolism is a networkof unknown chemical reactions taking place only inside the protocell. The net reaction is supposed to be exothermic,since living matter is hotter than abiotic matter (under the same external thermodynamical conditions).Let us compare this scheme to actual evolved cells. The growth of bacteria in a nutrient rich medium follows aspecies dependent periodic process [5, 12]. At regular time intervals, each cell splits to form two daughter cells. Thisrequires the synchronization of numerous biochemical and mechanical processes inside the cell, involving cytoskeletalstructures positioned at the locus of the future cut (septum). However, in the history of life, such complex structuresare a high-tech luxury and must have appeared much later than the ability to split. Protocells must have used asimple splitting mechanism to ensure their reproduction, before the appearance of genes, RNA, enzymes and all thecomplex organelles present today even in the most rudimentary forms of autonomous life [34].In this article, we present a simple model for the growth and self-replication of a protocell, following the laws ofirreversible thermodynamics near equilibrium. Our guide is the rate of entropy production, which is minimal in asteady state [32, 20]. A key point of our approach is that the heat produced by the metabolism of the protocell isapproximately proportional to its volume, whereas the heat flow that it can loose is proportional to the area of itsmembrane. In a rod-shaped cell (bacillus) growing linearly, these two quantities are approximately proportional so thateach increment of the membrane area should be sufficient, ideally, to evacuate the heat produced by the correspondingincrement of the cell volume. However, the irreversible physical and chemical processes produce heat more quicklythan the growing tubular membrane can dissipate to the outside. This increases slowly the inner temperature andenhances the fluctuations of the shape of the membrane, of the various concentrations and of the local electric field.In a growing spherical protocell, the maximal heat flow that the membrane can expell to the outside withoutoverheating the inside puts an upper limit to the radius of the protocell. Indeed, the formation of two small protocellsfrom a big one releases work [33], so that large protocells are mechanically unstable. However, neither [33] nor [8]provides a path to follow to realise this deformation.In our model, we start from a cylindrical shape to simplify the computations. As the inner temperature increases,the growth of the outer leaflet of the membrane becomes more probable than the growth of the inner leaflet. If arandom thermal fluctuation lowers slightly the radius of this cylinder, then its area increases more quickly than duringthe steady state cylindrical growth (FIG. 2).
HERMALLY DRIVEN FISSION OF PROTOCELLS 3 ↓↓↓ hotcold
Figure 2.
Splitting a cylindrical protocell.
This reduction of the radius induces a loss of convexity of the membrane. This favors the outflow of heat and the ratioarea/volume increases slightly, compared to a convex cylindrical shape.The plan of the article goes as follows. In Section II, in order to formulate these ideas mathematically, we state allthe physical hypotheses of our model of protocells. In Section III, we define the various flows of matter and energyand their associated thermodynamical forces. In the linear approximation, the rate of entropy production is the scalarproduct of these flows and forces and is minimal in a steady state [32]. In Section IV, we derive a differential equationfor the evolution of the area and the integral of the mean curvature of the membrane, starting from the advancementof the chemical reaction for the synthesis of the membrane molecules. This linear equation admits a solution growingexponentially. In Section V, we use variational calculus [10] to prove that the local reduction of the radius of the cellincreases its length and its area, if its volume is kept constant. This intuitive property implies that heat is more easilyreleased when the protocell is squeezed. In Section VI, we propose a molecular mechanism for the increase of themean curvature of the membrane associated to this squeezing. The position of each amphiphilic membrane moleculeis reduced to a single degree of freedom : the distance from the polar head to the middle of the hydrophobic slice.We use a double well effective potential to describe the trapping of these molecules in the membrane. Due to thetemperature difference between the inner and outer sides, the membrane molecules go from the inner leaflet to theouter leaflet more often than in the opposite direction. This asymmetry forces the membrane to curve and shrinkaround the middle of the protocell and initiates its splitting. Our main mathematical result (Proposition VI.I) statesthat a stability condition, L mθ < L mm L θθ , can not be satisfied at high temperature, because the squared crossedconductance, L mθ , increases more quickly than the product of the diffusion coefficients, L mm for membrane moleculesand L θθ for heat. Hence, the cylindrical growth process is unstable when the temperature difference is sufficiently high.We conclude in Section VII with a proposition of an experimental test for our model. The appendices contain thedetailed computations of our model. The mathematical notions involved are elementary (linear differential equationsand geometry of surfaces). 2. Hypotheses of our model
Let us state more precisely the hypotheses underlying our model :(1) Our protocells are made of a membrane of average thickness 2 ε , bounding a cytosol of finite volume V ( t ).(2) The cytosol contains unknown specific molecules (reactants, catalysers, chromophores, . . . ) which participateto a network of chemical reactions. We suppose that the concentrations are constant and uniform in thevolume V .(3) The protocell starts with a cylindrical shape closed by two hemispherical caps of fixed radius, R . The totallength, ℓ ( t ) + 2 R + 2 ε , increases with time due to the synthesis of membrane molecules (FIG. 3). ROMAIN ATTAL ℓ ( t ) + 2 R + 2 ε ε R − ε ) 2( R + ε ) ℓ ( t ) Figure 3.
Geometry of an idealised cylindrical protocell.
This may seem a rather drastic hypothesis, but the computations could be made for a generic, approximatelyspherical shape using an expansion in spherical harmonics. This would add to the model an unnecessarymathematical complexity that would hide the main physical phenomena. The use of cylindrical, rotationinvariant shapes allows us to reduce the problem to one dimension. Moreover, this is a best case scenario forthe release of heat in steady state, since the ratio volume/area can be held constant in a steady growth.(4) Due to the surface tension of the membrane, its mean curvature has an upper bound, H max = R . Indeed,due to the attractive forces between the polar heads of the membrane molecules, and due to their geometry,they can not form structures arbitrarily small [27].(5) Food (nutrients and water) enters the protocell by mere passive diffusion through the membrane. Waste andheat also diffuse passively but in the opposite direction. Protocells did not use specialized membrane moleculesfor an active transport through the membrane.(6) The membrane molecules are synthesised inside the protocell in an unknown network of chemical reactions.It might use some encapsulated catalyzers or chromophores trapped in the volume and catching part of theambient light [23], but we will make no hypothesis on the details of this network.(7) These metabolic reactions generate heat to be evacuated and increase slowly the internal temperature, T ,whereas the external temperature, T , remains fixed.(8) The characteristic time of the variations of T ( t ) is much larger than the characteristic times of chemicalreactions and diffusion processes across the membrane.(9) The cytosol is homogenous and contains no organelles, no cytoskeleton, no enzymes, no RNA/DNA. Justsimple chemical reactants uniformly distributed. (Rashevsky’s model [33] allows for a slight radial variationof concentrations due to the diffusion of food and waste through the membrane).(10) The membrane is a bilayer made of unspecified amphiphilic molecules. We presume that their hydrophobictails are long enough (10-12 carbon atoms) to form a stable bilayer, but not too bulky in order to allow flip-flop(or translocation) processes between the two leaflets. We do not include sterol molecules because they are theproduct of a long biochemical selection process [27], and a high-tech luxury for protocells.(11) The inner leaflet ( L ) is at temperature T whereas the outer leaflet ( L ) is at temperature T < T . Thistemperature drop allows the bilayer to undergo coupled transport phenomena (food and waste diffusion,including water leaks, heat diffusion, flip-flop, etc.).(12) The membrane may contain other molecules, in small concentrations, but we don’t need them to transportfood, waste or any molecule through the membrane.The validity of these hypotheses will depend on the agreement of their predictions with the results of futureexperiments made with real protocells.3. Flows, forces and energy dissipation
In any living system, some processes release energy whereas other processes consume energy. Globally, the systemtakes usable energy from the outside and rejects unusable energy, in the form of heat and waste, that can be usedby other living systems. In order to describe such a system, we must define the various flows of matter and energyand the forces causing these flows. Any gradient of concentration, pressure, temperature, etc. will cause a currentof particles, fluid, heat, etc. These processes are generally irreversible and dissipate energy to inaccessible degrees offreedom. This dissipation of a conserved quantity is measured by the entropy function, which increases as time passes.The study of irreversible thermodynamical processes near equilibrium [29, 30, 35, 20] is based on the rate of entropyproduction, represented by a bilinear function of flows (chemical reaction speed, thermal current, particle current,electric current, etc.) and forces (chemical affinity, temperature gradient, concentration gradient, electric tension,etc.). In a first approximation, flows and forces are related linearly, as in Ohm’s law :(1) electric current = conductivity × electric field HERMALLY DRIVEN FISSION OF PROTOCELLS 5 and the power dissipated is a quadratic function of the tension :(2) power dissipated = tension × current= conductance × tension . Similarly, in viscous fluids :(3) power dissipated = friction coefficient × velocity . We suppose that the protocell metabolism is in a steady state not too far from equilibrium, so that the various flows, J i , and the thermodynamic forces, X k , are linearly related :(4) J i = X k L ik X k and the entropy rate is a quadratic function of X :(5) σ := XJ = X ik X i L ik X k . The coefficients L ik are called phenomenological because their computation depends on the chosen model of microscopicdynamics (kinetic theory) and their numerical value has to be compared to a measurement in the real world to(in)validate this model and the linearity hypothesis. An important property of the phenomenological coefficientsis provided by Onsager’s relations [29, 30, 20, 35]. Under the hypotheses of microscopic reversibility and parity ofthe variables under time reversal (in particular, in the absence of magnetic coupling and vorticity), the matrix L issymmetric :(6) L ik = L ki . This important law has been checked experimentally for various systems near equilibrium and is satisfied quite accu-rately in many cases.3.1.
Main irreversible processes.
To each irreversible physical or chemical process are associated a flow of matteror energy and a thermodynamical force, just as an electric current and an electric tension correspond to each branchof an electric network. If we identify the main processes that take place during the growth of a protocell, we cancompute the global rate of dissipation of energy, or entropy creation. According to Prigogine’s Theorem [32, 11], thisrate reaches a minimum when the system is in a steady state.In order to compute this dissipation, we need to define the various compartiments containing energy. In the sequelof this article, the subscript 0 (resp. 1) will denote the variables outside (resp. inside) the protocell. The physical andchemical processes are grouped as follows : f → f : food molecules (nutrients + water) diffuse into the protocell through the membrane. f → m + c + w : food is transformed into membrane, cytosol and waste, inside the protocell. This is a globalprocess, a superposition of catabolism and anabolism. Taking into account the stoichiometric coefficients, wecan write more precisely :(7) X i ν f i f i −→ X j ν m j m j + X k ν c k c k + X l ν w l w l where f i denotes the food molecules of type i , m j the membrane molecules of type j , c k the cytosol moleculesof type k and w l the waste molecules of type l . If N α is the number of molecules of type α , the advancementof this reaction, ξ , is defined by :(8) d ξ := d N α ± ν α where the stoichiometric coefficients, ν α , are counted positively for the products and negatively for the reac-tants. Note that our definition of ξ involves N α instead of the volumic concentration, C α = N α / V , becausethe volume is not fixed. w → w : waste molecules diffuse out of the protocell through the membrane. m ⇆ m : molecules of the membrane bilayer go from one side to the other. In modern cells, this process iscatalysed by enzymes (flippase for 0 → → ∼ (cid:23) ↼ −− ⇁ ( ) because the ratio N m /N m ofthe numbers of membrane molecules on each side is related to the mean curvature of the bilayer, which is thegeometric parameter monitoring the splitting process. q → q : electric charges can be transfered from one side of the membrane to the other, by an ionic bound on thepolar head of the membrane molecules. This electric current builds up an electric tension, U , counteracted ROMAIN ATTAL by possible ionic leaks through the membrane. If we suppose that the membrane molecules are monovalentfatty acids, each one can carry a monocation (H + , Na + , K + , . . . ). This cotransport process could be theancestor of the modern sodium-potassium pump. Anions also can participate to this transmembrane electriccurrent, by leaking throuh water pores [14].3.2. Flows associated to each irreversible process.
The main processes of our model are described by thefollowing flows in the protocell (see FIG. 4) : J f : the flow of food entering the protocell through its membrane (molecules per unit time per unit area). J w : the flow of waste exiting the protocell through its membrane (molecules per unit time per unit area). J θ : the heat flow exiting the protocell by diffusion through its membrane (energy per unit time per unit area). J mab : the flow of membrane molecules from a to b (molecules per unit time per unit area). The possible valuesof a and b are : c : the cytosol ;1 : the inner leaflet of the membrane ( L ) ;0 : the outer leaflet of the membrane ( L ) ;The net flow of membrane molecules is usually unidirectional, C → L → L , hence J mc > J m := J m − J m > J r : the speed of the synthesis reaction inside the cytosol (molecules per unit time per unit volume). ξ beingthe advancement of the reaction f → m + c + w , defined above, then J r is the time derivative of ξ :(9) J r := d ξ d t .J q : some ions can be transported from one side to the other, bounded to the polar head of the membranemolecules. L L (cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(cid:23)(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( J m J q ions J f f = food f J r −−→ m + c + wJ mc w = waste J w J θ heat diffusion E = environment T C = cytosol T ( t ) Figure 4.
Main flows of energy and matter in our model.
We then have the following linear flow diagram for the synthesis and motion of membrane molecules :(10) E J f −−→ C J r −−→ C J mc −−−→ L J m −−−→ L . This picture is however slightly misleading. Indeed, the amphiphilic molecules being in a liquid phase, their positionsfluctuate in each leaflet (transversal diffusion) and they undergo perpendicular motions (protrusion) and translocationsfrom one leaflet to the other. The pictures obtained by molecular dynamics simulations [14, 15, 4, 1] give us a moreprecise representation of real world membranes.3.3.
Thermodynamical forces.
The thermodynamical forces associated to these processes are defined as follows : X θ : the thermal force is the difference of the inverse temperatures inside and outside the protocell :(11) X θ := 1 T − T > .X f i : the chemical force driving the food molecules, f i , is the difference of the ratios − µ f i /T outside and insidethe protocell :(12) X f i := µ f i T − µ f i T . The influx of food is guided by mere diffusion through the membrane (dedicated channel and intrinsic proteinsdid not exist yet in protocells). Since food is consumed inside the protocell, [ f i ] < [ f i ] . For a sphericalprotocell, the profile of the concentration of each molecule (as a function of the distance to the center) can be HERMALLY DRIVEN FISSION OF PROTOCELLS 7 computed by solving the diffusion equation [33]. An important result of this computation is the existence of adiscontinuity in the concentration of each molecule, f i , proportional to the radius, R , of the protocell, to therate of the reaction, q i (concentration/time), and inversely proportional to the permeability, h i (length/time),of the membrane for this molecule : [ f i ] − [ f i ] ∝ q i Rh i . X w j : the force driving the waste molecules to the outside of the protocell is the difference of chemical potentialsdivided by the temperature :(13) X w j := µ w j T − µ w j T . Note that X w j and X f i must have different signs for waste and food to go in opposite directions. X r : the chemical force driving the synthesis reactions (metabolism) is the chemical reaction affinity, A r , of theglobal process ( f → m + c + w ), divided by the inner temperature of the protocell :(14) X r := A r T . This affinity is a linear combination of the chemical potentials of the synthesis equation, weighted by thestoichiometric coefficients, counted positively for the reactants ( f ) and negatively for the products ( m, c, w ) :(15) A r = X i ν f i µ f i − X j ν m j µ m j − X k ν c k µ c k − X l ν w l µ w l .X m ′ : The membrane molecules are synthesised in the cytosol at temperature T . Their hydrophobic tail enforcesthe spontaneous organisation of these molecules into a bilayer. We suppose that the temperature varies onlyacross the membrane. The driving force of this isothermal process is the affinity of the reaction m c → m ,divided by the inner temperature, T :(16) X m ′ = A mc T = µ mc − µ m T . Here, µ mc is the chemical potential of the free membrane molecules inside the cytosol and µ m is their chemicalpotential in the inner leaflet. The heat released to the inner leaflet during this process is :(17) Q mc = µ mc − µ m = T X m ′ .X m : the membrane molecules are transfered from the inner layer, at temperature T , to the outer leaflet, attemperature T < T , releasing the heat Q m into the environmental thermostat, at temperature T . Thethermodynamical force of this process is :(18) X m = µ m T − µ m T .X q : the thermodynamical force driving the ions of species i , of charge z i e , across the membrane is the differenceof electrochemical potentials [2] :(19) X qi = ˜ µ i − ˜ µ i = (cid:0) µ i + z i eψ (cid:1) − (cid:0) µ i + z i eψ (cid:1) = µ i − µ i + z i eU . where ψ denotes the electrostatic potential and U := ψ − ψ is the electric tension across the membrane.Among these forces, only X θ is a linear function of the small temperature difference, ∆ T = T − T . The othershave, generically, a supplementary constant term, of order 0 in ∆ T .3.4. Conductance matrix.
The phenomenological coefficients, L ik , which couple all the irreversible processes of ourlinear model, can be put in a 7 × L = L θθ L θf L θw L θm L θm ′ L θq L θr L fθ L ff L fw L fm L fm ′ L fq L fr L wθ L wf L ww L wm L wm ′ L wq L wr L mθ L mf L mw L mm L mm ′ L mq L mr L m ′ θ L m ′ f L m ′ w L m ′ m L m ′ m ′ L m ′ q L m ′ r L qθ L qf L qw L qm L qm ′ L qq L qr L rθ L rf L rw L rm L rm ′ L rq L rr . In a first approximation, some coefficients can be set equal to zero :
ROMAIN ATTAL (21) L ≃ L θθ L θf L θw L θm L θq L fθ L ff L wθ L ww L mθ L mm L mq
00 0 0 0 L m ′ m ′ L qθ L qm L qq
00 0 0 0 0 0 L rr . The diagonal coefficients of L are positive but we let L • r = 0 = L r • because the synthesis reactions take place in thecytosol and are decoupled from the transport processes across the membrane. Similarly, we let L • m ′ = 0 = L m ′ • ,because the transfer of membrane molecules from the cytosol to the inner leaflet is decoupled from the other processes.Since the diffusion processes of different molecules (food, waste, ions or membrane constituents) across the membraneare supposed to be decoupled, we put L fw = 0 = L wf , L fm = 0 = L mf and L wm = 0 = L mw . L θθ is the thermal diffusion coefficient across the membrane. L m ′ m ′ is the diffusion coefficient for the transport ofmembrane molecules from the cytosol to the inner leaflet of the membrane. L ff , L ww , and L mm , are the conductancecoefficients of food, waste and membrane molecules through the membrane. We suppose that all these diagonalcoefficients are constant and uniform across the cytosol or the membrane, because protocells could not rely on localspecialised channel molecules (intrinsic proteins, in evolved cells) to supply their food and evacuate their waste. Wealso suppose that food and waste molecules are electrically neutral and that the electric current is entirely due to thetransport of small ions with the help of the translocation process and water pores.The off-diagonal coefficients, L fθ = L θf , L wθ = L θw , L mθ = L θm and L qθ = L θq , depend on the heat capacity ofthe molecules transported and on the rate constants of this transport. They couple the transport of matter and theheat flow. For our purpose, the most interesting off-diagonal coefficient is L θm . It can be viewed as the ratio of heatflow, J θ , to the affinity X m when T = T and in the absence of food and waste driving forces :(22) L θm = (cid:18) J θ X m (cid:19) ( X θ ,X f ,X w ,X m ′ )=0 . In this case, the thermal flow is due only to the asymmetry of the membrane, induced by its bending. This phenomenonis similar to the Dufour effect [20]. If one can prove experimentally that a bending of the membrane induces a heatflow through it, this means that L θm = 0, hence, by Onsager’s reciprocity relations, L mθ = 0, i.e. a heat flow modifiesthe bending. Indeed, we also have the relation :(23) L mθ = (cid:18) J m X θ (cid:19) ( X m ,X f ,X w ,X m ′ )=0 . Hence, L mθ measures the effect of a slight temperature difference (between both sides of the membrane) on the inducedflow of molecules between the leaflets, which implies a modification of its mean curvature. This phenomenon is similarto the thermodiffusion or Soret effect [20]. It is reciprocal to the previous effect and might be easier to observe andmeasure.3.5. Entropy production and stability.
Just as the power dissipated by Joule effect in an ohmic conductor is(24) Power dissipated = Current × Voltage= Conductance × Voltage , the rate of dissipation of energy, or entropy creation, in a general chemical system out of equilibrium is a quadraticfunction of the thermodynamical forces acting in the system [32, 20] :(25) Rate of entropy produced= Flows × Forces= Forces × Conductance matrix × Forces . This relation rests on a linearity hypothesis supposed to be valid only in the neighbourhood of an equilibrium state.The main difference between the ohmic conductor and the chemical system is that, in the latter, the conductance isnot a single number but a matrix which, in the general case, couples all the currents. Taking into account the variousthermodynamical forces defined previously, the rate of entropy production inside the protocell has to the following
HERMALLY DRIVEN FISSION OF PROTOCELLS 9 expression :(26) σ ( X ) = L ff X f + L ww X w + L mm X m + L m ′ m ′ X m ′ + L rr X r + L θθ X θ + 2 X θ ( L θf X f + L θw X w + L θm X m )= L ff (cid:18) µ f T − µ f T (cid:19) + L ww (cid:18) µ w T − µ w T (cid:19) + L mm (cid:18) µ m T − µ m T (cid:19) + L θθ (cid:18) T − T (cid:19) + L m ′ m ′ (cid:18) µ mc − µ m T (cid:19) + L rr (cid:18) A r T (cid:19) + 2 (cid:18) T − T (cid:19) (cid:18) L θf (cid:18) µ f T − µ f T (cid:19) + L θw (cid:18) µ w T − µ w T (cid:19) + L θm (cid:18) µ m T − µ m T (cid:19)(cid:19) . The stability of this steady state is equivalent to the positivity of the matrix L , which is also the matrix of secondorder derivatives of σ in the coordinate system X = ( X θ , X f , X w , X m , X m ′ , X r ) :(27) L ik = 12 ∂ σ∂X i ∂X k . If P is a n × n matrix with real coefficients, the positivity of P , defined by :(28) u t P u > ∀ u ∈ R n implies the following inequalities :(29) P ii > ∀ i and P ii P jj > (cid:18) P ij + P ji (cid:19) ∀ i, j. These conditions are necessary but not sufficient to ensure the positivity of P . In the present case, L being symmetric,we have, in particular :(30) L ii > L θθ L ff > L θf L θθ L ww > L θw L θθ L qq > L θq L mm L qq > L mq L θθ L mm > L θm . If one of these inequalities is not satisfied, the growth process is destabilized. In Section VI, we will prove that thelast one can be reversed as the inner temperature of the protocell increases. In order to prove this proposition, wemust first write down evolution equations for the geometry of the cell.4.
Membrane geometry and growth equation
Just as the growth of a child depends on his diet, the evolution of the geometric parameters of a protocell dependson the flow of molecules to its membrane. This flow is determined by the food intake and by the rate of the synthesis ofthese structural molecules. In this section, we establish the differential equations governing the growth of the volumeand area of a cylindrical protocell by relating them to the flows of matter.4.1.
Conservation of matter and exponential growth.
The advancement, ξ , of the overall synthesis reaction, f → m + c + w , is the internal clock of the protocell. The corresponding flow of matter, J r = d ξ d t , is channeled to allthe other processes in the protocell. In particular, it determines the flux of matter to the inner leaflet and the growthspeed of the membrane. By writing the equations of conservation of matter, we can then determine the evolution ofthe size of the protocell.Let a ∈ { c, , } denote the possible position of a membrane molecule : either in the cytosol ( c ), or the inner leaflet(1) or the outer leaflet (0). Let N ma be the number of membrane molecules in each of them. The time derivatives ofthese functions are related to the flows defined previously :(31) d N mc d t = − J mc A + J rm V d N m d t = J mc A − J m A d N m d t = J m A . Similarly, the number of food (resp. cytosol and waste) molecules, N f (resp. N c and N w ), evolves according to thefollowing relations :(32) d N f d t = J f A − J rf V d N c d t = J rc V d N w d t = − J w A + J rw V where the flows J r • are defined by :(33) J rm := ν m d ξ d tJ rf := ν f d ξ d t = ν f ν m J rm J rc := ν c d ξ d t = ν c ν m J rm J rw := ν w d ξ d t = ν w ν m J rm . In a steady state, the concentration of membrane molecules in the cytosol is constant :(34) C mc := N mc V = cst.Let c m and c m be the average number of membrane molecules per unit area in each leaflet :(35) c m := N m A and c m := N m A . The conservation equations for m imply the evolution equations of the geometry of the protocell :(36) c m d A d t = J m A + A c m d A d t = J mc A − c m d A d tc mc d V d t = J rm V − J mc A . Let us introduce the following parameters :(37) 2 ε := average thickness of the membrane η := c m c m (layer density ratio ≃ τ := J m J mc (transmission rate through the membrane) t := c m J mc (inner leaflet characteristic time) τ c := J mc J rm (transmission rate to the membrane) t c := c mc J rm (cytosol characteristic time) . The transmission ratio, τ , can be written in terms of thermodynamical forces :(38) τ := J m J mc = L mm X m + L mθ X θ + . . .L m ′ m ′ X m ′ . Let U = V /ε and ˙ X = t d X d t . We obtain the following system of differential equations :(39) ˙ A = ητ A + A ) = ητ A ˙ A = − τ A + (cid:16) − τ (cid:17) A = (1 − τ ) A − B ˙ U = t t c U − c m εc mc A . HERMALLY DRIVEN FISSION OF PROTOCELLS 11
In matrix form :(40) ˙ X = ˙ A ˙ A ˙ U = ητ ητ − τ − τ − c m εc mc t t c A A U = M XM := ητ ητ − τ − τ − c m εc mc t t c and X := A A U . This growth equation is solved in Appendix B. The matrix M has a block diagonal form, hence A and A evolveindependently of U , whereas the equation for U contains terms linear in A and A . The upper left 2 × A and A are linear combinations of exponential functions of time (multiplied by an affine functionof t in the degenerate, non diagonalisable case). The rates of growth of these exponential functions are the eigenvaluesof this 2 × t t c for U .4.2. Cylindrical growth in steady state.
When we meet an ordinary differential equation, describing the timeevolution of a dynamical system, a first reflex is to search for constant solutions or at least steady state solutions,where the speed is constant. In the present case, we can look for a solution where the length increases steadily whereasthe radius is constant. This corresponds to the observed growth of some bacterial species in difficult environments[28]. When the sludge content of wastewater is too high or when the composition is lopsided, a higher percentage ofbacteria adopt a filamentous growth strategy which allows them to survive in harsher conditions, by catching foodmore easily.If the protocell grows like a cylinder of radius R , we have ε A = R B , hence d AA = d BB and(41) x := R ε = AB = d A d B = ˙ A ˙ B = (cid:0) ( η − τ + 1 (cid:1) A − B (cid:0) ( η + 1) τ − (cid:1) A + B = α + A − B α − A + B where α ± = ( η ∓ τ ±
1. Therefore, x satisfies the fixed point equation :(42) x = α + x − α − x + 1 i.e. α − x − ( α + − x + 1 = 0 . The discriminant of this quadratic equation is(43) ( α + − − α − = ( η − τ − η + 1) τ + 4= 4∆( η, τ )(cf. Appendix B) and its roots, x ± , are related to the eigenvalues, λ ± , of the matrix M (Eq. 40) :(44) x ± = 12 α − (cid:16) α + − ± p ( α + − − α − (cid:17) = ( α + − ± p ∆( η, τ )2 α − = 2 λ ± − α − . Consequently, the radius, R , of the cylinder whose length increases in a steady state is determined by the flows( J mc , J m , J rm ) and the concentrations ( C mc , c m , c m ), via the coefficients ( ε, η, τ ) :(45) R = εx ± = ε λ ± − α − = ε (cid:0) ( η + 1) τ ± √ ∆ (cid:1) (cid:0) ( η + 1) τ − (cid:1) = ε η + 1) τ ± p ( η − τ − ( η + 1) τ + 1( η + 1) τ − Thermal instability of cylindrical growth
As long as the protocell grows by increasing only its length, keeping a cylindrical shape of fixed radius, R , itsvolume and its membrane area grow proportionally, i.e. ˙ A = cst. × ˙ B . If the heat generated by the metabolic reactionswere exactly proportional to the volume increment, the increase of the area of the membrane would be sufficient toevacuate steadily the heat generated by the chemical reactions taking place inside the newly created volume. However,the heat generated by all these irreversible processes adds up to that coming from the exothermic metabolic reactions and the inner temperature must therefore increase. This overheating generates larger fluctuations of all the physicalparameters which destabilize the initial steady state of cylindrical growth. We will see below that the geometricalparameters ( A , B , V ) can follow a path leading to a more efficient release of heat, by reducing the radius R .5.1. The Squeezed Sausage Theorem (SST).
When we squeeze a sausage, its length increases as well as itsarea. Indeed, the stuffing being incompressible, the squeezing is an isovolumic deformation. The stuffing is pushedlongitudinally, away from the squeezed zone, and increases the length of the sausage, thanks to the elasticity of thegut. The area of the slice of reduced radius increases consequently to bound the same volume. Let us prove thismathematically.A length δx of cylinder of radius R has volume δ V and boundary area δA given by :(46) δ V = πR δxδ A = 2 πR δx Let us suppose that this cylindrical growth is perturbed by a small, local radius variation, which can be positive(anevrism) or negative (stenosis). We study here a triangular perturbation and, in the appendix, a smooth ( C ),rotation invariant perturbation of the cylinder. To keep it simple, we suppose that this perturbation is piecewise linearand symmetric, with an extremum δR at x = 0, and vanishes outside of the interval h − δx ′ , δx ′ i . FIG. 5 representsthe resulting isovolumic deformation according with the sign of δR .2 R δxδ V δ A δx ′ > δx ( R + δ R ) < R δ V ′ = δ V δ A ′ > δA ( R + δ R ) > R δ V ′ = δ V δx ′ < δxδ A ′ < δA Figure 5.
Isovolumic variation of the area of a cylinder under a small triangular deformation.
In the second and third pictures of FIG. 5, the Gaussian curvature is concentrated on the circular sections at x = 0 andat x = ± δx ′ (dotted lines), where the mean curvature has a finite discontinuity. The volume and lateral membranearea of this slice of thickness δx ′ (contained between the dotted lines) are therefore :(47) δ V ′ = π (cid:18) R + δR (cid:19) δx ′ δ A ′ = 2 π (cid:18) R + δR (cid:19) δx ′ . The straight slice and the deformed slice have equal volumes ( δ V = δ V ′ ) if their thicknesses satisfy :(48) δx ′ δx = (cid:18) δR R (cid:19) − ≃ (cid:18) − δRR (cid:19) . Hence the ratio of their areas is(49) δ A ′ δ A ≃ (cid:18) δR R (cid:19) (cid:18) − δRR (cid:19) ≃ − δR R . The heat flows through these surfaces are, respectively :(50) δq = L θθ (cid:18) T − T (cid:19) δ A δq ′ = L θθ (cid:18) T − T (cid:19) δ A ′ HERMALLY DRIVEN FISSION OF PROTOCELLS 13 hence their ratio is the same as for the areas :(51) δq ′ δq = δ A ′ δ A = 1 − δR R . When δR <
0, this ratio is larger than 1. Consequently, the inner volume being held fixed, a small stenosis of acylindrical protocell evacuates heat more efficiently than a small anevrism. This local reduction of the radius of theprotocell increases its mean curvature. For this deformation to happen, the outer leaflet must grow more rapidlythan the inner leaflet. Therefore, the equilibrium m ⇆ m must be shifted towards m in order to have δR < T increases slightly and m → m is exothermic. We propose that the translocation of membranemolecules to the outer leaflet [14, 15, 4, 1] can be triggered by the increase of the inner temperature, T ( t ). The areaof the outer leaflet then increases more quickly than the area of the inner leaflet, which leads to the bending of themembrane until the total splitting of the protocell into two daughters.5.2. Fluctuations, translocation and heat transfer.
In order to increase L mθ and destabilise the cylindricalgrowth, the transfer coefficient, τ , must also increase. In [14, 15], the authors present a detailed mechanism for thetransfer of membrane molecules between the leaflets. Due to the fluctuations of ionic densities in the neighbourhoodof the membranes, the local electric field fluctuates strongly enough to push molecules of water into the membrane,via the field-dipole interaction force (dielectrophoresis). When it is sufficiently strong, this force can create a transientwater pore that is stable enough to let some membrane molecules dive into this water pore and join the other side.The increase of the inner temperature can also enhance these ionic density fluctuations and favor this translocationprocess from the hot side to the cold side, since the hottest, most agitated molecules have a higher probability to diveinto the water pore than the colder molecules. This asymmetric flow of hot molecules to the cold side enhances theoutgoing heat flow and cools down the protocell.During this process, the shape of the hydrophobic tails is not important, as long as they remain in the hydrophobiczone, surrounded by siblings. The only energetic cost is for the hydrophilic head surrounded by these aliphatic chains,and some clandestine water molecules forming the water pore (not represented below). The shape of the tail isirrelevant since the energy depends only on the position of the polar head (FIG. 6). ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ hydrophilic zonehydrophilic zonehydrophobic zone Figure 6.
Translocation of a membrane molecule from one leaflet to the other.
Thermal balance.
Let us make a thermal balance of the whole growth process. After heating its cold nutrientmolecules from T to T and processing the isothermal inner chemical reactions ( J r ), our protocell disposes of its hotwaste (including some water flowing through the water pores) and loses heat by translocation of membrane moleculesfrom the inside to the outside, and by diffusion ( J θ ) without mass transfer. Let q i be the heat exported out of theprotocell by each molecule of type i . Cold entering molecules and hot outgoing molecules both have q i >
0. Let κ i bethe heat capacity of the molecules of type i . Let J h be the outgoing heat flow (energy/(time × area)). The heat flowexported by the cold entering food and water molecules is :(52) J f q f = J f κ f ( T − T ) . Similarly, the heat flow exported by the outgoing waste and water molecules is :(53) J w q w = J w κ w ( T − T ) . And the heat flow exported by the net translocation of membrane molecules is :(54) J m q m = J m κ m ( T − T )if we suppose that they immediately thermalise from T to T once they reach the outer leaflet. The contact of thehydrophobic tails inside the membrane allows for a diffusive heat flow :(55) J θ = X i L θk X k . The total heat flow is the sum of these terms :(56) J h := ( J f q f + J w q w + J m q m ) + J θ = (cid:0) J f κ f + J w κ w (cid:1) ( T − T )+ ( L mm X m + L mθ X θ + . . . ) κ m ( T − T )+ ( L θθ X θ + L θm X m + . . . ) . Since(57) T − T = T T X θ = T X θ − T X θ L mθ appears as a factor of X θ in the convective term, J m , whereas L θm is a factor of X m in the diffusive term, J θ .Moreover, X m increases linearly with X θ :(58) X m = µ m T − µ m T = µ ◦ m T − µ ◦ m T + k B ln (cid:18) a m a m (cid:19) = µ ◦ m X θ + k B ln (cid:18) a m a m (cid:19) where µ ◦ denotes the standard chemical potential, at temperature 298 K and pressure 1 atm [2]. The L mθ -dependentterm in J h becomes :(59) J h = L mθ (cid:18) κ m T X θ − T X θ + µ ◦ m X θ (cid:19) + . . . Consequently, as the cytosol heats up, J h increases more quickly by translocation ( κ m term) than by diffusion ( µ ◦ m term). Translocation is a particular kind of heat convection and by analogy with the Rayleigh-B´enard instability[35], we conjecture the existence of a transition from a diffusive regime to a convective regime, where translocationovertakes diffusion and expells heat more efficiently.6. Translocation between leaflets
The energetic barrier, of width 2 ε ′ and height E ∗ , is difficult to penetrate for the hydrophilic head since thisguarantees the stability of the bilayer under ordinary thermal fluctuations. When the ratio of concentrations, η = c m c m ,becomes too large compared to unity, the mechanical constraint on the inner leaflet is released by pushing molecules tothe outer leaflet. Conversely, when the outer leaflet is stretched and the inner leaflet compressed, η is slightly greaterthan unity (FIG. 7). (( ( (( (( (( ( (( ( ( ( ( ( ( ( (( ( ( ( ( ( ( c m compressed c m < c m stretched (cid:23) (cid:23) (cid:23)(cid:23) (cid:23)(cid:23) (cid:23)(cid:23) (cid:23)(cid:23) (cid:23)(cid:23) (cid:23)(cid:23) (cid:23)(cid:23) (cid:23) (cid:23)(cid:23) (cid:23) (cid:23)(cid:23) (cid:23) (cid:23) (cid:23)(cid:23) (cid:23) (cid:23) (cid:23)(cid:23) (cid:23) Figure 7.
Mechanical constraints modify the ratio, η , of leaflet concentrations. To facilitate this process, some water molecules can leak through the hydrophobic zone and ease the passage ofthe hydrophilic head. This leakage of water lowers the activation energy, E ∗ , and realizes an aqueous catalysis of thetranslocation process [14, 15, 4, 1]. If we suppose that the density, n p , of water pores in the membrane is constantfor fixed temperatures, T and T , then J m depends only on this density and on the net number, j mp , of membranemolecules translocated from L to L during the lifetime of the pores :(60) n p := number of water pores per unit area j mp := net number of translocationsthrough each water pore J m = n p j mp . This first approximation is based on the hypothesis that the pores have the same size, the same lifetime and the samenumber of net translocations during their short life. However, to be more realistic, we must take into account the factthat larger pores live longer and leak more (over the same duration) than smaller short lived pores. We integrate overthe interval of possible lifetimes ( t p ) the density of water pores of lifetime t p created per unit time ( n p ( t p )) multipliedby the net number ( ν mp ( t p )) of molecules each pore of lifetime t p translocates from the inside to the outside during HERMALLY DRIVEN FISSION OF PROTOCELLS 15 its existence :(61) J m = Z ∞ d t p n p ( t p ) ν mp ( t p ) . The increase of X θ enhances at the same time the rate of formation of pores, hence n p , and the net number oftranslocated molecules, due to larger thermal fluctuations. Therefore, J m increases more than linearly as a function of X θ . Consequently, the crossed conductivity coefficient, L mθ , increases with X θ . On the other side of the inequality, L θθ and L mm depend more weakly on the temperature. Indeed, the heat diffusion coefficient, L θθ , involves the (temperatureindependant) number of interacting degrees of freedom between the hydrophobic tails inside the hydrophobic layer,and the molecular diffusion coefficient :(62) L mm = T (cid:18) J m µ m − µ m (cid:19) ( X θ ,X f ,X w ,X m ′ )=0 depends mainly on the ratio of concentrations between the two leaflets, i.e. on η . In order to know if the initialinequality, L mθ < L θθ L mm , can be reversed, the temperature dependance of the convective coefficient, L mθ , must becomputed and compared to that of the diffusion coefficients, L θθ and L mm . This necessitates a microscopic model ofthe interactions of membrane molecules and water and a precise description of the translocation process, to go beyondthe linear response theory. In the sequel, we adopt a simple mean field approach where each molecule evolves in thesame energetic landscape as the others.6.1. An effective potential for translocation.
The exact shape and position of each membrane molecule is de-scribed by dozens of parameters specifying the position of each atom and the orientation of each interatomic bond.It would be cumbersome to take them all into account to describe mathematically the evolution of a single moleculeinside the membrane. However, we can make a simplifying approximation by remarking that the main energetic costis in the displacement of the hydrophilic head into the hydrophobic layer or the protrusion of this head outside ofthe membrane, which forces the tail to go into the hydrophilic zone. We can make a mean-field approximation byconsidering only the position, z , of the hydrophilic head as a dynamical variable, and defining an adequate effectivepotential energy, U ( z ), that traps the head inside the membrane. In the sequel of this article, we will use a doublewell effective potential to compute the net flow, J m , across a plane bilayer subject to a difference of temperatures. Bydifferentiation, we obtain the coefficients L mθ and L mm and, in particular, their dependence on temperature. Thismodel suggests that the inequality L mθ < L mm L θθ can be reversed if the inner temperature increases sufficiently. Ourhypotheses are the following ones :(1) The membrane molecules have length ε = ε ′ + ε ′′ , where ε ′′ is the size of the hydrophilic head and ε ′ is thelength of the hydrophobic tail.(2) The translocation process is described by only one parameter : the position of the center of mass of thehydrophilic head, varying between − ε and + ε .(3) On each side of the membrane, the distribution of velocities of the heads follows a Maxwell-Boltzmann law[35]. The probability of finding a molecule with velocity v perpendicularly to the membrane is :(63) p i ( v ) = r m πk B T i exp (cid:18) − mv k B T i (cid:19) . (4) The translocation requires an energy E ∗ and the head of the molecule evolves in an effective double wellpotential (FIG. 8). hydrophilic zonepolar heads+ water+ ions hydrophobic zonealiphatic chains hydrophilic zonepolar heads+ water+ ions z − ε − ε ′ T εε ′ T E ∗ U ( z ) Figure 8.
Potential energy of the hydrophilic head (5) The hydrophilic heads trapped in the well [ − ε, − ε ′ ] have temperature T , whereas those trapped in the well[ ε ′ , ε ] have temperature T . The thermalisation processes for the motion along the z axis occur only once thehead is trapped in the arrival well. This drastic hypothesis simplifies the computations and should be refinedin a more realistic model. In reality, the motions of the hydrophobic tails between z = − ε ′ and z = + ε ′ canthermalise the molecule during the travel across the membrane and this affects the translocation time.Only half of the molecules of kinetic energy E > E ∗ can escape from a well to the other side. The time it takesthem to go through the barrier is given by :(64) t f = Z + ε ′ − ε ′ d z r m E − E ∗ )= 2 ε ′ r m E − E ∗ )= 2 ε ′ r mmv − E ∗ . The flow of molecules of velocity belonging to the interval [ v, v + d v ], with v > v ∗ := q E ∗ m , going from side 1 to side0, is proportional to the surface density of molecules, c m , to the Maxwell-Boltzmann weight, p ( v )d v , of this velocityinterval, and to the reciprocal of the translocation time :(65) J m = Z + ∞ v ∗ d v c m p ( v ) t f = Z + ∞ E ∗ d E √ mE ε ′ r E − E ∗ ) m c m e − E/k B T q πk B T m = 12 ε ′ √ π Z + ∞ E ∗ d E √ mE r E − E ∗ k B T c m e − E/k B T . The net flow of molecules from leaflet 1 to leaflet 0 is :(66) J m := J m − J m = 12 ε ′ √ π Z + ∞ E ∗ d E √ mE r E − E ∗ k B T c m e − E/k B T − ε ′ √ π Z + ∞ E ∗ d E √ mE r E − E ∗ k B T c m e − E/k B T . HERMALLY DRIVEN FISSION OF PROTOCELLS 17
Computation of L mθ . The temperature T being fixed, we have :(67) L mθ = ∂J m ∂X θ = − ∂J m ∂T − = − c m ε ′ √ π Z + ∞ E ∗ d E r E − E ∗ mE ∂∂T − (cid:18) e − E/k B T √ k B T (cid:19) = c m ε ′ k B √ πmk B T Z + ∞ E ∗ d E r E − E ∗ E (cid:18) E − k B T (cid:19) e − E/k B T . We set u ∗ := E ∗ k B T and change the variable of integration from E to s := EE ∗ :(68) L mθ = c m E ∗ √ k B T ε ′ k B √ πm Z + ∞ d s r − s (2 su ∗ − e − su ∗ = α F ( u ∗ ) α := c m E ∗ √ k B T ε ′ k B √ πm where the function F is defined by :(69) F ( a ) := Z + ∞ d s r − s (2 as − e − as . We can now compute the relative variations of L mθ with respect to relative variations of temperature. Since L mθ depends on T through E ∗ and F ( u ∗ ), we have :(70) ∂ ln L mθ ∂ ln T = ∂ ln α ∂ ln T + ∂ ln F∂ ln T = 12 + ∂ ln E ∗ ∂ ln T + ∂ ln u ∗ ∂ ln T ∂ ln F∂ ln u ∗ = 12 + ∂ ln E ∗ ∂ ln T + (cid:18) ∂ ln E ∗ ∂ ln T − (cid:19) ∂ ln F∂ ln u ∗ = 12 + ∂ ln E ∗ ∂ ln T (cid:18) ∂ ln F∂ ln u ∗ (cid:19) − ∂ ln F∂ ln u ∗ . ∂ ln E ∗ ∂ ln T can not be computed in the present model, because it depends on the microscopic details of the formation ofwater pores. However, we know that E ∗ diminishes as T increases, since the water pores become more frequent (and,probably, larger and more durable) when the ionic density fluctuations increase [14, 15]. Consequently, we have :(71) ∂ ln E ∗ ∂ ln T < . In Appendix C, we prove that 1 + ∂ ln F∂ ln u ∗ is slightly negative at high temperature. Since ∂ ln E ∗ ∂ ln T is also negative, weobtain the following estimate :(72) ∂ ln L mθ ∂ ln T &
32 at high temperature.6.3.
Computation of L mm . L mm is obtained by differentiating J m with respect to X m = µ m − µ m T while keepingthe other thermodynamical forces equal to zero :(73) L mm = (cid:18) ∂J m ∂X m (cid:19) ( X θ ,X f ,X w ,X m ′ )=0 = T (cid:18) ∂J m ∂ ( µ m − µ m ) (cid:19) ( X θ ,X f ,X w ,X m ′ )=0 = 1 k B (cid:18) ∂J m ∂ ln( a /a ) (cid:19) ( X θ ,X f ,X w ,X m ′ )=0 . In our model, based on the double well effective potential, the activities of the membrane molecules in each leaflet areequal to their respective concentrations. A more accurate model, taking into account the attractive interactions insideeach leaflet, is necessary to improve this first approximation. Replacing a a by c m c m = η , we obtain :(74) L mm = 1 k B (cid:18) ∂J m ∂ ln η (cid:19) T = T . J m is a linear combination of the leaflet concentrations :(75) J m = ζ ( T ) c m − ζ ( T ) c m ζ ( T ) := E ∗ e − u ∗ ε ′ √ πmk B T Z + ∞ d x e − u ∗ x r xx + 1 u ∗ := E ∗ k B T .
If the temperatures of both leaflets are equal, then J m is simply proportional to the difference of their concentrations:(76) (cid:0) J m (cid:1) T = T = T = ζ ( T )( c m − c m )and its derivative with respect to ln η , while c m is held fixed, is :(77) (cid:18) ∂J m ∂ ln η (cid:19) c m =cst. = ζ ( T ) c m = k B L mm . Since(78) Z + ∞ d x e − ax r xx + 1 = 1 a − ln( a )2 + O (1) ( a → + )the high temperature expansion of L mm gives :(79) (cid:18) ∂ ln L mm ∂ ln T (cid:19) T = T = T = (cid:18) ∂ ln ζ∂ ln T (cid:19) T = T = T = 12 + o (1) . Estimation of L θθ . The heat diffusion coefficient, L θθ , depends only on the number of degrees of freedom thatinteract in the membrane bilayer. As long as the structure of the membrane is unchanged, the same hydrophobic tailsinteract similarly at any temperature. Therefore, we conjecture that L θθ is independant of the temperature in theliquid disordered phase [27]. Therefore :(80) ∂ ln L θθ ∂ ln T ≃ . Destabilisation.
Putting together the scaling laws for L mθ , L mm and L θθ , we obtain :(81) ∂∂ ln T (cid:18) L mθ L mm L θθ (cid:19) = 3 − − Proposition 6.1.
Since L mθ L mm L θθ grows as T / , the stability condition, L mθ < L mm L θθ , can not be satisfied at hightemperature. The exact value of T for which this transition occurs can not be computed in our simple model, but the onlycharacteristic temperature being E ∗ k B , the critical temperature must be of this order of magnitude.This destabilisation of the steady growth regime is comparable with the onset of heat convection in a fluid subjectto a strong temperature gradient. In fine , the self-replication of protocells could be interpreted as a convectivephenomenon inside their membrane, triggered by their metabolic activity.7.
Conclusions and perspectives
We have proposed a toy model of protocell growth, fission and reproduction. The scenario thus described canbe viewed as the ancestor of mitosis. The main force driving this irreversible process is the temperature differencebetween the inside and the outside of the protocell, due to the inner chemical activity. We propose that the increase ofthe inner temperature, due to a rudimentary inner metabolism, enhances the transfer of membrane molecules from theinner leaflet to the outer leaflet, as described in silico by models of molecular dynamics [14, 15]. Due to this transferof molecules, coupled to a heat transfer, the difference of their areas and the total mean curvature of the mediansurface increase. The cylindrical growth becomes unstable and any slight local reduction of the radius of the initialcylinder increases until the protocell is cut into two daughter protocells, each one containing reactants and catalysersto continue the growth and fission process. The cut occurs near the hottest zone, around the middle. This modelis based on the idea [23] that the early forms of life were simple vesicles containing a particular network of chemicalreactions, precursor of modern cellular metabolism :Protolife = Cellularity + Inner Metabolism.With a large supply of reactants in the so-called prebiotic soup [31, 16, 23], and with an optimal salinity and pH, theseingredients are sufficient to induce an exponential growth of prebiomass and make possible the exploration of a largenumber of chemical reactions in these miniature chemical factories. The possibility to sythesize complex molecules(sterols, RNA, DNA, proteins, etc.) comes later, once these factories self-replicate and thrive.
HERMALLY DRIVEN FISSION OF PROTOCELLS 19
In order to test our model experimentally, we have to manipulate vesicles that can be heated from within in acontroled way. Let us imagine, in a solution maintained at temperature T , vesicles containing molecules of type A able to absorb visible radiation, with which the surrounding molecules do not interact. Let us suppose that A re-emits radiation in the near infrared. The heat thus generated inside the vesicle creates a controled temperaturedifference, T − T >
0, between both sides of the membrane. If L mθ is large enough, we should observe a bending ofthe membrane of the vesicles due to the transfer of the hottest molecules from the inner leaflet to the outer leaflet.Another experimental test of our model can be made by observing eukaryotic cells, where the mitochondria arethe main source of heat. It seems possible to measure their temperature variations using fluorescent molecules [3].Although the very notion of temperature at this scale and far from a thermodynamical equilibrium is not clear, themeasurement of the temperature variations inside the cell during its life cycle could be correlated with the onset ofmitosis and with the shape of mitochondrial network [21].Our model is obviously oversimplified since the polar heads of membrane molecules are treated as an ideal gasin a box. In particular, we haven’t taken into account the interaction between these molecules and the surroundingsolution. This calls for the development of a better model to treat the effect of these interactions on the temperaturedependence of the conductance coefficients. The scaling law of the ratio L mθ L mm L θθ at temperatures higher than E ∗ k B isthe key argument that explains the splitting of the protocell. Future investigations and experiments will decide of theplausibility of this proposition. Acknowledgments : We thank Jorgelindo Da Veiga Moreira (Universit´e de Montr´eal), Marc Henry (Universit´e deStrasbourg), Olivier Lafitte (Institut Galil´ee, Universit´e Paris XIII), Kirone Mallick (Institut de Physique Th´eorique,CEA, Saclay), Laurent Schwartz (AP-HP) and Jean-Yves Trosset (SupBiotech, Villefuif) for their advice and helpfuldiscussions.
Appendix A. The mean curvature of the membrane
Let Σ t be a family of surfaces, indexed by a time parameter t ∈ [ t , + ∞ [. We suppose that each Σ t is a smooth,orientable and closed (compact, without boundary) hence diffeomorphic to the standard 2-sphere. At each point P ∈ Σ t , the Taylor expansion of the distance from Q ∈ Σ t to the tangent plane, T P Σ t , defines a quadratic form whoseeigenvalues (homogenous to the inverse of a length) do not depend on the coordinate system in the neighbourhood of P . We denote them R − and R + . The mean curvature of Σ t at P is the arithmetic mean of the principal curvatures :(82) H := 12 (cid:18) R + + 1 R − (cid:19) and the gaussian curvature is their product :(83) K := 1 R + R − . In the case of a cylinder, R + = + ∞ and R − = R = its radius, hence H cyl. ( P ) = R and K ( P ) = 0 at every point P ∈ Σ t (except on the end hemispheres).Let Σ t and Σ t be the surfaces obtained by shifting Σ t in the normal direction, over an infinitesimal distance ε on both sides of Σ t . Let A ( t ) (resp. A ( t )) be the average area of the outer (resp. inner) layer of the membrane,measured at the hydrophilic heads, and A = ( A + A ) the average area of the median surface, where the hydrophobictails join. The difference of their areas, A − A , is given by the first term of Weyl’s Tube Formula [13] :(84) A − A = 4 ε Z Σ t H d A + O ( ε ) . Let B be the infinitesimal variation of area along the outer normal :(85) B := 2 ε Z Σ t H d A. A and A can also be written as functions of A and B :(86) A = A + B and A = A − B . Our dynamical variables are the area of the median surface, A ( t ) = R Σ t d A , the volume of the cytosol, V ( t ), and thevariation of area, B ( t ) = 2 ε R Σ t H d A . In the next section, we will establish their evolution equations as a consequenceof the balance equations for the number of membrane molecules.Remark : In the case of a cylinder of radius R , we have H = R and(87) A − A = 2 B = 4 εH A = 2 ε A R . Since 2 ε A + A = 2 ε A is also the volume, v , of this normal thickening of Σ t , we have :(88) v = 2 B H = 4 B R . Appendix B. Solutions of the growth equation
In this appendix, we solve the growth equation, using basic linear algebra and standard results about linear differ-ential equations [19]. The matrix form of the growth equation is(89) ˙ X = ˙ A ˙ A ˙ U = ητ ητ − τ − τ − c m εc mc t t c A A U = M X.
As long as no flow vanishes, the determinant of M is non-zero :(90) det( M ) = ητ t t c = c m J m J rm c m c mc J mc and the protocell grows exponentially :(91) X ( t ) = e tt M X (0) . In general, the two leaflets of the membrane grow at different speeds. Indeed, the characteristic polynomial of M is :(92) det( M − λ Id)= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ητ − λ ητ − τ − τ − λ − c m εc mc t t c − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) λ − λ (cid:18) η − τ (cid:19) + ητ (cid:19) (cid:18) t t c − λ (cid:19) . Its roots are t t c and the two roots, λ ± ( η, τ ), of the polynomial λ − λ (cid:0) η − τ ) (cid:1) + ητ :(93) λ ± ( η, τ ) := 12 (cid:18) η − τ ± p ∆( η, τ ) (cid:19) ∆( η, τ ) := 14 ( η − τ − ( η + 1) τ + 1 . • If η = 1, i.e. if both leaflets have the same density, then ∆ is an affine function of τ :(94) ∆(1 , τ ) = 1 − τ and λ ± (1 , τ ) = 1 ± √ − τ . If, moreover, τ = , i.e. the inner leaflet transmits half of the incoming membrane molecules to the outer leaflet, then(95) ∆ (cid:18) , (cid:19) = 0 and λ ± (cid:18) , (cid:19) = 12and both leaflets grow at the same speed. • If η = 1, then ∆( η, τ ) is a quadratic function of τ , bounded from below, of discriminant(96) δ = ( η + 1) − ( η − = 4 η > τ ± ( η ) = 2 η + 1 ± √ η ( η − = 2 (cid:18) √ η ± η − (cid:19) = 2( √ η ∓ . Physically, η ≃ τ ≃ . If η = 1 + h , with 0 < h ≪ τ + ≃ h ≫ > τ − and(98) τ − ≃ (cid:0) h (cid:1) ≃ − h . Consequently, τ stays < τ − (FIG. 9). τ − physicalregion non-physical region∆ > < > τ + ≫ τ − Figure 9.
Exponential growth necessitates to keep τ < τ − . HERMALLY DRIVEN FISSION OF PROTOCELLS 21
Mathematically, we have three possibilities :(1) τ < τ − ( η ) or τ > τ + ( η ) → ∆( η, τ ) > λ + ( η, τ ) = λ − ( η, τ ) (real numbers) ;(2) τ = τ − ( η ) or τ = τ + ( η ) → ∆( η, τ ± ) = 0 λ + (cid:0) η, τ ± ( η ) (cid:1) = λ − (cid:0) η, τ ± ( η ) (cid:1) = √ η √ η ± τ − ( η ) < τ < τ + ( η ) → ∆( η, τ ) < λ + ( η, τ ) = λ − ( η, τ ) (complex numbers).B.1. Case 1 : η = 1 and τ > τ + ( η ) or τ < τ − ( η ) . In these intervals, N is diagonalisable and a basis of eigenvectorsof N is given by :(99) B ± = λ ± ( η,τ ) ητ − ! = A − A λ ± ( η, τ ) ητ A = B + λ ± ( η, τ ) ητ A i.e. B + and B − grow exponentially, with a rate of growth λ ± /t , respectively :(100) B ± ( t ) = B ± (0) exp (cid:18) λ ± ( η, τ ) t t (cid:19) . The area of the inner leaflet is :(101) A ( t ) = B + ( t ) − B − ( t ) λ + ητ − λ − ητ = ητ √ ∆ (cid:16) B + (0) e tλ + /t − B − (0) e tλ − /t (cid:17) . The area of the outer leaflet is :(102) A ( t ) = (2 λ + − ητ ) B − ( t ) − (2 λ − − ητ ) B + ( t ) λ + − λ − = 2 λ + − ητ √ ∆ B − (0) e tλ − /t − λ − − ητ √ ∆ B + (0) e tλ + /t . And U ( t ) is obtained from A ( t ) :(103) e tt /t c dd t (cid:16) U ( t ) e − tt /t c (cid:17) = − c m εc mc A ( t )(104) U ( t ) e − tt /t c = − ητ c m B + (0) εc mc √ ∆ (cid:16) λ + t − t t c (cid:17) e t (cid:16) λ + t − t tc (cid:17) + ητ c m B − (0) εc mc √ ∆ (cid:16) λ − t − t t c (cid:17) e t (cid:16) λ − t − t tc (cid:17) + cst. U ( t ) = ητ c m εc mc √ ∆ e tλ − /t λ − t − t t c − e tλ + /t λ + t − t t c ! + cst. e tt /t c where the integration constant is determined by U (0).B.2. Case 2 : η = 1 and τ ∈ { τ + ( η ) , τ − ( η ) } . In this singular case, the upper-left 2 × N := 12 (cid:18) ητ ητ − τ − τ (cid:19) = T (cid:18) Λ ± ( η ) 01 Λ ± ( η ) (cid:19) T −
12 ROMAIN ATTAL where Λ ± ( η ) is the single eigenvalue of N when τ is fixed equal to τ + ( η ) or τ − ( η ) :(106) Λ ± ( η ) := λ (cid:0) η, τ ± ( η ) (cid:1) = 2 + ( η − τ ± ( η )4= 2 + η − √ η ∓ η ± √ ηη − √ η √ η ± . An easy computation gives us :(107) 2Λ + ητ − − √ ηη + √ η and 2Λ − ητ − √ ηη − √ η .N has a unique proper line, generated by the vector(108) B ±∗ := ± ητ ± − ! = 12 ∓ √ ηη ±√ η ! = A (cid:18) ∓ √ ηη ± √ η (cid:19) A ∗ means that λ + = λ − , whereas the upper ± depends on the choice between τ = τ + ( η ) and τ = τ − ( η ).Since(109) B ±∗ ( t ) = B ±∗ (0) e t Λ ± /t we obtain :(110) A ( t ) + (cid:18) η ∓ η ± √ η (cid:19) A ( t ) = 2 B ±∗ (0) e t Λ ± /t . Since B ±∗ = T (cid:0) (cid:1) , the vector B ±∗ is the right column of T . The left column of T is the vector B ±• = (cid:0) xy (cid:1) which satisfiesthe equation ( N − Λ ± ) B ±• = B ±∗ , or in extenso :(111) (cid:16) ητ − Λ ± (cid:17) x + (cid:16) ητ (cid:17) y = 12 − τ x + 2 − τ − ± y = Λ ± ητ − . Taking x = 0 and y = ητ gives a solution :(112) (cid:0) N − Λ ± (cid:1) B ±• = (cid:18) ητ − Λ ± ητ − τ − τ − Λ ± (cid:19) (cid:18) ητ (cid:19) = (cid:18) − τ − ± ητ (cid:19) = (cid:18) ± − ητητ (cid:19) = B ±∗ . The matrix T and its inverse, T − , are therefore :(113) T = (cid:18) ητ − ητ ητ (cid:19) and T − = (cid:18) ητ − ητ (cid:19) . Since B ±• = A ητ ± ( η ) , we have B ±• ( t ) = B ∗ (0) tt e t Λ ± /t and :(114) A ( t ) = ητ ± ( η ) B ±∗ (0) tt e t Λ ± /t B.3.
Case 3 : τ − ( η ) < τ < τ + ( η ) . In this interval, ∆ < M has two distinct complex conjugated eigenvalues, λ and λ , functions of η and τ . Let α, β ∈ R be the real and imaginary parts of λ :(115) α := 2 + ( η − τ > β := √− ∆2 > λ ( η, τ ) = α + i β ( i = − . Let V (resp. V ) be a complex eigenvector of N , of eigenvalue λ (resp. λ ), for instance :(116) V := B + λητ A and V := B + λητ A then the real and imaginary parts of V , defined by V ′ := ( V + V ) and V ′′ := i ( V − V ), form a basis of R on which N acts as an orthogonal matrix [19] :(117) 12 (cid:18) ητ ητ − τ − τ (cid:19) = U (cid:18) α β − β α (cid:19) U − V ′ = U (cid:18) (cid:19) = (cid:18) αητ − (cid:19) V ′′ = U (cid:18) (cid:19) = (cid:18) βητ (cid:19) i.e. the matrix U has V ′ and V ′′ as columns :(118) U = (cid:18) αητ − βητ (cid:19) = − ( η +1) τ ητ √− ∆2 ητ ! . Let s = tt . Since our evolution operator, the exponential of sN , is :(119) e sN = e αs U (cid:18) cos( βs ) sin( βs ) − sin( βs ) cos( βs ) (cid:19) U − we have :(120) V ′ ( s ) = e sN V ′ (0) = e αs U (cid:18) cos( βs ) sin( βs ) − sin( βs ) cos( βs ) (cid:19) (cid:18) αητ − (cid:19) = e αs ητ (cid:18) αητ − βητ (cid:19) (cid:18) ητ cos( βs ) + (2 α − ητ ) sin( βs ) − ητ sin( βs ) + (2 α − ητ ) cos( βs ) (cid:19) = e αs ητ ητ cos( βs ) + α − ητ sin( βs ) (2 α − ητ )(2 β + ητ )2 ητ cos( βs ) + (cid:16) (2 α − ητ ) ητ − β (cid:17) sin( βs ) ! . Similarly, we have the expression of V ′′ ( s ) :(121) V ′′ ( s ) = e sN V ′′ (0)= βe αs ητ sin( βs ) (cid:16) α − ητ ητ (cid:17) sin( βs ) + βητ cos( βs ) ! . Finally, A and B are obtained from V ′ and V ′′ by the linear relations :(122) B ( s ) = βV ′ ( s ) − αV ′′ ( s ) β − α A ( s ) = ητα − β (cid:0) V ′ ( s ) − V ′′ ( s ) (cid:1) . Appendix C. Smooth perturbation of cylindrical growth
In this appendix, we compute the variation of the area and of the total mean curvature of a surface of revolutionunder a small variation of its generating curve. We will work in an orthonormal system of coordinates ( x, y, z ). Let ussuppose now that Σ is a revolution surface whose generating curve, rotated around the axis { y = 0 = z } , is given by :(123) p y + z = R ( x ) = R + δR ( x )with | δR ( x ) | ≪ R . The function δR represents an infinitesimal normal perturbation around the cylindrical shape.The variable x satisfies 0 ≤ x ≤ ℓ and the deformed cylinder is glued smoothly with two hemispherical caps of radius R . In other words, we suppose that(124) δR (0) = δR ( ℓ ) = 0 δR ′ (0) = δR ′ ( ℓ ) = 0 . Let us compute the variations of area, δ A , of length, δℓ , and of total mean curvature, δ H , for a fixed volume.C.1. Isovolumic variation of the area. A is a functional of the length, ℓ , the radius, R , and its derivative, R ′ :(125) A ( ℓ, R, R ′ ) = Z ℓ d x πR p R ′ . Its variation under infinitesimal changes of ℓ and R is :(126) δ A = 2 πR δℓ + 2 π Z ℓ d x δR (cid:18)p R ′ − dd x (cid:18) RR ′ √ R ′ (cid:19)(cid:19) . Since(127) dd x (cid:18) RR ′ √ R ′ (cid:19) = RR ′′ + R ′ √ R ′ − R ′ R ′′ RR ′ (cid:0) R ′ (cid:1) / = (1 + R ′ ) − / (cid:0)(cid:0) RR ′′ + R ′ (cid:1)(cid:0) R ′ (cid:1) − RR ′ R ′′ (cid:1) = (1 + R ′ ) − / (cid:0) RR ′′ + R ′ + R ′ (cid:1) we obtain(128) δ A = 2 πR δℓ + 2 π Z ℓ d x δR (cid:0) R ′ (cid:1) − / (cid:0) − RR ′′ − R ′ (cid:1) . Similarly, the volume, V , is a functional of ℓ and R :(129) V ( ℓ, R ) = 4 πR Z ℓ d x πR and its variation under infinitesimal changes of ℓ and R is :(130) δ V = πR δℓ + 2 π Z ℓ d x R δR. If V is held constant, then δ V = 0 and :(131) (cid:0) δℓ (cid:1) V =cst. = − R Z ℓ d x R δR. Inserting this expression of δℓ into that of δ A , we obtain the isovolumic variation of area :(132) ( δ A ) V =cst. = 2 π Z ℓ d x δR (cid:18)(cid:0) R ′ (cid:1) − / (cid:0) − RR ′′ − R ′ (cid:1) − RR (cid:19) . Theorem C.1.
The isovolumic variational derivatives of the length and of the area of a (nearly cylindrical) closedrevolution surface are negative : (133) (cid:18) δℓδR (cid:19) V = cst. < and (cid:18) δ A δR (cid:19) V = cst. < . In other words, since the stuffing is incompressible whereas the gut is elastic, the length and the area of a squeezedsausage increase. We call this simple statement the
Squeezed Sausage Theorem (SST).C.2.
Isovolumic variation of the total mean curvature.
The circles { x = cst. } and the meridians, obtainedby rotating the generating curve of equation z = R ( x ), form an orthogonal system of geodesics [7], and the meancurvature of Σ is given by :(134) H = 12 R √ R ′ + R ′′ (cid:0) R ′ (cid:1) / ! . The lateral area of a slice of width d x , perpendicular to the axis of the surface, is :(135) d A = 2 πR p R ′ d x and the total mean curvature is :(136) H := Z Σ H d A = Z caps H d A + Z ℓ πR p R ′ d x = 4 πR · R + 2 π Z ℓ (cid:18) R R ′′ R ′ (cid:19) d x = 4 πR + πℓ + π Z ℓ R R ′′ R ′ d x. HERMALLY DRIVEN FISSION OF PROTOCELLS 25
Since H is a functional of ℓ , R , R ′ and R ′′ , its variation under a change δR of the radius of gyration and a change oflength δℓ , is obtained after a double integration by parts [10] :(137) δ H = πδℓ + π Z ℓ d x R ′′ R ′ + dd x RR ′ R ′′ (cid:0) R ′ (cid:1) ! + d d x (cid:18) R R ′ (cid:19) δR. Instead of computing each term of the integrand, let us make the approximation R ′ ≪
1, valid when the initialcylinder is only slightly deformed. The expression of δ H then simplifies to(138) δ H ≃ πδℓ + πδ Z ℓ d x RR ′′ ≃ πδℓ + 2 π Z ℓ d x R ′′ δR. Using the expression of δℓ = − R R ℓ d x R δR when V is held constant, we obtain :(139) (cid:0) δ H (cid:1) V =cst. ≃ π Z ℓ d x (cid:18) R ′′ − RR (cid:19) δR. As long as R | R ′′ | ≪ R , the isovolumic variational derivative of H with respect to R is negative :(140) (cid:18) δ H δR (cid:19) V =cst. < R ′ ≪ R | R ′′ | ≪ R. When δR approaches − R and the protocell is ready to split, the two radii of curvature are small compared to R buthave opposite sign, hence the Gaussian curvature around the septum is large and negative. After the cut, when thetwo caps are formed, the mean curvature and the Gaussian curvature are positive again. Appendix D. Asymptotic expansion of F ( a )The change of variable t = p a ( s −
1) in the integral defining F gives us :(141) F ( a ) = e − a a Z + ∞ d t f ( a, t ) f ( a, t ) := 2 t e − t (cid:18) p t + a − √ t + a (cid:19) . Let(142) G ( a ) := Z + ∞ d t f ( a, t ) = ae a F ( a ) . The function f (0 , · ) is integrable over the half line [0 , + ∞ [ and(143) G (0) = Z + ∞ d t f (0 , t )= Z + ∞ d t t e − t (2 t − Z + ∞ d u e − u (2 u −
1) = 1 . Let us compute the asymptotic expansion of G ( a ) when a → + :(144) G ( a ) − G (0)= Z + ∞ d t (cid:0) f ( a, t ) − f (0 , t ) (cid:1) = 2 Z + ∞ d t t e − t (cid:18) (cid:0)p t + a − t (cid:1) − (cid:18) √ t + a − t (cid:19)(cid:19) = Z + ∞ t d t e − t (cid:0)p t + a − t (cid:1) (cid:18) t + 1 √ t + a (cid:19) = Z + ∞ d u e − u (cid:0) √ u + a − √ u (cid:1) (cid:18) √ u + 1 √ u + a (cid:19) = 2 Z + ∞ d u e − u p u ( u + a ) + Z + ∞ d u e − u (1 − u ) − Z + ∞ d u e − u r uu + a = 2 Z + ∞ d u e − u p u ( u + a ) − − Z + ∞ d u e − u r uu + a . Hence :(145) G ( a ) = ϕ ( a ) − ϕ ′ ( a )where(146) ϕ ( a ) := 2 Z + ∞ d u e − u p u ( u + a )= 2 a Z + ∞ d x e − ax p x ( x + 1) .ϕ ( a ) being the Laplace transform of the function x a p x ( x + 1), its expansion as 0 + is given by integrating theexpansion of p x ( x + 1) at + ∞ term by term :(147) p x ( x + 1) = x + 12 − x + O ( x − ) ϕ ( a ) = 2 a (cid:18) a + 12 a − Z + ∞ d x e − ax x + O (1) (cid:19) = 2 + a − a a ) + O ( a ) . Similarly, for ϕ ′ ( a ) we have :(148) r xx + 1 = 1 − x + 38 x + O ( x − ) ( x → + ∞ ) ϕ ′ ( a ) = a Z d x e − ax r xx + 1+ a Z + ∞ d x e − ax r xx + 1= a Z + a (cid:18) e − a a − Z + ∞ d x e − ax x + O (1) (cid:19) = 1 − a ln( a )2 + O ( a ) . Consequently :(149) G ( a ) = 1 + a ln( a )2 + O ( a )and(150) F ( a ) = e − a a + e − a ln( a )2 + O ( a )= 1 a + 12 ln( a ) + o (1) . The asymptotic expansion of ∂ ln F∂ ln a is therefore :(151) ∂ ln F∂ ln a = − a ln( a )2 + O ( a ) . HERMALLY DRIVEN FISSION OF PROTOCELLS 27
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