Key biology you should have learned in physics class: Using ideal-gas mixtures to understand biomolecular machines
KKey biology you should have learned in physics class:Using ideal-gas mixtures to understand biomolecular machines
Daniel M. Zuckerman ∗ Department of Biomedical Engineering,Oregon Health & Science University, Portland, OR 97239 (Dated: March 25, 2020)
Abstract
The biological cell exhibits a fantastic range of behaviors, but ultimately these are governed bya handful of physical and chemical principles. Here we explore simple theory, known for decadesand based on the simple thermodynamics of mixtures of ideal gases, which illuminates severalkey functions performed within the cell. Our focus is the free-energy-driven import and export ofmolecules, such as nutrients and other vital compounds, via transporter proteins. Complementaryto a thermodynamic picture is a description of transporters via “mass-action” chemical kinetics,which lends further insights into biological machinery and free energy use. Both thermodynamicand kinetic descriptions can shed light on the fundamental non-equilibrium aspects of transport.On the whole, our biochemical-physics discussion will remain agnostic to chemical details, but wewill see how such details ultimately enter a physical description through the example of the cellularfuel ATP. a r X i v : . [ q - b i o . B M ] M a r . INTRODUCTION How many of us in the physics community have felt grateful we did not need to masterthe uncountable lists of special cases which seem to be the sum-total content of biologytextbooks? We physicists prefer a “beautiful science” based on just a few principles, fromwhich everything else can be derived with a bit of mathematics. But is it possible thereare just a few basic principles of biology that physicists can learn and use as a springboardto gain non-trivial insight into the field? Here, I’m not referring to the ultimate principleof evolution, but rather to well-understood physical principles, embodied in equations, thatunderpin so much of the function of biological cells. The idea is not original but builds onprior work, especially that of physicist Terrell Hill and numerous others. .This article will study protein “machines” and how they facilitate free energy-driventransport of molecules in and out of a cell. In essence, every protein performs a job rangingfrom catalysis (enzymes) to responding to environmental signals (receptors) to locomotion(motors). We will focus on transporters, which are membrane proteins that transportnumerous molecular substrates in or out of the cell, typically in conditions requiring freeenergy – i.e., in the direction opposite to which the substrate would flow spontaneously if achannel were available. We will address a number of questions: What are suitable elementarythermodynamic and kinetic descriptions of transport? What are the assumptions for thesedescriptions, and how are they justified? How can we understand transport in the contextof both equilibrium and non-equilibrium physics? How do coarser descriptions of a systemarise in principle from more fundamental microscopic theory? And finally, in a cell biologycontext, how far beyond transporters can we expect the simple approaches to be fruitful?In exploring transporters, we will largely leave aside details of the chemistry and struc-tural biology, but not the physical essentials. After all, proteins don’t “know” biology. Theindividual transporter is inanimate, merely a large molecule that obeys the laws of physicsand chemistry. Consistent with its physico-chemical nature, a protein is only capable offour actions: (i) binding to another molecule; (ii) catalysis of chemical reactions; (iii) confor-mational change of its shape; and (iv) diffusion or thermally driven passive motion. Each ofthese, especially (i) - (iii), have evolved to operate in chemically precise ways – e.g., bindingto only a small set of molecules or catalyzing a specific set of chemical reactions. Note thatconformational changes typically alter the function of a protein – e.g., opening or closing a2inding cavity or permeation channel.The real “magic” and power of protein machine-like behavior comes from the evolved coupling between two or more of the basic actions. In the class of transporter proteinsto be analyzed here, conformational changes are triggered by binding events. As sketchedin Fig. 1, if a protein has more than one conformational state, then binding to anothermolecule – generically called a ligand – can shift the free energy landscape to favor a differentconformation. Many different chemical mechanisms can cause such a shift, but a simpleexample would be if the ligand effectively glues together two previously floppy regions of theproteins, perhaps via favorable electrostatics.The coupling of binding and conformation changes enables transporters to function asmolecular machines which use an external source of (free) energy to do work. The source offree energy often is a gradient of an ion – more precisely, a difference in chemical potential ofthe ion across a membrane – or from an “activated” molecule like ATP . (As we will see inthe Discussion below, even the activation of ATP can only be understood in a thermodynamiccontext.) The work done by a transporter typically is to pump a substrate molecule to acellular compartment which represents “uphill” motion against its own chemical-potentialgradient.Although an exact theoretical description of transporter function would be highly com-plex, we can understand the essential ideas using very basic thermodynamics or kinet-ics. This is where the ideal gas comes in — following a long history in the field ofbiochemistry. We can describe the thermodynamics of transporter systems using thesimple equations for a mixture of ideal gases in which there is no potential-energy cost toswitch among the components. The transporter not only couples, say, the ion “gases” acrossthe membrane but also can enforce stoichiometric exchange with the gas of the substratemolecule being transported. That is, if we let A (subscripted as a ) represent the ion andB (subscript b ) the substrate, the transporter can be viewed as catalyzing the followingexchange across a membrane separating two arbitrary regions which we’ll name “in” and“out” for concreteness: A out + B out (cid:10) A in + B in . (1)A process like this where both molecules move in the same direction, as sketched in Fig.1, is called “co-transport” or “symport” and is exemplified by Na + ions (A), which havemuch higher extra-cellular concentration, being used to drive the import of glucose (B) or3nother sugar into the cell . Well-studied transporters using this mechanism include Mhp1which “salvages” metabolic precursors from the environment for a microbacterium andvSGLT which imports sugars in a flagellum-powered seawater bacterium . Numerous othertransporter processes occur, as discussed below, but we’ll focus solely on (1) for clarity.Our theoretical approach may seem cumbersome at first but will turn out to be fullytractable using undergraduate-level tools. To start, we consider the Helmholtz free energy F of an ideal mixture of all four components noted in (1): F tot ( N in a , N out a , N in b , N out b ; V in , V out , T ) (2)where N represents the number of particles of the indicated species and V gives the volumeof the subscripted compartment. Note that the volumes and temperatures are held constantthroughout; only the N values can vary. However, because of constraints implicit in ourmodel, in fact there is only one degree of freedom, not four, in Eq. (2)! First, we will assumethere are a fixed total number of each chemical species A and B. Second, the process (1)implies that number of inside and outside molecules change in a coupled way. In the end,as we will see below, the math is greatly simplified.In addition to the thermodynamic description, it is very valuable to consider the com-plementary time-dependent viewpoint of chemical kinetics implied by the “reaction” (1).After all, in nature, it is dynamical, microscopic processes which lead to macroscopic ther-modynamics and not the other way around. The most common description in chemistryand biochemistry, known as “mass action” kinetics and originated in the 1800s , providesa precise dynamical analogue to the ideal gas because molecular reactants are considered tobe non-interacting except for their possibility to transform into the likewise non-interactingproducts. This ideality is, of course, an approximation but such a useful one that it is es-sentially unquestioned throughout biochemistry . The explicitly non-equilibrium nature ofa simple kinetic description will echo insights gained from a free-energy picture.In the context of the physics education literature, the present contribution rests on severalpoints. First, the author is unaware of a pedagogical, physics-based treatment of biochemi-cal transport in the literature. Second, the dual thermodynamic/kinetic perspectives offer amodel for analyzing other (bio)physical processes. Third, the material is presented in a man-ner consistent with mainstream biochemistry, including notation as appropriate, providingthe reader a toe-hold into the primary literature of that field.4he remainder of this paper will start by describing biological transporters in sufficientdetail to motivate the statistical physics description that follows. The complementary kineticdescription will be given and shown to be consistent with the thermodynamics. Finally, thediscussion section will provide important connections to other biological processes. Appen-dices delve into more advanced aspects of the formulation and suggest further undergraduate-level calculations of interest. The principles described here in the context of transport arequite general and lend insight into numerous biological processes . II. TRANSPORTERS IN CELL BIOLOGY
To set transporters somewhat more broadly in the context of cell biology, not only arethere transporters to import every kind of nutrient into the cell but there are numerousion-only transporters geared toward maintaining the transmembrane electrostatic potentialand the well-regulated balance among different ion species in the cell. Transporters are alsocritical in inter-nerve cell communication, not only mediating electrostatic “action poten-tials” but also vital to absorbing neurotransmitters from synapses. In short, transportersare not a detail of cell biology, but fundamental. They are the object of extensive currentresearch.We focus here on “secondary active transporters,” which means the free energy drivingthe transport is derived from the “gradient” of an ion across a membrane. As noted above,these transporters can use the difference in chemical potentials of the ion inside and outsidethe cell (or cellular compartment) to power the transport of a specific substrate moleculeagainst its gradient — from low to high chemical potential. The full chemical potentialincludes all chemical and physical factors, including electrostatics, but it will not provenecessary to delve into these details here. Chemical specificity for a given ion and substrateare provided by the particular amino acids in the transporter proteins which tend to providea fully complementary shape and electrostatic environment. . Roughly speaking, there is adifferent transporter protein or protein complex for each substrate molecule, although thereare notable exceptions. . In a thermal environment, unsurprisingly, neither the ligandspecificity nor the coupling efficiency of a transporter is perfect, but we shall put offconsidering those aspects until the Discussion section.5 II. BASIC THERMODYNAMICS OF TRANSPORT
To elucidate the principles of transporter function, we will pursue nearly exact treat-ments of simple models - a mixture of ideal gases and subsequently mass-action kinetics.The essence of the ideal-gas theory is long-established in the physics community and broadlyaccepted (if only implicitly) in the field of biochemistry. Mass-action kinetics are the cor-nerstone of quantitative biochemistry and represent an intuitive physical approach, aswill be seen. The author hopes the typical physicist or biochemist will encounter somethingnew, and hopefully informative, in the combination of approaches to be employed here.Why are we justified in using ideal-gas theory? As is often the case when digging intoa real-world problem, we start with the simplest sufficient model. You will see that theideal gas provides just that. But less obviously, there are good theoretical reasons to expectsome insensitivity to details of molecular interactions. These points are elaborated upon inAppendix A and are of great importance to readers who want to go a bit deeper into thebiophysics.To tackle transporters, we need to treat two types of molecules A and B, each of whichcan be inside or outside the cell. As suggested by the free energy (2), in the ideal-gaspicture, we must account for four independent “gases.” But of course, these components arenot truly independent because transit of an A particle from outside to inside simultaneouslychanges the A counts in both compartments, not to mention the coupling to B transit via(1). This coupling can be fully accounted for in a simple additive formulation — which weshall derive from the full-system partition function later on, for completeness.
A. Refresher: Defining equilibrium, configurations, and states
Equilibrium is a key reference point in understanding any aspect of statistical physics orbiochemistry, even if our main goal is to understand non-equilibrium phenomena. As we’llsee below, equilibrium theory will enable us to understand almost all of the free-energeticsof our system.To discuss equilibrium, we first need some nomenclature. A configuration is defined tobe a single point in phase space – i.e., the set of all particle positions and momenta for aclassical system . On the other hand, a state is taken to be a collection of configurations,6hich can be defined in almost any way that is convenient although not all definitions areequally useful. Somewhat confusingly, a “steady state” does not refer to a chosen set ofconfigurations but rather to a constant-in-time distribution of all configurations.Equilibrium is a special steady state, defined by more than time-invariant properties.
Not only must the population of each species (e.g., B outside the cell) and its spatial distri-bution be constant in time, but further there must be no flows in the system, on average.That is, if we observe the system over a long period of time, although a given species mayget transported numerous times, there should be an equal number of transits in each di-rection. Likewise, although molecules may diffuse spatially, there should be no “rivers” ofuni-directional motion. All this can be encapsulated in the notion of detailed balance , whichmeans an equal and opposite average flow of particles/probability per unit time betweenany pair of configurations. If you think about it, detailed balance implies unchanging prob-abilities because each configuration gains as much as it loses. Note that if detailed balanceholds among “microscopic” configurations, then it also must hold among coarser states. In equilibrium, the probability of a system configuration is governed by its Boltzmannfactor (see below) a property which is critical to defining the partition function and the freeenergy.
B. Refresher: A single ideal gas
To introduce some notation and remind ourselves of ideal-gas statistical mechanics, let’sstart from the partition function for an equilibrium system of N classical, non-interactingparticles of mass m at temperature T confined to a fixed volume V . A partition function isjust a sum and/or integral over the equilibrium Boltzmann factors for every possible systemconfiguration defined by positions and velocities: consult your favorite statistical mechanicsbook for reference (e.g., Refs. 14, 22, and 23).Fortunately, the ideal gas partition function can be written down and evaluated easily.Because ideal particles experience no forces from one another or the container walls, thepotential energy is a constant (taken to be zero) independent of the positions of the particles.Hence the total energy needed for the partition function is solely kinetic. Letting r i =( x i , y i , z i ) denote the position of particle i and analogously defining v i as the velocity vectorwith magnitude v i , we can write the classical ideal gas partition function and evaluate the7ntegrals exactly yielding Z idl ( N, V, T ) = 1 N ! (cid:16) mh (cid:17) N (cid:90) d v · · · d v N (cid:90) d r · · · d r N exp (cid:34) − N (cid:88) i =1 (1 / m v i (cid:14) k B T (cid:35) = 1 N ! (cid:18) Vλ (cid:19) N , (3)where h is Planck’s constant, k B is Boltzmann’s constant, and λ ( T ) = h/ √ πmk B T .The Helmholtz free energy is then obtained as F idl ( N, V, T ) = − k B T ln Z idl (cid:39) k B T (cid:2) N ln N − N − N ln( V /λ ) (cid:3) (cid:39) N k B T ln (cid:18) NV /λ (cid:19) , (4)where we have employed Stirling’s approximation as usual. Importantly for biochem-ical applications, note that the fundamental dependence here is F idl /N ∼ ln( N/V ) — i.e.,the log of the number density
N/V , also called concentration . We shall be assuming constanttemperature throughout, appropriately for biochemistry, so the temperature dependence isnot directly pertinent.
C. The simplest mixture: Two ideal gases exchanging across a membrane
As a second step toward modeling a biological transporter, we’ll increase the complexity ofour system incrementally by considering two separated ideal gases of the same molecule type(A) which can exchange particles across a membrane, as in Fig. 2. To suggest a connectionwith cell behavior, we’ll call the two compartments “inside” and “outside” with volumes V in and V out as well as corresponding particle numbers N in a and N out a constrained to sum to N tot a . We can explicitly write out the total free energy as the sum of the two ideal-gas freeenergies from (4); the additivity is justified in Appendix B. We obtain F tot a = N in a k B T ln (cid:18) N in a V in /λ (cid:19) + N out a k B T ln (cid:18) N out a V out /λ (cid:19) = N in a k B T ln (cid:18) N in a V in /λ (cid:19) + ( N tot a − N in a ) k B T ln (cid:18) N tot a − N in a V out /λ (cid:19) (5)= F tot a ( N in a ; N tot a , V in , V out , T ) , where we used N out a = N tot a − N in a . Because N tot a , V in , V out , T are considered constants, thisfree energy has only a single adjustable “parameter” N in a as shown in Fig. 2.8ur system will self-adjust, via in-out exchange of A particles, until a free energy min-imum is reached. It is straightforward to calculate this equilibrium point by setting thederivative of F tot a to zero. We first find that dF tot a dN in a = k B T (cid:20) ln (cid:18) N in a V in /λ (cid:19) + 1 (cid:21) − k B T (cid:20) ln (cid:18) N tot a − N in a V out /λ (cid:19) + 1 (cid:21) , (6)where we have used the total, not partial, derivative notation here because we have explicitlyincluded all dependence on N in a . Setting the derivative to zero then yields N in a V in = N tot a − N in a V out = N out a V out (at equilibrium) , (7)which is a condition of equal inside and outside concentrations. This result should not besurprising, since there is no driving force or interaction favoring inside vs. outside. Nev-ertheless, our calculation illustrates a generally useful procedure: we can usefully combineideal-gas free energies, write them in terms of a single parameter, and then minimize theresult to find the equilibrium point. We can repeat these steps in more complicated scenarios.There is still one very important bit of physics we should take away from this calculation.In particular, the principle that the free energy is minimized with respect to an adjustableparameter ( N in a , in our case) immediately tells us that the free energy is higher at (non-equilibrium) N in a values which do not satisfy (7). That is, there is usable energy availablewhen the system is away from the minimum but none is available at the minimum—i.e., atequilibrium—itself, so long as N tot a , V in , V out , T are held constant. See Fig. 2. After all, thefree energy literally means the energy available to do work , but more precisely it is theavailable energy referenced to the minimum accessible value.As we will see below, if different components of a system are suitably coupled, the freeenergy of one component can drive work done on the other. Active transport across amembrane is the perfect case in point! Biological cells have developed myriad ways of“transducing” free energy to perform the tasks necessary for life , and the Discussion willsketch a few examples beyond transport. D. Transport thermodynamics: A four-part mixture of particle-exchanging idealgases
We are now ready to construct the ideal thermodynamics for the 1:1 biological transporterexecuting the process (1). Although there are four components in the free energy (2) to9ccount for both species inside and outside the cell, our calculations turn out to be quitesimple. For mathematical convenience, we will make several simplifications which do notaffect the key physics we wish to understand. We assume that both species, A and B, havethe same total number of particles, N tot a = N in a + N out a = N tot b = N in b + N out b , (8)and that each species has the same mass, so that λ a = λ b = λ . We will enforce thestoichiometric coupling (1) of our transporter—that one and only one of each type of particleis transferred at one time, in either direction—by constraining the particle counts inside andoutside to change in lockstep N in b = N in a + N , (9)where N is an offset, akin to an initial condition, that ultimately will prove critical tounderstanding the biological transporter. The preceding assumptions are very convenientand will not change the fundamental conclusions in any way.Once again, we write the full free energy (2) as the sum of individual ideal-gas expressions(4). Because of the constraints just noted, F tot will depend only on a single adjustableparameter, which we choose to be N in b because it will most directly help us understandtransport. (We could have chosen any of the other three particle numbers: there is no effecton the final result.) Summing the free energies based on Appendix B and substituting basedon constraints, we have F tot = N in a k B T ln (cid:18) N in a V in /λ (cid:19) N out a k B T ln (cid:18) N out a V out /λ (cid:19) + N in b k B T ln (cid:18) N in b V in /λ (cid:19) + N out b k B T ln (cid:18) N out b V out /λ (cid:19) = ( N in b − N ) k B T ln (cid:18) N in b − N V in /λ (cid:19) + ( N tot b − N in b + N ) k B T ln (cid:18) N tot b − N in b + N V out /λ (cid:19) + N in b k B T ln (cid:18) N in b V in /λ (cid:19) + ( N tot b − N in b ) k B T ln (cid:18) N tot b − N in b V out /λ (cid:19) = F tot ( { N i } , V in , V out , T ; N in b ) , (10)where { N i } is shorthand for the full set of particle variables. The adjustable parameter N in b is set off from others to remind us of its importance.10ollowing the procedure of the preceding section, we minimize F tot with respect to theadjustable parameter. Differentiation yields dF tot dN in b = k B T (cid:20) ln (cid:18) N in b − N V in /λ (cid:19) + 1 − ln (cid:18) N tot b − N in b + N V out /λ (cid:19) −
1+ ln (cid:18) N in b V in /λ (cid:19) + 1 − ln (cid:18) N tot b − N in b V out /λ (cid:19) − (cid:21) . (11)We obtain a deceptively simple but key result by setting the derivative to zero, and substi-tuting for more interpretable { N i } variables: N in a /V in N out a /V out = N out b /V out N in b /V in . (12)Quite simply, this relation implies that if we increase the outside concentration of A (our“driving” molecule) then the system tends to an equilibrium with higher inside concentrationof B (the “substrate” we hope to see transported). This is the thermodynamic signatureof a pump — a.k.a. a biochemical transporter! We have obtained a simple physics result,arguably one that is obvious in retrospect since the coupling (1) means that A and B willmove together, but it is of profound importance in biochemistry. Further, as will be describedin the Discussion, the analysis presented here is paradigmatic for biochemical processes thatare much more challenging to intuit.We can re-frame the preceding comments on driving in terms of stored free energy, build-ing on our initial discussion in Sec. III C. The minimum free energy condition (12) representsequilibrium, and if we displace the system away from this minimum the system will tendto move back toward it. Such driving can be seen as a consequence of the system having ahigher free energy, above the minimum. That excess energy can be used to do work, namely,pumping B molecules from outside to inside — a process which will occur even if the insideconcentration of B already exceeds the outside concentration of B! Pumping will occur, forexample, if the left-hand ratio of A concentrations in (12) starts at a value of 1 /
10 while theright-hand B ratio starts at 1 /
2. Based on the transporter’s 1:1 coupling (1), both A and Bmolecules will move from outside to inside until the ratios of (12) match.In the bigger picture, given the simple physics involved, we should avoid the mistakennotion that the biochemistry of transporters is trivial. In fact, we have taken the non-trivialfor granted in our whole development. The “magic” of the biochemistry lies firstly in thefunction of a protein (or protein complex) which actually enforces the reaction/condition (1).Second, the cell continually uses its energy resources to maintain a highly non-equilibrium11radient of sodium ions across the plasma membrane, effectively a battery . Once these highly non-trivial features are arranged, it is fair to say the rest is simple.On a more technical level, note that the equilibrium result (12) is not generally the sameas the prior equilibrium finding (7) applied to both A and B species separately. Because thetransporter enforces stoichiometric movement of particles, (12) is a constrained equilibriumpoint , rather than a global equilibrium, as explained further in Appendix C. IV. KINETIC DESCRIPTION OF TRANSPORT
We can gain a deeper understanding of transport, which fundamentally is a non-equilibrium phenomenon, using a chemical-kinetics description. This will not involveany chemical details or structures of biomolecules, but rather the simplest possible time-dependent description of transport via basic differential equations. The approach we take iscompletely standard and is sometimes called a “mass-action” description which refersto the simple concentration dependencies assumed for transition probabilities.Mass-action kinetics, as we will see, are a fairly precise analog of ideal-gas thermody-namics in the sense that both assume particles are non-interacting and both lead to thesame equilibrium point. However, the kinetic picture assumes a “reaction” (transport, inour case) probability per unit time that depends on the product of concentrations of any“reactants”. In other words, the particles don’t interact ... until they do. Further confirma-tion for the ideal-gas/mass-action relationship comes from analyzing our kinetic descriptionat its stationary point, which yields the same relationship for the equilibrium concentrationas was derived from the free energy picture.The starting point for a kinetic description will be the “reaction” (1) performed by ourtransporter, which we repeat here for convenience:A out + B out (cid:10) A in + B in . (1)The mass-action formulation quantifies the time dependence of concentrations (number den-sities) of the chemical species, which will be characterized via biochemical notation,[ X ] = N X /V , (13)in molar (M) units. We also require the forward and reverse reaction rate constants for (1),12 io and k oi , which are reaction probabilities per mole per unit time — and have units of(M s) − . Rate constants are independent of both time and concentration, by assumption.We are almost ready to write down the key equation. First note that in the mass-actionpicture, the overall rates for the two directions of the “reaction” (1) are[A out ][B out ] k oi (out → in) [A in ][B in ] k io (in → out) , (14)where you should note the distinction between overall rate and rate constant. These ex-pressions can be understood intuitively. . Consider first a single A molecule in a fixedvolume V , along with N b fully independent B molecules. The probability of an A-B en-counter is proportional to to N b /V = [B]. If we now include a total of N a A molecules,that encounter probability will increase by a factor of N a /V = [A], at least in the limit ofsmall N a . The mass-action assumption is that the overall reaction probabilities (per mole)are of the simple product form (14) regardless of the A and B concentrations. Thus, thereaction probability can always increase regardless of whether, for instance, B molecules arealready fully surrounded/caged by A molecules in a spatially realistic picture — an assump-tion which essentially mirrors the non-interacting nature of ideal particles which can occupythe same location without energetic cost. Also implicit in the mass-action expressions (14)is an isotropic assumption that the concentrations remain uniform in space – i.e., that anyspatial fluctuations rapidly dissipate via diffusion.The governing mass-action kinetic equation for a given species then reflects the differencebetween the overall forward and reverse rates (14). We will focus on the “substrate” B in because biochemical pumping should increase this concentration under cellular conditions: d [B in ] dt = [A out ][B out ] k oi − [A in ][B in ] k io . (15)The first term on the left is the rate of “formation” of B in in the mass-action formulation,while the second term is the rate of removal/destruction. Eq. (15) is the only differentialequation needed for our system because the analogous equations for the other three speciescan readily be derived from (15) based on the relationships (8) and (9) between the speciescounts. For example, d [A in ] /dt = d [B in ] /dt and d [B out ] /dt = − d [B in ] /dt .Note that a chemical-kinetics equation such as (15) is a deterministic and averaged de-scription, which is sufficient for many purposes such as ours. However, the actual behaviorwill be stochastic, and a given system is not expected to precisely follow the average. Such13uctuating outcomes will instead be governed by a chemical master equation, as discussedin Appendix D.We turn to the stationary (steady-state) behavior of (15), which will turn out to constrainthe rate constants. Setting the time derivative to zero and re-arranging terms leads to arelation among the steady-state concentrations:[A in ][A out ] = [B out ][B in ] k oi k io (at steady state = equilibrium) . (16)Although a steady-state does not necessarily correspond to equilibrium, in our case it doesbecause there are no inputs or outputs of energy or matter to our system .It turns out we implicitly have more information about the ratio of rate constants occur-ring in (16). We have assumed our A and B particles are non-interacting and further do notexperience any external field (e.g., electrostatic) which might discriminate inside vs. outside.(Had there been such a field, there would have been an energy term for it in our free energyformulation.) Therefore the equilibrium point cannot favor inside or outside and we must k io = k oi . In a kinetic picture, this means the transporter does not favor one direction overthe other, consistent with out ideal gas perspective.Once we recognize that k oi /k io = 1 by our prior assumptions, we see that (16) is equiv-alent to our previous result (12) derived thermodynamically. This helps to confirm thehypothesized relationship between mass-action kinetics and ideal-gas thermodynamics.Although our differential equation (15) naturally allows examination of transient behav-ior, that will not be our focus here. We’ll simply point out that if the system is initiatedaway from its steady state, it will relax toward that steady state over time. See Fig. 4. Therelaxation will be exponential in a simple system like ours.In the context of biochemical transport, which may reasonably be considered to occurat steady state, it is very instructive to consider non -equilibrium steady states driven byprocesses external to our system. In particular, for transporters, the driving A molecule (of-ten an ion) is generally maintained far from the equilibrium point it would attain uncoupledto B because it is continually pumped out of the cell using free energy, described below,from ATP hydrolysis . Most precisely, we can say that in a cellular context, the chemi-cal potential of A is much higher outside than inside, so there is a thermodynamic driving“force” on A in the out → in direction. In general, the chemical potential depends on every-thing in a molecule’s environment, including electrostatics and van der Waals interactions.
14n our ideal system with no interactions, however, only the species concentrations affect thechemical potential, so we model a driving force by assuming the outside concentration of Agreatly exceeds the inside value: [A out ] (cid:29) [A in ].What happens to B when there is a driving force on A? The answer is intuitive: becauseB transport is coupled to A via (1), then B will also be driven from outside to inside thecell. The key point is that this can occur even when B is driven against its own gradient —from lower to higher chemical potential. This driving is readily quantified by returning tothe fundamental differential equation, armed with the knowledge that k io = k oi . Based on(15), [B in ] will increase whenever right-hand side is positive:[A out ][A in ] > [B in ][B out ] . (17)Thus, if molecule A is sufficiently far from its own equilibrium of equal concentrations (7), itcan drive B from low to high concentrations. This is the essence of gradient-driven transport,and is easily appreciated simply based on the sign of the time-derivative for the species ofinterest. V. DISCUSSIONA. Yes, physics matters
The primary goal of this article, broadly speaking, is to introduce a physics-trained audi-ence to essential cell biology concepts framed strictly using undergraduate-level physics. Thetake-home message should be that physics is essential to understanding cell biology, a pointthat has long been appreciated at least implicitly by subsets of the biological community —e.g., the fields of biochemistry , bioenergetics , and some cell biology authors . Advancedphysics is not required to understand some of the most important phenomena, and furtherexamples are given below. The humble ideal gas has great power in the right context.At the same time, some topics which are under-emphasized in typical undergraduate,and even graduate, physics curricula have been featured. These include: (i) the value ofreciprocal kinetic and thermodynamic descriptions; (ii) the fundamental importance of non-equilibrium (NE) phenomena and the ease with which NE basics can be presented; and (iii)insight into the meaning and approximation of that taken-for-granted phrase, “free energy15inimization.” In other words, the application of familiar ideas to a new problem can deepenour understanding of old material.By no means is this article intended to be a survey or overview of the importance ofphysics in understanding biology, nor a presentation of the most interesting biology one canunderstand with physics. Far from it. The hope was to go deep enough into a single problemfor readers to appreciate that there is a deep and substantial role for physics in biologicalstudy. However, it’s worth considering which additional problems can be addressed with thesimple ideas discussed here. B. Beyond simple co-transport
We have focused our attention on a 1:1 symporter, or co-transporter, which carries outthe process (1), but the cell uses many variations on this theme. Other transporters fall intothe class of “antiporter” or exchanger, which generate a contrasting process:A out + B in (cid:10) A in + B out . (18)The treatment of 1:1 antiport is analogous to our analysis above. In both symporters and antiporters, different stoichiometries occur, so that two ions (A)might be required to transport one sugar molecule (B), for instance. All such transporters,which employ free energy stored in the inside-outside chemical potential difference, arecalled “secondary active transporters” to distinguish them from “primary” transporters thathydrolyze ATP to perform transport. Primary active transporters may also involve multiplesubstrates in different stoichiometries .Beyond stoichiometric variation, there is a growing awareness that transporters may notalways function in simple stoichiometric fashion or by simple mechanisms.
That is, theratio of substrate to ion (B to A) molecules moved per transport cycle may not need to be aninteger. Mechanistically, this likely results from the “slippage” phenomenon quantified byphysicist Terrell Hill in his seminal book. In other words, in a detailed map of the networkof possible processes occurring within a transporter, some may result in apparently futileleakage of either substrate or ion down its electro-chemical gradient. Slippage, as well as thequestion of mechanistic heterogeneity, are active research topics.It is fair to say there are processes far more remarkable than transport occurring in the cell16hich can be modeled using a straightforward physical approach. Perhaps the most excitingis a phenomenon called “kinetic proofreading” (KP) which was independently discovered bya physicist and a biochemist . It is fair to say that KP is one of the fundamental “secretsof life” , but unfortunately remains too much of a secret: it is not a textbook subject, andis little known in either the biological or physical communities.Quite simply, KP can be described as a generic strategy of using free energy to preserveinformation, or more precisely, to achieve higher biochemical discrimination than would bepossible without the extra energy use. For example, KP is what permits our cells to translateproteins from mRNA with an error rate of about 10 − instead of 10 − . Without it, youwould not be reading this article; our species could not exist. KP can be understood usingundergraduate-level physics akin to what is described above , and it also has receivedmore general physics treatments. This is a great topic for anyone seeking to delve deeperinto physical biology.
C. Last word: Chemical details and the example of ATP free energy
Adenosine tri-phosphate (ATP, Fig. 5) is surely one of the most important and mostmisunderstood molecules. It plays a key role in transport, as the driver of a wide class of“primary” active transporters (which are not driven by ion gradients). We learn in highschool that ATP is the “fuel” of the cell, which is roughly true but somewhat misleading.Our earlier discussion was somewhat more precise in referring to ATP as “activated” ,but we should understand the physics of this.Like every other molecule and process in the cell, ATP must obey the laws of thermalphysics. We can use our ideal-gas picture to illustrate the activation of ATP quantitatively.ATP provides free energy by its hydrolysis reaction, which simply means water is necessaryfor its decomposition: ATP + H O (cid:10) ADP + P i (19)where ADP is adenosine di-phosphate and P i is the separated inorganic phosphate. Althoughthis reaction is sometimes shown as uni-directional, proceeding from ATP to ADP only,every chemical reaction is reversible. For simplicity, we’ll omit water and phosphate fromour analysis, which won’t affect our conclusions; it is straightforward to include them ifdesired. 17he reaction (19) is extremely slow in the absence of a suitable catalyst, which biologi-cally is very important. If ATP hydrolysis happened rapidly in solution, the reaction wouldquickly reach its equilibrium point and ATP would no longer store free energy — see below.In a biological context, both directions of (19) only occur in the presence of a biologicalcatalyst, typically an enzyme, thus facilitating the coupling of hydrolysis to useful work,such as transport, biochemical synthesis, or locomotion. The chemical details are buried in the rate constants, which we will call k td and k dt ,respectively, for the forward and reverse directions of (19). To gain some insight, we writedown the mass-action equation for ATP, omitting water and phosphate for simplicity: d [ATP] dt = k dt [ADP] − k td [ATP] , (20)which has an equilibrium point [ADP][ATP] = k td k dt . (21)Because of the proximity of the charged phosphate groups in ATP, as shown in Fig. 5, it isintuitively expected that this equilibrium will greatly favor ADP, which is indeed the case.In our simplified description (20), this means that k td (cid:29) k dt . This great imbalance is dueto the chemical details.The “activation” of ATP is not due to the tendency for hydrolysis per se but ratherbecause the reaction (19) is kept so far from equilibrium in the cell. That is, ATP doesnot intrinsically store free energy. After all, without external input of energy, the reaction(19) will go to equilibrium — and no free energy will be stored, as in our discussion of idealgases and transporters. Instead, the cell continually uses energy from the metabolism ofglucose to synthesize ATP, in turn making the cellular concentration ratio much smallerthan the equilibrium point (21). It is in this sense that ATP is activated; it is significantlydisplaced from equilibrium.
In equilibrium, by contrast, no free energy is stored regardlessof chemical details.In sum, ATP cannot be understood without physics, but that physics is very basic andaccessible. 18
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Appendix A: Justifying the ideal gas model
Why are we justified in using a “gas” formulation in the first place when our particles(molecules or ions) are embedded in aqueous solution? This is an approximation, of course,but what has been assumed? First, it is legitimate to focus on only a subset of molecules(omitting water, for example) so long as we correctly account for excluded degrees of freedomvia effective interactions, governed formally by the potential of mean force (PMF) . Soour approximation is that the PMF among particles of interest is constant, with zero inter-particle force. But if some of our particles are (charged) ions, is this reasonable? Oneargument is that we are only attempting to learn qualitative features of these transportersystems. Thus we follow the usual “spherical cow” physics strategy.From a more fundamental physics point of view, the fact is that “integrating out” in-termediary solution molecules significantly decreases effective/PMF interactions in typicalcases. No doubt you are already familiar with the high dielectric constant of water, (cid:15) ≈ q and q separated by a distance r are21laced in water, the effective Coulomb energy of interaction changes from q q /r to q q /(cid:15)r ,decreasing by almost two orders of magnitude. In fact, the direct Coulomb interaction be-tween the charges does not change at all, but rather the additional interactions betweenthe charges and water reduces the average force. If water electrostatics and conformationalmotions were included explicitly, using (cid:15) = 1 would lead to the same observed behavior —namely, forces weakened by a factor of (cid:15) — among the non-water charges. The phenomenon of electrostatic screening resulting from a mixture of positive and nega-tive mobile ions is even more dramatic. Excess ions exponentially damp Coulombic interac-tions, fundamentally breaking the “long-ranged” inverse-distance dependence.
The ap-proximate Debye-H¨uckel potential behaves as q q e − κr /r with a “screening length” 1 /κ ∼ Appendix B: Partition-function derivation of the mixture free energy
In our analysis, we assumed the free energies for individual components were simply ad-ditive terms in the total free energy. This is not exactly true. Here we examine the approxi-mation which was implicitly made and the consequences, which turn out to be insignificant.Ultimately, the conclusions we have drawn are completely accurate in a qualitative senseand even quantitatively reasonable. Certainly the additivity assumption for the componentfree energies is no worse than assuming non-interacting molecules in the first place!We will consider a two-component system because that is sufficient to understand theissues at play. Specifically, we’ll restrict ourselves to a system with N tot a A molecules whichcan freely exchange among inside and outside compartments — the same setup and notationas was considered in Sec. III C.To write the partition function, first recall that the partition function generally is a22um/integral over the Boltzmann factor of all possible configurations of the system.
In our case, with two ideal gases in separate compartments, we not only have to sum overthe coordinate and momentum degrees of freedom as usual, but also over the discrete statesrepresented by different occupancy numbers N in a and N out a of inside vs. outside compart-ments. That is, we really have a sum over the full-system partition functions Z ( N in a , N out a )for every pair of values N in a and N out a = N tot a − N in a . We therefore write Z tot a ( N tot a , T, V in , V out ) = N tot a (cid:88) N in a =0 Z ( N in a , N out a , V in , V out , T )= N tot a (cid:88) N in a =0 Z idl ( N in a , V in , T ) Z idl ( N tot a − N in a , V out , T ) (B1)where we used the fact that the partition function of two independent systems is simply theproduct of the individual partition functions, which follows from the factorizability of theBoltzmann factor for independent coordinates. Substituting from (3) for Z idl , we can writethe total partition function exactly with an expression that seems unwieldy at first: Z tot a ( N tot a , T, V in , V out ) = N tot a (cid:88) N in a =0 N in a ! (cid:18) V in λ (cid:19) N in a N tot a − N in a )! (cid:18) V out λ (cid:19) ( N tot a − N in a ) . (B2)Before evaluating this expression, let’s pause to understand the underlying physics. Weshould think of the partition function (B2) as a sum over the (un-normalized) probabilities w for each possible N in a value: Z tot = (cid:80) N in a w ( N in a ). In other words, the w values give therelative probabilities of the N in a values and hence define a distribution over N in a . The keypoint is that statistical mechanics predicts a distribution of N in a values, each occurring withthe appropriate equilibrium probability. Necessarily, there will be a single largest probability,but fundamentally the distribution governs the observed behavior.Returning to the equation, we can dramatically simplify (B2) if we multiply and divideby N tot a ! and observe that the sum is exactly of binomial form, leading to Z tot a ( N tot a , T, V in , V out ) = 1 N tot a ! (cid:18) V in + V out λ (cid:19) N tot a . (B3)By comparison to (3), we see that this is simply the partition function for a single ideal gasof N tot a particles confined to a volume V in + V out . Indeed, each ideal, independent particle inour combined system ultimately can access both V in and V out , so the result makes sense.23wo observations are important before we address the original question about free energyadditivity. First, note that the final free energy doesn’t depend on N in a at all. This issomething you should expect because we have effectively “integrated out” — really, summedover — N in a . More interesting, the final partition function (and hence, free energy) fullyaccounts for all possible N in a values, which are weighted in by their relative probabilities.That is, in a full statistical mechanics description, the system isn’t limited to a single optimalvalue, as is the case implicitly based on free-energy minimization.Returning to (B1), we can now understand the precise approximation which has beenmade. First, what does summing free energies imply about the underlying partition func-tions? Well, note that if a partition function is exactly equal to a product of two other parti-tion functions, e.g., Z ab = Z a Z b , then the free energy is exactly a sum: F ab = − k B T ln Z ab = − k B T ln Z a + ( − k B T ln Z b ). On the other hand, the exact expression (B1) for the systemwe’re considering is a sum over partition-function products. When we write the free energyas a simple sum, we are estimating the partition sum in (B1) by the maximum term , whichis a standard approximation in statistical thermodynamics. . Specifically, our approxi-mation amounts to Z tot a ( N tot a , T, V in , V out ) ≈ Z idl ( N in ∗ a , V in , T ) Z idl ( N tot a − N in ∗ a , V out , T ) , (B4)where N in ∗ a is the value which maximizes the right-hand side of this expression. Althoughapproximating (B2) by a single term seems unreasonable at first, note that since there are N tot a + 1 terms total and each must be less than the maximum, the error in the logarithm of the sum required for the free energy should be of lower order than the dominant term inthe thermodynamic limit, N tot a → ∞ .Most important of all is to realize that our key results about transporters — which haveto do with the type of equilibrium points that exist and the thermodynamic driving whichis present away from equilibrium — are not affected at all by the details of the maximum-term approximation. After all, even if we did not make the approximation, there wouldstill be a minimum free-energy point specifying an optimum N in ∗ a value; and further, thisoptimum would exactly correspond to equal inside and outside concentrations in the specialcase V in = V out based on symmetry arguments. It’s clear the essence of our findings wouldstill hold. 24 ppendix C: Constrained vs. Global Equilibrium It is useful to compare the condition obtained when A is coupled to B by the transporter,namely (12), to the previous result (7) when A is the only species and hence uncoupled fromB. The coupling of A to B in the presence of the transporter based on (1) shifts the resultingequilibrium for A, and vice versa, of course.The A-B coupling means that we have not necessarily obtained the global free energyminimum, but what can be termed a constrained minimization. That is, if we consider N tot a , N tot b , V, T as constants, there are two degrees of freedom ( N in a and N in b ) but we didnot allow all possible pairs of these variables. Because of our transporter, the pair wasconstrained to lie on a line specified by (8) rather than being able to explore the entire( N in a , N in b ) plane. The free energy was minimized among the available values on this line,leading to (12).What would we have obtained if both degrees of freedom could vary independently? Inthat case, each molecular species would separately equilibrate to the equal-concentrationpoint (7). This is the global free energy minimum among all ( N in a , N in b ) points, which gen-erally won’t be accessible for the transporter-coupled case unless N = 0. Appendix D: More advanced perspective on chemical kinetics via trajectories andthe Master Equation
Our discussion in the main text examined chemical kinetics solely in terms of ordinarydifferential equations (ODEs) like (15) in the mass-action picture. ODEs, as you may know,are deterministic and lead to a unique solution for given initial conditions. In our case, thatwould mean the time-evolution of the concentrations [A in ] , [A out ], etc. are defined functionsof time, as in Fig. 4 (even if we couldn’t solve the equations analytically). The deterministicODE behavior represents the average behavior and we do indeed expect this to be uniquebased on specified initial conditions.A more microscopic picture of chemical kinetics accounts for different possible outcomesbased on the inherent stochasticity of the system, which is nicely illustrated by returning tothe system of Sec. III C. If particles of an ideal gas of A particles are free to move between“in” and “out” compartments, as in Fig. 2, then if we imagine making a “movie” of the25ystem at time points separated by a fixed time interval ∆ t , we would generate a sequenceof of pairs of particle numbers ( N in a , N out a ). For example, starting from an equal distributionof particles we might obtain the sequence (cid:16) N in a ( t ) , N out a ( t ) (cid:17) = (50 , , (50 , , (49 , , (48 , , (48 , , (49 , , . . . . (D1)If we watched our system a second time, a different sequence likely would occur. That is,there is a distribution of possible trajectories just as for any stochastic system , althoughformalizing the trajectory picture is well beyond the scope of this article.The (chemical) “master equation” (CME) description is intermediate between the ODEand trajectory pictures, though closer to the latter. The CME assumes an ensemble picturewhere multiple independent systems are studied simultaneously, characterized by a time-varying distribution of discrete states p ( N in a , N out a ; t ). The p for each state is simply thefraction of systems in that state, implying the normalization (cid:80) N tot a N in a =0 p ( N in a , N out a ) = 1 at all t . The CME picture can be understood from trajectories such as (D1). Imagine generatingmany one-step trajectories started from the state (50, 50). We could estimate the distribu-tion of outcomes by counting the resulting states, noting that for a very small time step,only states reachable by translocating a single A particle (or none) occur as in trajectory(D1). Mathematically, we can encode this in a differential equation which, for the state (50,50), is given by d p (50 , dt = ˆ k io p (51 ,
49) + ˆ k oi p (49 , − (cid:16) ˆ k io + ˆ k oi (cid:17) p (50 , , (D2)where the circumflex is used to distinguish these single-particle rate constants from the co-transport counterparts k io and k oi of (15). The different terms in (D2) account for systemsfrom the ensemble which both arrive at (positive sign) or leave (negative) the state (50 , m and n denote state indices – that is, each state like(50, 50) has a single index – with k mn and k nm the associated transition rates ( k mn = k m → n ),obtaining dp m ( t ) dt = (cid:88) n (cid:54) = m k nm p n ( t ) − (cid:88) n (cid:54) = m k mn p m ( t ) . (D3)This equation says that the probability of state m increases from incoming transitions anddecreases from outgoing transitions, just as in (D2). Note that some rate constants may26e strictly zero, for processes which cannot occur in a single step. The CME governs thetime evolution of the distribution over states and allows for outcomes besides the averagebehavior of a simple kinetics description. Note that the the set of rate constants of theCME can be used to generate trajectories consistent with (D3) via the stochastic simulation(Gillespie) algorithm.
The CME description unpacks the average behavior of simple chemical kinetics, similarto the way that statistical mechanics is the microscopic theory for thermodynamics. Note,however, that the discrete states in the CME picture are themselves averages over configu-rational coordinates treated in statistical mechanics. Part of the challenge, and beauty, oftheory is appreciating the relationship between different levels of averaging.
Appendix E: Questions and exercises for enrichment
We have seen the very basics of biochemical physics for understanding cellular processes.Readers may be interested in further issues.1. Apply the analysis above to an antiporter governed by (18). Write down the free energyand solve for the equilibrium point. Also write down the governing kinetic equationsand show these have the same equilibrium point as the thermodynamic calculation.2. Derive the equilibrium effects of varying stoichiometry. For example, assume two Amolecules (ions) are needed to transport a single B molecule. What relation now holdsin place of (12)?3. How would the formulation given have to be modified to account for electrostatics?Assume that A and B are charged molecules and there is a potential difference ∆ φ between inside and outside. Further assume charges are sufficiently well screened (seeAppendix A) so there are no direct molecule-molecule, or ion-ion, interactions. Whatequilibrium relation now holds in place of (7)?4. A thermodyanmic explanation was provided for the driving force available from ATPin the cell. Look up an ATP-driven process in a cell biology book and make a kineticargument for the driving – i.e., suggest the sequence of events likely to happen givendifferent ATP, ADP concentrations and knowledge of the equilibrium point.27. Look up the cellular processes that are thought to employ kinetic proofreading anddiscuss what properties they share and why the proofreading might be important.6. Numerous numerical experiments can arise from the material. A comparison could bemade of the CME stochastic formulation (Appendix D) to the deterministic chemicalkinetics prescription; see Ref. 29. More microscopically, a particle-based simulation could be performed to test the validity of mass-action assumptions with varying pa-rameters such as density, diffusion and reaction rates.28 ppendix F: Figure Captions BA OutsideInside
A A AB BA B BB BA
OutsideInside
A A AB BA B BBBA
OutsideInside
A A AB BA B BB (a)
EmptyA, B boundB only bound C o n f o r m a t i o n a l F r ee E n e r g y Conformational coordinate (abstract)Outward facing Inward facingOccluded (b)
FIG. 1. Highly schematic representation of the co-transport/symport process. (a) Binding of Aand B molecules to the outward-facing conformation of a transporter (light blue) embedded in amembrane (gray) triggers a conformational change that leads to an outward facing conformationwhere A and B unbind. The system then resets. All steps are reversible, with directionalitydepending on relative concentrations inside and out. (b) Binding events alter the free energylandscape of the transporter, favoring different conformations during the cycle. (cid:1840) (cid:3028)(cid:2925)(cid:2931)(cid:2930) , (cid:1848) (cid:2925)(cid:2931)(cid:2930) , (cid:1846)(cid:4667)(cid:4666)(cid:1840) (cid:3028)(cid:2919)(cid:2924) , (cid:1848) (cid:2919)(cid:2924) , (cid:1846)(cid:4667) (cid:4666)(cid:1840) (cid:3028)(cid:2925)(cid:2931)(cid:2930) , (cid:1840) (cid:3029)(cid:2925)(cid:2931)(cid:2930) , (cid:1848) (cid:2925)(cid:2931)(cid:2930) , (cid:1846)(cid:4667)(cid:4666)(cid:1840) (cid:3028)(cid:2919)(cid:2924) , (cid:1840) (cid:3029)(cid:2919)(cid:2924) , (cid:1848) (cid:2919)(cid:2924) , (cid:1846)(cid:4667) (a) (a)
390 400 410 420 430 440 450 460 0 20 40 60 80 100 F ( N a i n ) / k B T N a in (b) FIG. 2. Free energy minimization for a single ideal gas in a container with a permeable divider. (a)The ideal gas particles are divided between two compartments, separated by a permeable dividerwhich enables the system to sample all possible particle allotments between the compartments. (b)The free energy (5) is plotted as a function of N in a with N tot a = 100, V in = V out and V in /λ = 1 forconvenience. (cid:1840) (cid:3028)(cid:2925)(cid:2931)(cid:2930) , (cid:1840) (cid:3029)(cid:2925)(cid:2931)(cid:2930) , (cid:1848) (cid:2925)(cid:2931)(cid:2930) , (cid:1846)(cid:4667)(cid:4666)(cid:1840) (cid:3028)(cid:2919)(cid:2924) , (cid:1840) (cid:3029)(cid:2919)(cid:2924) , (cid:1848) (cid:2919)(cid:2924) , (cid:1846)(cid:4667) (a)
780 790 800 810 820 830 840 850 860 0 20 40 60 80 100 F ( N b i n ) / k B T N b in N = 0N = 20N = -20 (b) FIG. 3. Two ideal gases constrained by a transporter. (a) A mixture of A (red filled circles) andB (blue open circles) particles occupy two compartments separated by an impermeable membrane(gray) with a single embedded transporter (light blue). The transporter allows free passage ofA and B particles, but only in a 1:1 ratio. (b) The free energy for the system is plotted as afunction of the single “free parameter” N in b which self-adjusts to minimized the free energy. Thefree energy is shown for different N = N in b − N in a values, each of which leads to a different minimum— i.e., constrained equilibrium. We have set N tot a = N tot b = 100, V in = V out and V in /λ = 1 forconvenience. IG. 4. Relaxation to equilibrium in the symporter system (schematic). Initially, all A moleculesare outside, while B molecules are split evenly between inside and outside. The driving force fromthe A molecules, which are further from their own equilibrium, pumps B molecules from outside toinside until the constrained equilibrium condition (12) is satisfied. Here, we assume N tot a = N tot b and V in = V out .FIG. 5. The universal fuel of the cell, ATP. The bonds connecting the charged phosphate groupsare said to be high-energy. However, the true source of free energy obtained from ATP is due tothe concentrations of ATP and its hydrolysis products being maintained far from equilibrium..FIG. 5. The universal fuel of the cell, ATP. The bonds connecting the charged phosphate groupsare said to be high-energy. However, the true source of free energy obtained from ATP is due tothe concentrations of ATP and its hydrolysis products being maintained far from equilibrium.