The principal aim of this work is to show that the microscopically detailed, quantitative relationship between irreversibility andproduction illustrated above has significant, general thermodynamic consequences for far-from-equilibrium, macroscopic processes such as biological self-replication. Building on past results in nonequilibrium statistical mechanics,we will first derive a generalization of the Second Law of Thermodynamics for macroscopic irreversible processes. Subsequently, we will use this result to demonstrate a lower bound on the heat output of a self-replicator in terms of its size,rate, internaland durability. We will furthermore show, through analysis of empirical data, that this bound operates on a scale relevant to the functioning of real microorganisms and other self-replicators.

Progress comes from recognizing that the heat expelled into the bath over a transition fromtois given by Δ, and moreover that the quantity βΔis the amount by which theof the heat bath changes over the course of this transition. Thus, we may write (2) sets up a microscopically detailed relationship between the irreversibility of a transition (that is, how much more likely it is to happen in the forward direction than in the reverse direction) and the amountis increased in the surroundings over the course of the forward trajectory. Moreover, it should be stressed that while this result is derived from a statement of detailed balance (which only holds at equilibrium) it is itself valid for any transition between two microstates, and thus applies to the relaxation dynamics of undriven systems arbitrarily far from equilibrium.

For the beginnings of a way forward, we should consider a system of fixed particle numberand volumein contact with a heat bath of inverse temperature β. If we give labels to the microstates of this system, etc. and associate energies, etc., respectively with each such microstate, then the underlying time-reversal symmetry of Hamiltonian dynamics tells us that the following detailed balance relation holds at thermal equilibrium:Here,* is the momentum-reversed partner of, and(β) is the canonical partition function of the system. The transition matrix element π(; τ) is the conditional probability that the system is found to be in microstateat time= τ > 0 given that it started off in microstateat an earlier time= 0. Thus, the above relation states that when the system has relaxed to thermal equilibrium and achieved adistribution over microstates (), the probability currents connectingtoin the forward and time-reversed directions are equal.

Every species of living thing can make a copy of itself by exchanging energy and matter with its surroundings. One feature common to all such examples of spontaneous “self-replication” is their statistical irreversibility: clearly, it is much more likely that one bacterium should turn into two than that two should somehow spontaneously revert back into one. From the standpoint of physics, this observation contains an intriguing hint of how the properties of self-replicators must be constrained by thermodynamic laws, which dictate that irreversibility is always accompanied by an increase of entropy. Nevertheless, it has long been considered challenging to speak in universal terms about the statistical physics of living systems because they invariably operate very far from thermodynamic equilibrium, and therefore need not obey a simple Boltzmann probability distribution over microscopic arrangements. Faced with such unconstrained diversity of organization, it is quite reasonable to worry that the particular mechanistic complexity of each given process of biological self-replication might overwhelm our ability to say much in terms of a general theory.

Having constructed a probabilistic definition of our macrostates of interest, we may now also define associated quantities that will allow us to give a macroscopic definition to irreversibility. In particular, we can writeandThe first of these probabilities π() gives the likelihood that a system prepared according tois observed to satisfyafter time τ. The second probability π() gives the likelihood that afterinterval of τ that the same system would be observed again to satisfy. Putting these two quantities together and taking their ratio thus quantifies for us the irreversibility of spontaneously propagating fromtowheredenotes an average over all paths from somein the initial ensembleto somein the final ensemble, with each path weighted by its likelihood. Defining the Shannonfor each ensemble in the usual manner (≡ −∑ln), we can construct Δ, which measures the internalchange for the forwardSince⩾ 1 +for all, we may rearrange (6) to writeand immediately arrive atIfandcorresponded to identical groups that each contained all microstates, we would have π() = π() = 1, and the above relation would reduce to a simple statement of the Second Law of Thermodynamics: on average, the change in theof the system Δplus the change in theof the bathmust be greater than or equal to zero, that is, the average totalchange of the universe must be positive. What we have shown here using Crooks' microscopic relation, however, is that theirreversibility of a transition from an arbitrary ensemble of states) to a future ensemble) sets a stricter bound on the associatedproduction: the more irreversible the macroscopic process (i.e., the more negative), the more positive must be the minimum totalproduction. Moreover, since the formula was derived under very general assumptions, it applies not only to self-replication but to a wide range of transitions between coarse-grained starting and ending states. In this light, the result in Eq. (8) is closely related to past bounds set onproduction in information-theoretic terms,as well as to the well-known Landauer bound for the heat generated by the erasure of a bit of information.

Suppose now that we let some time interval τ pass while keeping our system in contact with a heat bath of inverse temperature β (In subsequent expressions for π( i → j ) the implicit τ will be omitted). At that point, we could introduce a second criterion for coarse-grained observation of the system (e.g., “The system contains precisely two healthy, exponential-growth-phase bacteria at the start of cell division.”), which we might call II , and in the event that we observed this criterion to be satisfied, we could immediately define the probability distribution p ( j | II ) as being the likelihood of the system being in microstate j given that a system initially prepared in I subsequently propagated over a certain period of time τ and was then found to be in macrostate II .

By fixing the starting and ending points of our trajectory ((0) =(τ) =), we can average the exponential weight of the forward heat over all paths fromtoand obtain. This microscopic rule (of which Eq. (2) is clearly the special, undriven case) must have macroscopic consequences, and to investigate these we have to formalize what it means to talk about our system of interest in macroscopic terms. In order to do so, we first suppose there is some coarse-grained, observable conditionin the system, such as the criterion “The system contains precisely one healthy, exponential-growth-phase bacterium at the start of a round ofdivision.” If we prepare the system under some set of controlled conditions and then find that the criterion foris satisfied, we can immediately associate witha probability distribution) which is the implicit probability that the system is in some particular microstate, given that it was prepared under controlled conditions and then observed to be in macrostate

Our goal is to determine thermodynamic constraints on a macroscopic transition between two arbitrarily complex coarse-grained states. To do so, we have to think first about how probability, heat, andare related at the microscopic level. (2) established such a relationship in one special case, but the linkage turns out to be far more general, for it has been shownthat for all heat bath-coupled, time-symmetrically driven nonequilibrium systems whose dynamics are dominated by diffusive motions that lack any sense of ballistic inertia, any trajectory starting at(0), and going through microstates) over time τ obeysHere, β ≡ 1/sets the temperatureof the bath in natural units, π[)] is the probability of the trajectory) given(0), and Δis the heat released into the bath over the course of). This formula captures the essential microscopic relationship between heat and irreversibility: the more likely a forward trajectory is than its time-reverse, the more theof the universe is increased through the exhaustion of heat into the surrounding bath. Furthermore, it should be emphasized that this result holds for systems subject to time-symmetric external driving fields. This means that the steady-state probability distribution of the system need not be thedistribution, nor even a distribution whose steady-state probability flux satisfies detailed balance, in order for this fundamental relationship between irreversibility andproduction to hold.

A somewhat subtler point to make here is that self-replicators can increase their maximum potential growth rates not only by more effectively fueling their growth via Δ q , but also by lowering the cost of their growth via δ and Δ s int . The less durable or the less organized a self-replicator is, all things being equal, the less metabolic energy it must harvest at minimum in order to achieve a certain growth rate. Thus, in a competition among self-replicators to dominate the population of the future, one strategy for “success” is to be simpler in construction and more prone to spontaneous degradation. Of course, in the limit where two self-replicators differ dramatically in their internal entropy or durability (as with a virus and a rabbit) the basis for comparison in terms of Darwinian fitness becomes too weak. Nevertheless, in a race between competitors of similar form and construction, it is worthwhile to note that one strategy for reducing the minimal “metabolic” costs of growth for a replicator is to substitute in new components that are likely to “wear out” sooner.

Several comments are in order. First of all, let us consider comparing two different replicators with fixed Δ s int and δ that have different heat values Δ q and Δ q ′ . If Δ q > Δ q ′ , then clearly g max > g max ′ ; the replicator that dissipates more heat has the potential to grow accordingly faster. Moreover, we know by conservation of energy that this heat has to be generated in one of two different ways: either from energy initially stored in reactants out of which the replicator gets built (such as through the hydrolysis of sugar) or else from work done on the system by some time-varying external driving field (such as through the absorption of light during photosynthesis). In other words, basic thermodynamic constraints derived from exact considerations in statistical physics tell us that a self-replicator's maximum potential fitness is set by how effectively it exploits sources of energy in its environment to catalyze its own reproduction. Thus, the empirical, biological fact that reproductive fitness is intimately linked to efficient metabolism now has a clear and simple basis in physics.

The above result is certainly consistent with our expectation that in order for this self-replicator to be capable of netwe have to require that> δ, which in turn sets alower bound on the totalproduction associated with self-replication. More can be seen, however, by rearranging the expression at fixed δ, Δ, and Δ, to obtainIn other words, the maximum netrate of a self-replicator is fixed by three things: its internal(Δ), its durability (1/δ), and the heat that is dissipated into the surrounding bath during the process of replication Δ

For= 0) ≫ 1, we expect the behavior of the system to follow a deterministic exponentialpath of the form) =(0). Whatever the exact value of, the probability in a short period of timethat one particular replicator gives birth π() should be, while the probability of a newly born copy decaying back whence it came in the same-length interval time (π()) should be δ. Thus, plugging into (8) , we haveAs an aside, it should be noted here that we have avoided including in the above a spurious multiplicative factor from the number of particles in the system. As an example, one might have thought in the case of converting one particle into two that the probability rate of reverse back to one particle ought to be 2δ, with a resulting bound onproduction of the form ln (/δ) − ln 2. To see why the ln 2 ought not to be included, it is important to recognize that particles in a classical physical system have locations that distinguish them from each other. Thus, we either can think of the process in question as bounding theproduction of self-replicationmixing of particles (in which case the mixingterm cancels the factor of 2) or else we can define our coarse-grained states in the transition in question so that we only consider the probability current for the reversion of the replicator that was just born. Either way, the relevant bound comes out to ln (/δ).

The first thing to notice here is that exponentialof the kind just described is a highly irreversible process: in a selective sweep where the fittest replicator comes to dominate in a population, the future almost by definition looks very different from the past. Happily, we are now in the position to be quantitative about the thermodynamic consequences of this irreversibility. Thus, let us suppose there is a simple self-replicator living at inverse temperature β whose population≫ 1 obeys a masterof the formwhere) is the probability of having a population ofat time, and> δ > 0. For simplicity, we assume that a decay event mediated by the rate parameter δ amounts to a reversion of the replicator back into the exact set of reactants in its environment out of which it was made, and we also assume that forming one new replicator out of such a set of reactants changes the internalof the system by an amount Δ, and on average puts out an amount of heat Δinto the surroundings. We will furthermore ignore in this discussion the additional layer of complexity that comes with considering the effects of spontaneous mutation in evolving populations.

The generalization of the Second Law presented above applies in a wide range of far-from-equilibrium, heat-bath-coupled scenarios, from the operation of computers to the action of molecular motors. The result turns out to be particularly valuable, however, as a lens through which to take a fresh look at the phenomenon of biological self-replication. Interest in the modeling of evolution long ago gave rise to a rich literature exploring theof self-replication for population dynamics and Darwinian competition. In such studies, the idea of Darwinian “fitness” is frequently invoked in comparisons among different self-replicators in a non-interacting population: the replicators that interact with their environment in order to make copies of themselves fastest are more “fit” by definition because successive rounds of exponentialwill ensure that they come to make up an arbitrarily large fraction of the future population.

On a related note, it may also be pointed out that a naïve objection might be raised to the above account ofon the grounds that its conclusions about the maximum possiblerate ofwould seem to disagree with the empirical fact thatis obviously capable of undergoing replication much more rapidly than one phosphodiester linkage per hour; the processive holoenzymepolymerase III is well-known for its blazing catalytic speed of ∼1000 base-pairs per second.The resolution of this puzzle lies in realizing that in thereplication scenario being raised here, the polymerase assembly is first loaded ontoin an ATP-dependent manner that irreversibly attaches the enzyme to the strand using a doughnut-shaped clamp composed of varioussubunits. Thus, while the polymerase can catalyze the elongation of thechain extremely rapidly, one also must take into account that theofhydrolysis should happen much more readily with an enzyme tethered to the strand than it does for isolatedin solution. This example therefore underlines the care that must be shown in how the coarse-grained statesandare defined in the computation of irreversibility.

The particular example of a singlebase ligation considered here is instructive because it is a case where the relationship between irreversibility andproduction further simplifies into a recognizable form. It is reasonable to define our coarse-graining of states so that the starting and ending products of thisare each in a local, thermal equilibrium, such that all of their degrees of freedom are Boltzmann-distributed given the constraints of their chemical bonds. In such a circumstance, detailed balance alone requires that the ratio of the forward and reverse rates be locked to the Gibbs' free energy change for thevia the familiar formula ln [] = −βΔ≡ −βΔ+ Δ= βΔ+ ΔIn this light, the relationship between durability andrate has an elegant description in terms of transition state theory: an activation barrier that is lower in the forward direction will be lower in the reverse direction as well. The key point here, however, is that whereas the relationship between free energy andrates obtains only under local equilibrium assumptions, the inequality we have derived here boundingproduction in terms of irreversibility applies even in cases where many degrees of freedom in the system start and end the replication process out of thermal equilibrium.

The key point here is that if a self-replicatingcatalyzes its ownwith a rate constantand is degraded with rate constant δ, then the molecule should be capable of exhibiting exponentialthrough self-replication with a doubling time proportional to 1/(− δ). However, thermodynamics only sets a bound on the ratio/δ, which means that− δ can be made arbitrarily large while the “metabolic cost” of replication remains fixed. Thus, surprisingly, the greater fragility ofcould be seen as a fitness advantage in the right circumstances, and this applies even if the system in question is externally driven, and even still if the replicators maintain non-Boltzmann distributions over their internal degrees of freedom. Moreover, we would expect that the heat bound difference betweenandshould increase roughly linearly in the number of bases ℓ ligated during thewhich forces the maximum possiblerate for areplicator to shrinkwith ℓ in comparison to that of itscompetitor in an all things equal comparison. This observation is certainly intriguing in light of past arguments made on other grounds thatand notmust have acted as the material for the pre-biotic emergence of self-replicating

To underline this point, we may consider what the bound might be if this samewere somehow achieved usingwhich in aqueous solution is much more kinetically stable against hydrolysis thanIn this case, we would have, which exceeds the estimated enthalpy for the ligationand is therefore prohibited thermodynamically. This calculation illustrates a significant difference betweenandregarding each molecule's ability to participate in self-catalyzed replicationfueled by simple triphosphate building blocks: the far greater durability ofdemands that a much higher per-base thermodynamic cost be paid inproductionin order for for therate to match that ofin an all-things-equal comparison.

A simple demonstration of the role of durability in replicative fitness comes from the case of polynucleotides, which provide a rich domain of application for studying the role ofproduction in the dynamics of self-replication.One recent study has usedevolution to optimize therate of a self-replicatingmolecule whose formation is accompanied by a single backbone ligationand the leaving of a single pyrophosphate group.It is reasonable to assume that the reverse of thisin which the highly negatively charged pyrophosphate would have to attack the phosphate backbone of thewould proceed more slowly than a simple hydrolysis by water. The rate of such hydrolysis has been measured by carrying out a linear fit to the early-time (20 day) exponential kinetics of spontaneous degradation for the RNAase A substrate UpA, and the half-life measured is on the order of 4 years.Thus, with a doubling time of 1 h for the self-replicator, we can estimate for this system that. Since the ligationtrades the mixingof substrate for that of pyrophosphate at comparable ambient concentrations, we can also assume in this case that the change in internalfor theis negligible. Thus, we can estimate the heat bound asSince experimental data indicate an enthalpy for thein the vicinity ofit would seem this molecule operates quite near the limit of thermodynamic efficiency set by the way it is assembled.

BACTERIAL CELL DIVISION Section: Choose Top of page ABSTRACT INTRODUCTION MACROSCOPIC IRREVERSIBILI... GENERAL CONSTRAINTS ON SE... SELF-REPLICATING POLYNUCL... BACTERIAL CELL DIVISION << CLOSING REMARKS REFERENCES CITING ARTICLES

Ligation of polynucleotides provides a relatively simple test case where tracking the formation of a new replicator can be reduced to monitoring the progress of a single molecular event (i.e., the formation of a phosphodiester linkage); however, the macroscopic relationship between irreversibility and entropy production applies equally in cases that are much more complex. Indeed, we shall now argue that these basic thermodynamic constraints are even relevant to the growth and division of whole single-celled organisms.

E. coli bacterium, immersed in a sample of rich nutrient media in contact with a heat bath held at the bacterial cell's optimal growth temperature ( 1 / β ≡ T ∼ 4.3 × 10 − 21 J ). 17,18 17. H. P. Rothbaum and H. M. Stone, J. Bacteriol. 81, 172 (1961). 20, 1099 (2010). 18. P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright, and S. Jun, Curr. Biol., 1099 (2010). https://doi.org/10.1016/j.cub.2010.04.045 cell is in exponential growth phase at the beginning of its division cycle, and that, while the volume and mass of the entire system are held fixed so that no particles are exchanged with any external bath and the walls of the container do not move, the composition and pressure of the nutrient media surrounding the bacterium mimics that of a well-aerated sample open to the earth's atmosphere. If we summarize the experimental conditions described above with the label I, we can immediately say that there is some probability p(i|I) that the system is found in some particular microstate i (with energy E i ) given that it was prepared in the macroscopic condition I by some standard procedure. Although this probability might well be impossible to derive ab initio, in principle it could be measured (at least in a thought-experiment) through lengthy experimental consultation with a microbiologist. We begin by considering the preparation of a large system initially containing a singlebacterium, immersed in a sample of rich nutrient media in contact with a heat bath held at the bacterialoptimaltemperature ().We can assume furthermore that theis in exponentialphase at the beginning of its division cycle, and that, while the volume and mass of the entire system are held fixed so that no particles are exchanged with any external bath and the walls of the container do not move, the composition and pressure of the nutrient media surrounding the bacterium mimics that of a well-aerated sample open to the earth's atmosphere. If we summarize the experimental conditions described above with the label, we can immediately say that there is some probability) that the system is found in some particular microstate(with energy) given that it was prepared in the macroscopic conditionby some standard procedure. Although this probability might well be impossible to derive, in principle it could be measured (at least in a thought-experiment) through lengthy experimental consultation with a microbiologist.

entropy production, we must consider the entropy changes associated with the bacterium's ingestions and excretions during growth and division to be part of the internal entropy change of the system ΔS int . The somewhat unexpected result is that if, for example, the bacterium subsisted on metabolic reactions that were accompanied by large increases in entropy (through, for example, anaerobic gas evolution 19 121, 517 (2006). 19. U. von Stockar, T. Maskow, J. Liu, I. W. Marison, and R. Patino, J. Biotechnol., 517 (2006). https://doi.org/10.1016/j.jbiotec.2005.08.012 entropy change for forming a new bacterium could actually be positive. Three preliminary points are in order as a result of the fact that the “system” being defined here is not only the bacterium, but also the surrounding broth containing the food it will eat and the oxygen it will breath, etc. First of all, looking ahead to our expression forproduction, we must consider thechanges associated with the bacterium's ingestions and excretions duringand division to be part of thechange of the system Δ. The somewhat unexpected result is that if, for example, the bacterium subsisted on metabolicthat were accompanied by large increases in(through, for example, anaerobic gas evolution), the total internalchange for forming a new bacterium could actually be positive.

entropy S i n t ≡ − ln p ¯ ≡ − ∑ p ln p and the familiar idea we have of entropy from equilibrium statistical mechanics is not obvious, and requires elaboration. In Boltzmann-distributed systems, it has long been clear that the internal Shannon entropy has a natural connection to the heat exchanged with the surroundings because of the set relationship between internal energy and probability. Only more recently has it been demonstrated that S int continues to obey a general thermodynamic relation for arbitrary nonequilibrium transitions: ΔS int ⩽ ΔQ ex /T. Here, ΔQ ex is the so-called excess heat, which measures the extra heat evolution on top of what would accompany the various dissipative steady-states traversed during the transition. 20 86, 3463 (2001). 20. T. Hatano and S. Sasa, Phys. Rev. Lett., 3463 (2001). https://doi.org/10.1103/PhysRevLett.86.3463 entropy remains the measure of “statistical disorder” in the system that it is for equilibrium systems. The easiest way to see this is consider a case of uniform starting and ending distributions, that is, where p(i|I) = p I and p(j|II) = p II for all states i and j that have non-zero probability in their respective ensembles. Clearly, in this case ΔS int = ln [p I /p II ] = ln [Ω II /Ω I ], that is, it simply measures how many more or fewer states there are in II than in I. Moreover, it is more generally the case that the Shannon entropy is effecting a log-scale comparison of volumes in phase space before and after the transition. Accordingly, the factors affecting this quantity far from equilibrium should be the same as they are near equilibrium, namely the number of different possible positions and velocities available to particles in the system when they are arranged to belong to the ensemble in question. Thus changes in partial volumes of gases are, for example, still relevant to the question of how this internal entropy has changed. Second, it must be pointed out that the connection between the internal Shannonand the familiar idea we have offrom equilibrium statistical mechanics is not obvious, and requires elaboration. In Boltzmann-distributed systems, it has long been clear that the internal Shannonhas a natural connection to the heat exchanged with the surroundings because of the set relationship between internal energy and probability. Only more recently has it been demonstrated thatcontinues to obey a general thermodynamic relation for arbitrary nonequilibrium transitions: Δ⩽ Δ. Here, Δis the so-called excess heat, which measures the extra heat evolution on top of what would accompany the various dissipative steady-states traversed during the transition.Perhaps more importantly, we expect that even far from equilibrium, the Shannonremains the measure of “statistical disorder” in the system that it is for equilibrium systems. The easiest way to see this is consider a case of uniform starting and ending distributions, that is, where) =and) =for all statesandthat have non-zero probability in their respective ensembles. Clearly, in this case Δ= ln [] = ln [Ω/Ω], that is, it simply measures how many more or fewer states there are inthan in. Moreover, it is more generally the case that the Shannonis effecting a log-scale comparison of volumes in phase space before and after the transition. Accordingly, the factors affecting this quantity far from equilibrium should be the same as they are near equilibrium, namely the number of different possible positions and velocities available to particles in the system when they are arranged to belong to the ensemble in question. Thus changes in partial volumes of gases are, for example, still relevant to the question of how this internalhas changed.

E. coli (which is not carrying out any sort of externally driven photosynthetic process) the propagator for the system π(i → j; τ) over any interval of time τ can be taken to obey Boltzmann distribution over microstates in which detailed balance holds. What makes the scenario of interest here a far-from-equilibrium process, then, is only that its initial conditions p(i|I) correspond to a highly non-Boltzmann distribution over microstates. The bacterial growth that takes place at the very beginning of the long process of relaxation to equilibrium is in this sense a mere transient, the fuel for which comes from chemical energy stored in the system's starting microscopic arrangement. By the time detailed balance sets in, we expect all bacteria initially in the system will have long since overgrown their container, starved, perished, and disintegrated into their constituent parts. Finally, it should be noted that for the case of a microbe such as(which is not carrying out any sort of externally driven photosynthetic process) the propagator for the system π(; τ) over any interval of time τ can be taken to obey (2) . Put another way, while the bacterium by itself might appear to be driven by external currents of chemical reactants and products, the system as a whole (which includes the nutrient media) is not driven at all, and simply exchanges heat with a surrounding thermal reservoir. Thus, quite counterintuitively, we expect that the eventual steady-state for this system (as for any system of fixed volume and particle number left in contact with a heat bath for an infinite amount of time) will be adistribution over microstates in which detailed balance holds. What makes the scenario of interest here a far-from-equilibrium process, then, is only that its initial conditions) correspond to anon-Boltzmann distribution over microstates. The bacterialthat takes place at the very beginning of the long process of relaxation to equilibrium is in this sense a mere transient, the fuel for which comes from chemical energy stored in the system's starting microscopic arrangement. By the time detailed balance sets in, we expect all bacteria initially in the system will have long since overgrown their container, starved, perished, and disintegrated into their constituent parts.

Now consider what would happen in our system if we started off in some microstate i in I and then allowed things to propagate for a time interval of τ div , the typical duration of a single round of growth and cell division. From the biological standpoint, the expected final state for the system is clear: two bacteria floating in the media instead of one, and various surrounding atoms rearranged into new molecular combinations (e.g., some oxygen converted into carbon dioxide). We can label the ensemble of future states corresponding to such a macroscopic outcome II. Given, that the system was initially prepared in I and did subsequently end up in II, any microstate j will have some finite likelihood which we can call p(j|II).

cell division introduced above, our ensemble II is a bath of nutrient-rich media containing two bacterial cells in exponential growth phase at the start of their division cycles. In order to make use of the relation in II → I), the likelihood that after time τ div , we will have ended up in an arrangement I where only one, newly formed bacterium is present in the system and another cell has somehow been converted back into the food from which it was built. Of course, cells are never observed to run their myriad biochemical reactions backwards, any more so than ice cubes are seen forming spontaneously in pots of boiling water. Nevertheless, it is implicit in the assumptions of very well-established statistical mechanical theories that such events have non-zero (albeit absurdly small) probabilities of happening, 1 1. C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed. ( Springer , 2003). can measure experimentally. This is possible because the rough physical features of the system are sufficient for making plausible estimates of thermodynamic quantities of interest; since we are ultimately interested in bounding the heat generated by this process, we are only concerned with the impact of the probabilities we estimate on a logarithmic scale; we would have to change our probability estimate by many orders of magnitude to see any effect on the corresponding heat bound. For the process of bacterialintroduced above, our ensembleis a bath of nutrient-rich media containing two bacterialin exponentialphase at the start of their division cycles. In order to make use of the relation in (8) , we need to estimate π(), the likelihood that after time τ, we will have ended up in an arrangementwhere only one, newly formed bacterium is present in the system and anotherhas somehow been converted back into the food from which it was built. Of course,are never observed to run their myriadbackwards, any more so than ice cubes are seen forming spontaneously in pots of boiling water. Nevertheless, it is implicit in the assumptions of very well-established statistical mechanical theories that such events have non-zero (albeit absurdly small) probabilities of happening,and these likelihoods can be bounded from above using the probability rates of events wemeasure experimentally. This is possible because the rough physical features of the system are sufficient for making plausible estimates of thermodynamic quantities of interest; since we are ultimately interested in bounding the heat generated by this process, we are only concerned with the impact of the probabilities we estimate on a logarithmic scale; we would have to change our probability estimate by many orders of magnitude to see any effect on the corresponding heat bound.

The first piece is relatively easy to imagine: while we may not be able to compute the exact probability of a bacterium fluctuating to peptide-sized pieces and de-respirating a certain amount of carbon dioxide and water, we can be confident it is less likely than all the peptide bonds in the bacterium spontaneously hydrolyzing. Happily, this latter probability may be estimated in terms of the number of such bonds n pep , the division time τ div , and the peptide bond half-life τ hyd .

grows at a rate of r = n pep /τ div peptide bonds formed per unit time. 18 20, 1099 (2010). 18. P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright, and S. Jun, Curr. Biol., 1099 (2010). https://doi.org/10.1016/j.cub.2010.04.045 peptide bonds will get synthesized in the normal course of growth, so that the probability of subsequently undergoing spontaneous disintegration of the whole cell in the remainder of the time τ div shrinks exponentially. Between these two countervailing effects there must be an optimal time that maximizes the probability of disintegration; thus, assuming each peptide bond spontaneously hydrolyzes independently from the others with probability rate ∼ τ h y d − 1 , we may therefore model the total cell hydrolysis probability p hyd in time t as ln p h y d ≃ ( n p e p + r t ) ln [ t / τ h y d ] . (13) This quantity is maximized for t max satisfying τ div /t max + ln [t max /τ hyd ] + 1 = 0, which we can compute numerically for chosen values of the two input timescales. Following this, we simply have to evaluate | ln p h y d | ≃ | n p e p ln [ t max / τ h y d ] | = n p e p ( τ d i v / t max + 1 ) . (14) We have now dealt with the demise of one of the cells. Handling the one that stays alive is more challenging, as we have assumed this cell is growing processively, and we ought not make the mistake of thinking that such a reaction can be halted or paused by a small perturbation. The onset of exponential growth phase is preceded in E. coli by a lag phase that can last several hours, 17 17. H. P. Rothbaum and H. M. Stone, J. Bacteriol. 81, 172 (1961). cell for rapid division fueled by the available metabolic substrates. 21 45, 289 (2002). 21. D. E. Chang, D. J. Smalley, and T. Conway, Mol. Microbiol., 289 (2002). https://doi.org/10.1046/j.1365-2958.2002.03001.x cell in question as an optimized mixture of components primed to participate in irreversible reactions like nutrient metabolism and protein synthesis. An added complication comes from the fact that the bacterium normallyat a rate of/τbonds formed per unit time.In order to make a model for the probability of a bacterium disintegrating over a set period of time, we need to specify when during that period it starts to disintegrate, and here two different factors collide. The less time we wait for the disintegration to begin, the less probable it is that it will happen in that period of time. However, the longer we wait, the more newbonds will get synthesized in the normal course ofso that the probability of subsequently undergoing spontaneous disintegration of the wholein the remainder of the time τshrinks exponentially. Between these two countervailing effects there must be an optimal time that maximizes the probability of disintegration; thus, assuming eachbond spontaneously hydrolyzes independently from the others with probability rate, we may therefore model the totalhydrolysis probabilityin timeasThis quantity is maximized forsatisfying τ+ ln [/τ] + 1 = 0, which we can compute numerically for chosen values of the two input timescales. Following this, we simply have to evaluateWe have now dealt with the demise of one of theHandling the one that stays alive is more challenging, as we have assumed thisisprocessively, and we ought not make the mistake of thinking that such acan be halted or paused by a small perturbation. The onset of exponentialphase is preceded inby a lag phase that can last several hours,during which gene expression is substantially altered so as to retool thefor rapid division fueled by the available metabolic substrates.It is therefore appropriate to think of thein question as an optimized mixture of components primed to participate in irreversiblelike nutrient metabolism andsynthesis.

div ) in the progression of these reactions is very small indeed: if each enzymatic protein component of the cell were to reject each attempt of a substrate to diffuse to its active site (assuming a diffusion time of small molecules between proteins of τ d i f f ∼ 10 − 8 s 22,23 90, 3835 (1993). 22. D. Brune and S. Kim, Proc. Natl. Acad. Sci. U.S.A., 3835 (1993). https://doi.org/10.1073/pnas.90.9.3835 27, 1158 (1988). 23. S. C. Blacklow, R. T. Raines, W. A. Lim, P. D. Zamore, and J. R. Knowles, Biochemistry, 1158 (1988). https://doi.org/10.1021/bi00404a013 p pause |∝|n pep (τ div /τ diff )| to exceed |ln p hyd | by orders magnitude. We must, however, consider an alternative mechanism for the most likely II → I transition: it is possible that a cell could grow and divide in an amount of time slightly less than τ div . 18 20, 1099 (2010). 18. P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright, and S. Jun, Curr. Biol., 1099 (2010). https://doi.org/10.1016/j.cub.2010.04.045 cell of the recent division were to spontaneously disintegrate back into its constituent nutrients (with log-probability at most on the order of ln p hyd ), we would complete the interval of τ div with one, recently divided, processively growing bacterium in our system, that is, we would have returned to the I ensemble. Thus, via a back-door into I provided to us by bacterial biology, we can claim that ln π ( II → I ) ≤ 2 ln p h y d ≃ − 2 n p e p ( τ d i v / t max + 1 ) . (15) Having obtained the above result, we can now refer back to the bound we set for the heat produced by this self-replication process and write β ⟨ Q ⟩ ≥ 2 n p e p ( τ d i v / t max + 1 ) − Δ S i n t . (16) This relation demonstrates that the heat evolved in the course of the cell making a copy of itself is set not only by the decrease in entropy required to arrange molecular components of the surrounding medium into a new organism, but also by how rapidly this takes place (through the division time τ div ) and by how long we have to wait for the newly assembled structure to start falling apart (through t max ). Moreover, we can now quantify the extent of each factor's contribution to the final outcome, in terms of n pep , which we estimate to be 1.6 × 109, assuming the dry mass of the bacterium is 0.3 picograms. 24 24. F. C. Neidhardt, E. coli and Salmonella: Cellular and Molecular Biology ( ASM Press , 1990), Vol. 1. We can therefore argue that the likelihood of a spontaneous, sustained pause (of duration τ) in the progression of theseis very small indeed: if each enzymaticcomponent of thewere to reject each attempt of a substrate to diffuse to its active site (assuming a diffusion time of small molecules betweenof), we would expect |ln|∝|(τ/τ)| to exceed |ln| by orders magnitude. We must, however, consider an alternative mechanism for the most likelytransition: it is possible that acouldand divide in an amount of time slightly less than τIf, subsequent to such an event, the daughterof the recent division were to spontaneously disintegrate back into its constituent nutrients (with log-probability at most on the order of ln), we would complete the interval of τwith one, recently divided, processivelybacterium in our system, that is, we would have returned to theensemble. Thus, via a back-door intoprovided to us by bacterial biology, we can claim thatHaving obtained the above result, we can now refer back to the bound we set for the heat produced by this self-replication process and writeThis relation demonstrates that the heat evolved in the course of themaking a copy of itself is set not only by the decrease inrequired to arrange molecular components of the surrounding medium into a new organism, but also by how rapidly this takes place (through the division time τ) and by how long we have to wait for the newly assembled structure to start falling apart (through). Moreover, we can now quantify the extent of each factor's contribution to the final outcome, in terms of, which we estimate to be 1.6 × 10, assuming the dry mass of the bacterium is 0.3 picograms.

E. coli bacterium growing at its maximum rate on lysogeny broth (a mixture of peptides and glucose) is β⟨Q⟩ = 220n pep . 17 17. H. P. Rothbaum and H. M. Stone, J. Bacteriol. 81, 172 (1961). entropy change for cell division to come from the equimolar conversion of oxygen to carbon dioxide (since carbon dioxide has a significantly lower partial pressure in the atmosphere), and from the confinement of amino acids floating freely in the broth to specific locations inside bacterial proteins. We can estimate the contribution of the first factor (which increases entropy) by noting that ln ( υ C O 2 / υ O 2 ) ∼ 6 . The liberation of carbon from various metabolites also increases entropy by shuffling around vibrational and rotational degrees of freedom, but we only expect this to make some order unity modification to the entropy per carbon atom metabolized. At the same time, peptide anabolism reduces entropy: by assuming that in 1% tryptone broth, an amino acid starts with a volume to explore of υ i = 100 n m 3 and ends up tightly folded up in some υ f = 0.001 n m 3 sub-volume of a protein, we obtain ln (υ f /υ i ) ∼ −12. In light of the fact that the bacterium consumes during division a number of oxygen molecules roughly equal to the number of amino acids in the new cell it creates, 17,25 17. H. P. Rothbaum and H. M. Stone, J. Bacteriol. 81, 172 (1961). 11, 269 (1969). 25. C. L. Cooney, D. I. Wang, and R. I. Mateles, Biotechnol. Bioeng., 269 (1969). https://doi.org/10.1002/bit.260110302 S int ⩽ 10n pep . The total amount of heat produced in a single division cycle for anbacteriumat its maximum rate on lysogeny broth (a mixture ofand glucose) is β⟨⟩ = 220We expect the largest contributions to the internalchange forto come from the equimolar conversion of oxygen to carbon dioxide (since carbon dioxide has a significantly lower partial pressure in the atmosphere), and from the confinement of amino acids floating freely in the broth to specific locations inside bacterialWe can estimate the contribution of the first factor (which increasesby noting that. The liberation of carbon from various metabolites also increasesby shuffling around vibrational and rotational degrees of freedom, but we only expect this to make some order unity modification to theper carbon atom metabolized. At the same time,anabolism reducesby assuming that in 1% tryptone broth, an amino acid starts with a volume to explore ofand ends up tightly folded up in somesub-volume of awe obtain ln (υ/υ) ∼ −12. In light of the fact that the bacterium consumes during division a number of oxygen molecules roughly equal to the number of amino acids in the newit creates,we can arbitrarily set a generous upper bound of −Δ⩽ 10

cell division time of 20 min, 17,18 17. H. P. Rothbaum and H. M. Stone, J. Bacteriol. 81, 172 (1961). 20, 1099 (2010). 18. P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright, and S. Jun, Curr. Biol., 1099 (2010). https://doi.org/10.1016/j.cub.2010.04.045 peptide bonds of τ hyd ∼ 600 years at physiological pH, 26 118, 6105 (1996). 26. A. Radzicka and R. Wolfenden, J. Am. Chem. Soc., 6105 (1996). https://doi.org/10.1021/ja954077c t max ∼ 1 min and 2n pep (τ div /t max + 1) = 6.7 × 1010 ≃ 42n pep , a quantity at least several times larger than ΔS int . We often think of the main entropic hurdle that must be overcome by biological self-organization as being the cost of assembling the components of the living thing in the appropriate way. Here, however, we have evidence that this cost for aerobic bacterial respiration is relatively small, 19 121, 517 (2006). 19. U. von Stockar, T. Maskow, J. Liu, I. W. Marison, and R. Patino, J. Biotechnol., 517 (2006). https://doi.org/10.1016/j.jbiotec.2005.08.012 reaction as it churns out copies that do not easily disintegrate into their constituent parts. In order to compare this contribution to the irreversibility term in (8) , we assume atime of 20 min,and a spontaneous hydrolysis lifetime forbonds of τ∼ 600 years at physiological pH,which yields a∼ 1 min and 2(τ+ 1) = 6.7 × 10≃ 42, a quantity at least several times larger than Δ. We often think of the main entropic hurdle that must be overcome by biological self-organization as being the cost of assembling the components of the living thing in the appropriate way. Here, however, we have evidence that this cost for aerobic bacterial respiration is relatively small,and is substantially outstripped by the sheer irreversibility of the self-replicationas it churns out copies that do not easily disintegrate into their constituent parts.

More significantly, these calculations also establish that the E. coli bacterium produces an amount of heat less than six times (220n pep /42n pep ) as large as the absolute physical lower bound dictated by its growth rate, internal entropy production, and durability. In light of the fact that the bacterium is a complex sensor of its environment that can very effectively adapt itself to growth in a broad range of different environments, we should not be surprised that it is not perfectly optimized for any given one of them. Rather, it is remarkable that in a single environment, the organism can convert chemical energy into a new copy of itself so efficiently that if it were to produce even a quarter as much heat it would be pushing the limits of what is thermodynamically possible! This is especially the case since we deliberately underestimated the reverse reaction rate with our calculation of p hyd , which does not account for the unlikelihood of spontaneously converting carbon dioxide back into oxygen. Thus, a more accurate estimate of the lower bound on β⟨Q⟩ in future may reveal E. coli to be an even more exceptionally well-adapted self-replicator than it currently seems.