A thermodynamic paradigm for solution demixing inspired by nuclear transport in living cells

Living cells display a remarkable capacity to compartmentalize their functional biochemistry spatially. A particularly fascinating example is the cell nucleus. Exchange of macromolecules between the nucleus and the surrounding cytoplasm does not involve crossing a lipid bilayer membrane. Instead, large protein channels known as nuclear pores cross the nuclear envelope and regulate the passage of other proteins and RNA molecules. Together with associated soluble proteins, the nuclear pores constitute an important transport system. Beyond simply gating diffusion, this system is able to generate substantial concentration gradients, at the energetic expense of guanosine triphosphate (GTP) hydrolysis. Abstracting the biological paradigm, we examine this transport system as a thermodynamic machine of solution demixing. Building on the construct of free energy transduction and biochemical kinetics, we find conditions for stable operation and optimization of the concentration gradients as a function of dissipation in the form of entropy production. In contrast to conventional engineering approaches to demixing such as reverse osmosis, the biological system operates continuously, without application of cyclic changes in pressure or other intrinsic thermodynamic parameters.

Living cells display a remarkable capacity to compartmentalize their functional biochemistry spatially. A particularly fascinating example is the cell nucleus. Exchange of macromolecules between the nucleus and the surrounding cytoplasm does not involve crossing a lipid bilayer membrane. Instead, large protein channels known as nuclear pores cross the nuclear envelope and regulate the passage of other proteins and RNA molecules. Together with associated soluble proteins, the nuclear pores constitute an important transport system. Beyond simply gating diffusion, this system is able to generate substantial concentration gradients, at the energetic expense of guanosine triphosphate (GTP) hydrolysis. Abstracting the biological paradigm, we examine this transport system as a thermodynamic machine of solution demixing. Building on the construct of free energy transduction and biochemical kinetics, we find conditions for stable operation and optimization of the concentration gradients as a function of dissipation in the form of entropy production. In contrast to conventional engineering approaches to demixing such as reverse osmosis, the biological system operates continuously, without application of cyclic changes in pressure or other intrinsic thermodynamic parameters. Demixing of solutions is a difficult thermodynamic problem with important practical consequences[1]. Examples include the desalination of seawater, medical dialysis, and chemical purification. In all of these processes, free energy is consumed to balance the entropy of mixing, which tends to keep a solution mixed. Typical engineering approaches to demixing involve application of hydrostatic pressure (reverse osmosis), solution exchange (dialysis), or phase change (crystallization or distillation)[2, 3]. Moreover, these perturbations must be applied cyclically. In this context living cells adopt a fundamentally different paradigm by establishing and maintaining concentration gradients at steady-state without employing a cyclical protocol [4,5].
A prominent example of molecular separation is the eukaryotic cell nucleus, wherein the concentrations of many proteins and RNA differ significantly from those in the cell body (cytoplasm). These concentration gradients are maintained by a transport system that shuttles molecular cargo in and out via large protein channels known as nuclear pores [6,7]. This system has been under intensive study in the biological and biophysical literature [8,9], with particular emphasis on single-molecule interactions at the pore itself [10][11][12][13][14][15].
Simple thermodynamic considerations make clear that equilibrium pore-molecule interactions are insufficient to support concentration gradients in solution. Demix- * Electronic address: chinghao@bu.edu † Electronic address: pankajm@bu.edu ‡ Electronic address: michael.elbaum@weizmann.ac.il ing between two compartments cannot happen spontaneously but must be coupled to a free-energy source to counteract the entropic forces favoring mixing [16]. Different from cyclic engines of man-made design in which energy is pumped into the system by changing external control parameters such as pressure and volume cyclically, cellular machines rather operate under fixed intrinsic parameters, with free-energy necessary for transport comeing from a non-equilibrium distribution of energetic small molecules such as ATP or GTP.
Nuclear transport relies on a special class of proteins, called transport receptors (i.e. "importin"), that shuttle large molecules between the cytoplasm and nucleus through small portals perforating the nuclear membrane, the so-called nuclear pores [8]. Nuclear pores allow water molecules, charged ions, and small molecules to freely diffuse across the nuclear membrane. However, the channel permeability of nuclear pores begins to drop for proteins larger than about 20 kDa molecular weight and by about 80 kDa, proteins must be actively transported across the nuclear membrane by binding importin [12,17]. Recognition between transport receptors and their molecular cargo depends on the presence of particular amino acid sequences known as nuclear localization signals [13,18,19]. Moreover, the affinity between importin and cargo can be regulated by a third entity, a small GTP-binding protein called Ran [8,20]. When associated with GTP (RanGTP), the Ran binds strongly to importin; however, when Ran is associated with GDP (RanGDP), it binds importin weakly. Ran interconverts between these two forms through GTP hydrolysis and GTP/GDP exchange facilitated by the GTPase activat- , nuclear cargo accumulation is strongly favored and Ran GTP/GDP exchange cycle proceeds faster than without coupling(lower panel). The thickness of arrowed curves in Ran cycle indicates the strength of reaction flux while the length of the arrowed lines in cargo transport represents the rate at which the underlying processes occur. (B) Details of molecular demixing machine in the context of nuclear transport. Species labels are the same as in (A). In both panels, reactions corresponding to Ran cycle and cargo transport are highlighted by red and green boxes, respectively. The orange dashed box includes all reactions coupled by the importin-NTF2 system.
ing protein RanGAP and the Guanosine Exchange Factor RanGEF, respectively [21]. These two proteins are spatially localized in different cellular compartments. Ran-GAP is structurally bound to the cytoplasmic face of the nuclear pore and RanGEF is bound to chromatin. This generates a high concentration of RanGTP and RanGDP in the nucleus and cytoplasm, respectively (see Fig.1).
These biochemical details suggest that the demixing between the cytoplasm and nucleus is powered by transducing free energy from GTP hydrolysis through the interactions of transport receptor with Ran. The transport machinery has been formulated in terms of coupled chemical kinetics [22,23] but the energetics have not yet been addressed. In particular, we ask: How to relate the rate of dissipation to the desired concentration gra-dient? What's the physical definition of transport efficiency? Can we find an optimal working point and how is that reconcile with the nonequilibrium nature of this cell machine? To address these questions, it is helpful to reformulate this problem in a thermodynamic language and to infer from that the engineering aspects of nuclear transport machinery. For consistency with the literature we retain the nomenclature of nuclear transport, yet the aim is to understand the natural engineering in a more abstract sense that might ultimately be implemented synthetically.
In the thermodynamic formulation of biochemical machines, a central role is played by energy transduction in a "futile cycle" among the components. This is roughly analogous to heat flow in a Carnot cycle. GTP hydrolysis on Ran serves as the power source for nuclear transport (see Fig. 1). The importin receptor binds RanGTP and a second receptor known as nuclear transport factor 2 (NTF2) binds specifically RanGDP. The forward cycle takes RanGTP out to the cytoplasm and translocates RanGDP back to the nucleus. Detailed balance is broken by the distribution of RanGAP and RanGEF as described above, so that the reverse cycle is scarcely populated. The futile cycle is coupled to cargo transport via the importin, which can bind alternatively RanGTP or the cargo. The details of this biochemistry can be abstracted into a simple model where free-energy from the Ran cycle is transduced by importin to bias the steadystate free cargo concentrations in the nuclear and cytoplasmic compartments (Fig.1). Note that while NTF2 transports only RanGDP, importin molecules participate in both the futile cycle and the cargo accumulation.
The details of the biochemical reactions underlying nuclear transport are shown in Fig. 1B and can be modeled using law of mass action (see Appendix) [24][25][26]. The corresponding kinetic parameters can be found from the literature and earlier kinetic models [22,23], or can be estimated from simple scaling arguments (see Appendix for details of kinetic model). Numerical solutions are obtained by solving all the coupled rate equations using standard Runge-Kutta method. We emphasize that the present aim is not so much to model the biological implementation as to explore the generic function of the thermodynamic machine. Relations between parameters are therefore more important than specific values.
A useful measure of cargo demixing is the nuclear localization ratio, NL, defined as the ratio between nuclear and cytoplasmic cargo concentrations: [C] nu /[C] cyto . Note that the ratio defines a chemical potential, ∆µ = −k B T ln [C] nu /[C] cyto . Fig. 2 shows the NL ratio as a function of importin and NTF2 concentrations. The most striking feature is that NL is maximum for intermediate levels of importin. Futhermore, the importin concentration at which the NL ratio is maximized, [Im * ], is largely independent of the NTF2 concentration. To better understand this phenomena, we performed the same simulations at different cargo concentrations. We found that [Im * ] grows with [C] tot , and for all cargo concentra-tions it is largely independent of the NTF2 concentration (see Fig. 3B). At first sight it is surprising that augmenting the importin concentration, which literally increases the the number of molecules that can transport cargo at nucleus, actually decreases the localization ratio. This instead reveals that excessive importin weakens the coupling to the futile cycle, reflecting the dual role played by importin in the nuclear transport machinery. Concretely, importin not only transports cargo, but also couples the transport system to the Ran futile cycle that serves as the power source for demixing between the nucleus and cytoplasm. This suggest that these observations reflect thermodynamic constraints build into the design of the nuclear transport machinery.
To understand the thermodynamics of nuclear transport, we exploit the observation that dynamics of the Ran futile cycle can be formulated as a nonequilibrium Markov process. Since a nonequilibrium steady state (NESS) essentially necessitates breaking the detailed balance in the underlying Markov process, the system has a nonzero entropy production [16,27,28]. The entropy production (EP) of the Markov process is the energy per unit time (power) required to maintain this NESS, and, therefore, it is equivalent to the energy consumed by the biochemical circuit. In the Schnakenberg description, the EP for a NESS is given by [29] where p SS i is the steady state probability distribution of state i while W (i, j) denotes the transition probability from state i to state j.
In the context of the Ran futile cycle that powers nuclear transport, p SS i is simply the fraction of reactants that participate in the transition reaction starting from state i, while W (i, j) can be calculated from the relevant kinetics rate constants. This entropy production gives a direct measure of the power consumed by the underlying biochemical circuit (i.e. the Ran futile cycle  .7). Surprisingly, consuming more power actually decreases the efficacy of nuclear transport as measured by the NL ratio. Again this reflects the optimization for coupling of the futile cycle to the measured work (i.e. NL); on either side of optimum the extra dissipation is wasted without contributing to NL. We also calculated EP for various cargo concentrations as shown in Fig. 3C. As the importin concentration increases (for a given NTF2 level), EP first drops to a minimum and then peaks before slowly decaying. Nevertheless, the minimum entropy production tracks approximately, but not exactly, with [Im * ], the value at which the localization ratio peaks (see FIG.7). This might suggest a workable strategy for optimal demixing− having importin level at around where EP is minimal.
This non-monotonic behavior in EP is a direct result of the dual role importin plays as the inbound car- rier of cargo protein as well as the outbound carrier of RanGTP. Powering the futile cycle requires that importin bind RanGTP, whereas cargo transport requires importin to bind cargo. This gives rise to an effective competition between cargo and RanGTP to bind the importin molecules, albeit in different compartments (FIG.3A). A simple analysis reveals that importin is much more likely to be found bound to cargo than RanGTP and more or so at intermediate importin level (FIG.5). Moreover, this difference in the probability that importin is bound to cargo or RanGTP is maximal at intermediate levels of importin concentration where the EP is also maximum (FIG.3A ). Indeed, this can be inferred from similar behavior observed in the difference between two corresponding reaction fluxes (see FIG.5). Thus, a further increase of the importin concentration inhibits formation of the RanGTP-importin complex and suppresses the export of RanGTP to the cytoplasm. Since the coupling between energy source to the transport machine relies highly on this process (i.e. RanGTP export), Ran futile cycle must dissipate more energy to maintain NESS, leading to EP increase. Furthermore, the NL ratio declines because RanGTP can no longer effectively displace cargo from the importin in the nucleus (FIG.3A). Indeed, this basic picture is supported by the behavior of various fluxes as functions of importin concentration (see Appendix and FIG.4 and FIG.8).
The energy needed to power nuclear transport is extracted from GTP hydrolysis. For this reason, we asked how N L depends on the free GTP-GDP ratio, θ. This ratio enters the kinetics through an effective "free energy" F θ := ln (θ) ( See reaction 2, 5 in FIG.4A,B and appendix). A typical value is around θ ∼ 10 2 [21,30]. In- In summary, we have analyzed the biological paradigm for nuclear transport from a thermodynamic point of view. Consistent with prior understanding that protein cargo demixing is facilitated by hydrolysis of GTP, we identify the relevant components the constitute the futile cycle and draw the connection between dissipation of chemical energy and maintenance of the cargo concentration gradient at non-equilibrium steady states. We show that the efficacy of nuclear localization ratio peaks at intermediate importin level, which is not too far from power consumption minimal. This permits a thermodynamic definition of efficiency, rather than an ad hoc measure via counting of single molecule transport events as well as suggests a possible experimental test to our model.
Our thermodynamic analysis of nuclear transport also suggests several engineering design principles for designing demixing machines that operate at steady-state without a cyclic change in external parameters. The transport of cargo must be coupled to non-equilibrium fluxes that result from the transport of an energetic switch that is charged in one compartment (nucleus) and discharged in a separate compartment (the cytoplasm). This ensures that the nuclear transport machinery will be insensitive to fluctuations in the chemical potential of the raw energy source (GTP) that powers transport. Furthermore, for a given cargo concentration there exists an optimal concentration of importin in terms of localization NL per dissipated power EP. It is interesting to speculate whether living cells maintain operation near such a working point.
This work is part of a larger literature that seeks to examine basic biophysical processes from a thermodynamic perspective. It is now clear that thermodynamics fundamentally constrains the ability of cells to perform various task ranging from detecting external signals [31][32][33], to adaptation [34], to making fidelity decisions [28], generating oscillatory behavior [35], and of course generating forces and dynamic structures [36][37][38]. In all these examples, it is possible to map these tasks to Markov processes and compute the corresponding entropy production rate. This suggests that there may be general theorems about thermal efficiency in cells that are independent of the particular task under consideration [27,[39][40][41]. It will be interesting to explore if this is actually the case and to see if these principles can be applied to synthetic biology [39].
Acknowledgement PM and CHW were supported by a Simons Investigator in the Mathematical Modeling of Living Systems grant and a Sloan Fellowship (both to PM). Simulations were carried out on the Shared Computing Cluster (SCC) at BU. The 11 basic reactions constituting the whole transport process are depicted in Figure 4. Our model incorporates known mechanism of nuclear transport of cargo through binding with importin and the active consumption of energy through hydrolysis of GTP. Such process is facilitated by Ran's intrinsic GTPase activity, which is activated through Ran's interaction with the Ran GTPase activating protein (RanGAP). In addition, we also include the reverse conversion of RanGDP to RanGTP through the action of guanine Exchange Factor RCC1 (known as RanGEF) in our model. One distinct feature of our model is in addition to the standard model of nuclear transport whose biochemistry is summarized below, we also incorporates the backward reactions to account for the reversibility nature of this transport process [25].
• (Reaction 10, 9) At cytoplasm, say, compartment A, the complex form by importin protein (transport receptor) and the cargo C interacts with the nuclear pore complex and pass through the channel into the nucleus (compartment B).
• (Reaction 7, 4, 3) At the nucleus, RanGTP competes for binding with the receptor and cause the receptor to dissociate from the cargo. The new complex formed by RanGTP and receptor then translocates to the cytoplasm while the cargo is left inside the nucleus.
• (Reaction 2) Once at the cytoplasm, a protein called Ran Binding Protein (RanBP) separates RanGTP from receptor. GTPase activating protein (GAP) then binds to RanGTP, causing the hydrolysis of GTP to GDP and releasing energy.
• (Reaction 1, 6) The RanGDP produced in this process then binds the nuclear transport factor NTF2 which returns it to the nucleus.
• (Reaction 5) Now in the nucleus, RanGDP interacts with a guanine nucleotide exchange factor (GEF) which replaces GDP with GTP, resulting again a RanGTP from, and beginning a new cycle.

Kinetics equations
The whole process can be formulated by a set of kinetics equations involving both cargo protein translocation and Ran regulation. The molecular species in the kinetics equations are labelled according to Figure 4.
From this we can write down the following kinetics: Appendix B: Estimating the rate constants Here we list the kinetics rate constants used in the simulation. Some of them are directly available from literature while others are estimated as described below. 1 set by the diffusion-limited rate: k diff = 10 sec −1 nM −1 To simply notation in the following estimation analysis, we use the following labels Labels Species N NTF2 Im Importin (importin) RD RanGDP RT RanGTP N·RD NTF2+RanGDP complex N·RT NTF2+RanGTP complex Im·RD Importin+RanGDP complex Im·RT Importin+RanGTP complex fD (free) GDP fT (free) GTP

Reaction 5: Ran exchange mediated by RanGEF
The goal is to estimate the K D for the following reaction: namely, Consider the following two constituting reactions This implies (neglecting labels of steady states SS), Thus we can reexpress Eq.(B5) using Eq.(B6): The first term (i.e. k + α /k − α ) comes from guanine nucleotide exchange reaction and is of order one while the second (i.e. k + β /k − β ) is related to the free energy difference between binding and un-binding of NTF2+RanGTP complex which is much larger than 1: ∆F >> 1. This can also be understood using Eq.(B6) by noting that in the nucleus NTF2 seldom binds to RanGTP. Finally, since the free GTP to GDP ration, [f T ]/[f D], is buffered by cellular metabolism, we simply treat the last term as a free parameter θ. Note that there are far more free GTP than Ran on a molar basis and most of the Ran are in RanGTP form. After rescaling time by τ ← tc 0 k diff , with k diff = 10 sec −1 nM −1 and c 0 represent the diffusion-limited reaction rate and the characteristic molar concentration (set to 1nM), respectively, and approximating e ∆F ≈ 10 ∼ 100 , one can estimate (k is treated as a free parameter.

Reaction 2: Ran exchange mediated by RanGAP
We aim to approximate K D for such reaction: Similarly the estimation is based on the following two steps: This implies (neglecting labels of steady states SS), Thus we can reexpress Eq.(B13) using Eq.(B14): Considering two type of molecules A and B diffusing in a viscous environment. According the Fick's law the diffusion flux of one type of molecule assuming the other is at stationary is given as where µ = A, B and D µ is the diffusion constant of molecule µ. Assuming spherical symmetry one can integrate Fick's law to get the total number of molecules diffusing through a given surface area: where R is the sum of molecular radii of A and B. The factor k a := 4πR(D A + D B ) is exactly the reaction rate of the overall catalytic reaction under the assumption that the process is diffusion-limited (i.e. upon A and B are in contact, the intermediate complex AB immediately reacts to form the final product P): Finally, recall Stokes-Einstein relation: D = k B T /(6πηa) with molecule (spherical) particle radius a, we have where N A is the Avogadro's constant. The factor 10 3 appears because we convert the SI unit of volume m 3 to liter. Using η = 10 −1 (Pa·sec)= 10 −3 kg/m/sec, we get

Appendix D: Entropy Production of the Ran battery
The distinct feature of systems out of thermodynamics equilibrium is the continuous production of entropy. The rate of entropy change (in time) consists of two parts: (i) the internal entropy change and (ii) the exchange of entropy with the environment where S is the entropy of the system and Π is the rate of entropy production and Φ denotes the rate of entropy flow from the system to the outside. Within this context, the 2nd law of thermodynamics dictates Π ≥ 0 and the notion of steady states translates into Π = Φ: entropy produced is continuously given away to the environment. One can further distinguishes the equilibrium from the nonequilibrium steady states by Consider systems that can be described by a continuous time Markov process such that the probability flow can be written as a master equation: where W ij is the transition rate from state j to state i and P i (t) is the probability of state i at time t. An appropriate microscopic description for the nonequilibrium system amounts to (i) having well-defined entropy for the irreversible systems and (ii) the entropy production rate Π should respects the non-negativity and should vanish when system equilibrates (i.e. when it exhibits reversibility). For systems described by the master equation, thermodynamics equilibrium is essentially the detailed-balanced condition: P i W ij = P j W ji . The solution for the first is the Boltzmann-Gibbs entropy: while the entropy production rate is advanced by the Schnakenberg description [29]: In the stationary state, Eq.(D4) reduces to where P i is the stationary probability distribution. It's easy to check that Π − Π(t) = dS/dt → 0 in the stationary state.
In the context of Ran futile cycle, one can map the circuit in FIG. 4B to a nonequilibrium Markov process. A non-equilibrium steady state (NESS) essentially necessitates breaking the detailed balance in the underlying Markov process and therefore, the system(i.e. Ran futile cycle) has a nonzero entropy production that is continuously given away to the environment. Such entropy production is exactly the power consumed by the circuit to maintain NESS. From Eq.(D5) and setting k B = 1 and defining EP := Π where p SS i is the steady state probability distribution of state i while W (i, j) denotes the transition probability from state i to state j. In the setting of Ran battery, p SS i is simply the fraction of reactants participated in the transition reaction starting from state i while W (i, j) can be calculated from the relevant rate constants.                  [C]nu. The ratio between RanGDP to RanGTP is kept the same as in the main text. Other parameters used are the same as in FIG.2 and FIG.3