A thermodynamic paradigm for solution demixing inspired by nuclear transport in living cells
AA thermodynamic paradigm for solution demixing inspired by nuclear transport inliving cells
Ching-Hao Wang ∗ and Pankaj Mehta † Department of Physics, Boston University, Boston, MA 02215
Michael Elbaum ‡ Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot, Israel (Dated: September 22, 2018)Living cells display a remarkable capacity to compartmentalize their functional biochemistry. Aparticularly fascinating example is the cell nucleus. Exchange of macromolecules between the nu-cleus and the surrounding cytoplasm does not involve traversing a lipid bilayer membrane. Instead,large protein channels known as nuclear pores cross the nuclear envelope and regulate the passage ofother proteins and RNA molecules. Beyond simply gating diffusion, the system of nuclear pores andassociated transport receptors is able to generate substantial concentration gradients, at the ener-getic expense of guanosine triphosphate (GTP) hydrolysis. In contrast to conventional approachesto demixing such as reverse osmosis or dialysis, the biological system operates continuously, withoutapplication of cyclic changes in pressure or solvent exchange. Abstracting the biological paradigm,we examine this transport system as a thermodynamic machine of solution demixing. Building onthe construct of free energy transduction and biochemical kinetics, we find conditions for stableoperation and optimization of the concentration gradients as a function of dissipation in the formof entropy production.
PACS numbers: 75.50.Pp, 75.30.Et, 72.25.Rb, 75.70.Cn
Demixing of solutions is a difficult thermodynamicproblem with important practical consequences [1]. Ex-amples include the desalination of seawater, medical dial-ysis, and chemical purification. In all of these processes,free energy is consumed in order to balance entropy ofmixing. Typical engineering approaches to demixing in-volve application of hydrostatic pressure (reverse osmo-sis), solution exchange (dialysis), or phase change (crys-tallization or distillation) [2, 3]. In this context livingcells adopt a fundamentally different paradigm by es-tablishing and maintaining concentration gradients at steady-state under a fixed set of intrinsic thermodynamicparameters. This recalls the similar capacity to operatemechanochemical motors isothermally [4, 5].A prominent example of molecular separation is the eu-karyotic cell nucleus, wherein the concentrations of manyproteins and RNA differ significantly from those in thecell body (cytoplasm). These gradients are maintainedby a transport system that shuttles molecular cargo inand out via large protein channels known as nuclear pores[6, 7]. This system has been under intensive study inthe biological [8–12] and biophysical [13–17] literatures,with particular emphasis on single-molecule interactionsat the pore itself [18–21]. Simple thermodynamic consid-erations make clear that equilibrium pore-molecule inter-actions are insufficient to support concentration gradientsin solution. Demixing between two compartments can-not occur spontaneously, but must be coupled to a free ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] energy source [22]. At the same time, demixing does notrequire rectified translocation [23]. Concentration gradi-ents may be established in the presence of a balanced,bi-directional exchange [14, 24, 25].Nuclear pores represent an unusual transporter in thatthere is no membrane to cross. Water, ions, and smallmolecules diffuse freely across the nuclear envelope toequilibrate between the two compartments. Generally,the permeability drops between molecular weight 20 kDaand 40 kDa [26, 27]. Transport of larger macromoleculesrelies on a special class of proteins, called transport re-ceptors (i.e. “importin”), that usher their cargoes acrossthe nuclear pores by virtue of specific interactions withthe channel components. Recognition between importinsand their molecular cargo depends on the presence ofparticular amino acid sequences known as nuclear local-ization signals (NLS) [8, 28, 29]. The affinity betweenimportin and cargo is regulated by a small GTP-bindingprotein called Ran [30, 31]. When associated with GTP(RanGTP), Ran binds strongly to importin in a man-ner that is competitive to NLS binding. By contrast,Ran associated with GDP (RanGDP) binds importinvery weakly. Ran interconverts between these two formsthrough GTP hydrolysis and GTP/GDP exchange, facil-itated by the GTPase Activating Protein (RanGAP) andthe Guanosine Exchange Factor (RanGEF), respectively[32]. RanGAP is structurally bound to the cytoplasmicface of the nuclear pore and RanGEF is bound to chro-matin. Their activities generate a high concentration ofRanGTP in the nucleus and RanGDP in the cytoplasm(see FIG.1).Demixing is powered by transducing free energy fromGTP hydrolysis through the interactions of transport re- a r X i v : . [ q - b i o . S C ] F e b FIG. 1: (Color online) Demixing of cargo across the nuclearmembrane is driven by Ran coupled to NTF2 and importinsystem. (A) With such coupling (upper panel), nuclear cargoaccumulation is favored and Ran GTP/GDP exchange cy-cle proceeds faster than without coupling (lower panel). Thethickness of arrowed curves in Ran cycle indicates strengthof reaction flux; length of arrowed lines in cargo transportrepresents the rate at which the underlying processes occur.(B) Details of molecular demixing machine in the context ofnuclear transport. Species labels as above. Reactions corre-sponding to Ran cycle and cargo transport are highlightedby red and green boxes, respectively. The orange dashed boxincludes all reactions coupled by the importin-NTF2 system.See also Fig S1 in the Supplementary Material. ceptor with Ran. The transport machinery has been for-mulated in terms of coupled chemical kinetics [23, 33, 34]but the energetics have not yet been addressed. In par-ticular, we ask: How does the rate of dissipation (en-ergy consumption) relate to the achieved concentrationgradient? What is the proper definition of transport ef-ficiency? Is there an optimal working point given thenonequilibrium nature of this cellular machine? To ad-dress these questions, it is helpful to reformulate theproblem in a thermodynamic language. For consistencywith the literature we retain the biological nomenclature,yet the aim is to understand the natural engineering ina more abstract sense that might ultimately be imple-mented synthetically.In the thermodynamic formulation, a central role is played by energy transduction in a “futile cycle” amongthe components (see FIG. 1). This is roughly analogousto heat flow in a Carnot cycle. The importin receptorbinds RanGTP, and a second receptor known as nucleartransport factor 2 (NTF2) binds specifically RanGDP.The forward cycle takes RanGTP out to the cytoplasmwith importin and RanGDP back to the nucleus withNTF2. Detailed balance is broken by the distribution ofRanGAP and RanGEF as described above, so that thereverse cycle is scarcely populated.Free energy from the Ran cycle is transduced by im-portin to bias the steady-state free cargo concentrationsin the nuclear and cytoplasmic compartments. Detailsof the underlying biochemical reactions are shown inFIG. 1B and can be modeled on the basis of mass ac-tion. The corresponding kinetic parameters can be foundfrom the literature or estimated from simple scaling ar-guments (see FIG. S1 and Supplementary Material fordetails of the kinetic model[35]). Numerical solutions areobtained by solving all the coupled rate equations us-ing a standard Runge-Kutta method (The code used forsimulation is available in the Supplementary Material).We emphasize that the present aim is not so much tomodel the biological implementation as to explore thegeneric operation of the thermodynamic machine. Rela-tions between parameters are therefore more importantthan specific values.Energetics enter the model via the charging of Ranwith GTP and its subsequent hydrolysis to GDP (reac-tions 5 and 2 in FIG. S1, respectively). The flux throughthese two reactions must be equal in steady state. En-ergy is drawn from the non-equilibrium ratio of free GTPto GDP, θ , which is maintained by cellular metabolismand defines an effective “free energy” F θ := k B T log ( θ ).A typical value of θ is roughly a few tens to a hundred[32, 36]. Independent of the complex operational detailsof RanGEF and RanGAP with associated co-factors, wecan look at the steady states and relate the reactionsto θ . (See Supplementary Material for details.) On thenuclear side, the complex NTF2-RanGDP exchanges forNTF2 and RanGTP. The dissociation constant K D (for-ward divided by reverse flux) can be shown to be pro-portional to θ . Conversely, on the cytoplasmic side thecorresponding K D is proportional to 1 /θ . As a result,any enhancement of flux through the futile cycle in theforward reaction conferred by increasing θ (i.e. reaction5 in FIG.S1C) is balanced by the contradicting counter-part in preventing RanGTP release (i.e. reaction 2 inFIG.S1C).A useful measure of cargo demixing is the nu-clear localization ratio, NL, defined as the ratio be-tween nuclear and cytoplasmic cargo concentrations:[ C ] nu / [ C ] cyto . This ratio defines a chemical potential,∆ µ = − k B T log [ C ] nu / [ C ] cyto , that measures the excur-sion from equilibrium. FIG. 2A shows NL as a functionof importin and NTF2 concentrations. The most strik-ing feature is that NL is maximum for intermediate levelsof importin. The importin concentration at which NL is [Importin] (nM) N L r a t i o [C] tot =20 nM[C] tot =50 nM[C] tot =100 nM [Importin] (nM) EP / k B T [C] tot =20 nM[C] tot =50 nM[C] tot =100 nM A B [NTF2] tot = 100nM [C] tot = 100nM
FIG. 2: (Color online) Phase diagram of nuclear localization.(A) The cargo nuclear localization NL:= [ C ] nu / [ C ] cyto (colorshadings) is obtained by varying the total importin and totalNTF2 concentrations while keeping overall cargo level fixed at[ C ] tot = 100 nM. (B) A family of curves shows NL for severalcargo concentrations as a function of importin concentrationwith [ NT F tot = 100 nM. The 1D curve for [ C ] tot = 100nM is a cut across the plot of panel A. Locations of NL max-imum are marked by diamonds (see FIG.4C as well). Kineticrate constants used are given in the Supplementary Material.Total Ran concentration [ Ran ] tot =75 nM. maximized, [ Im ∗ ], grows with the total cargo load, [ C ] tot (see FIG. 2B). Furthermore, [ Im ∗ ] is largely independentof NTF2 concentration for different cargo concentrationconsidered (see FIG. S5). This suggests an inherent op-timization.At first sight it is surprising that augmenting theimportin concentration, which increases the number ofmolecules that can transport cargo to nucleus, may de-crease the localization ratio. The optimal dependenceof NL on importin reflects the dual role importin playsas the inbound carrier of cargo protein as well as the outbound carrier of RanGTP . Powering the futile cy-cle requires that importin bind RanGTP, whereas cargotransport requires importin to bind cargo. This estab-lishes a binding competition in the nucleus that is a char-acteristic feature of protein import (FIG.3A). In spiteof the higher affinity of RanGTP for importin, the cy-cle analysis shows that importin in the nucleus bindscargo more rapidly. As seen in FIG. 3BC, NL is max-imized close to the point at which the difference betweenthe reaction fluxes of importin-cargo formation ( Φ − :=˜Φ − [ Im ] nu = ( k − [ C ] nu )[ Im ] nu ) and importin-RanGTPformation (Φ +4 := ˜Φ +4 [ Im ] nu = ( k +4 [ RanGT P ] nu )[ Im ] nu ) is maximal. Intuitively, this is the realm where im-portin can bind cargo effectively while maintaining itscoupling to the reaction cycle that transduces energy forcargo transport.To understand the thermodynamics of nuclear trans-port, we formulate the transport system as a nonequi-librium Markov process. Since a nonequilibrium steadystate (NESS) necessarily breaks detailed balance in the X X X X X X X X X X X X CompartmentA CompartmentB
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NTF2 RanGTP RanGDP cargo Transport receptor cytoplasm nucleoplasm
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
NTF2 RanGTP RanGDP cargo Transport receptor cytoplasm nucleoplasm
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ - X X X X X X X X X X X X X X X X nucleoplasmcytoplasm A CB X X X X X X X X X X X X CompartmentA CompartmentB
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
NTF2 RanGTP RanGDP cargo Transport receptor cytoplasm nucleoplasm
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
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COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
NTF2 RanGTP RanGDP cargo Transport receptor
COMPARTMENT ACOMPARTMENT B + + ++++ - - ---- X X X X X X X X X X cytoplasmnucleoplasm ImportinRanGTPcargo ˜ ˜ +4 X : [Im] nu nM s c a l e d fl u x ˜ Φ + [ C ] tot = 10 nM [ C ] tot = 40 nM [ C ] tot = 70 nM [ C ] tot = 100 nM X : [Im] nu nM s c a l e d fl u x ˜ Φ − [ C ] tot = 10 nM [ C ] tot = 40 nM [ C ] tot = 70 nM [ C ] tot = 100 nM AB C
FIG. 3: (Color online) Competition between RanGTP andcargo to bind importin. (A) Schematic of the two competingreactions. (B) Reaction flux for importin-RanGTP formation˜Φ +4 ∼ k +4 [ RanGT P ] and (C) flux for importin-cargo forma-tion ˜Φ − ∼ k − [ C ] nu . Fluxes are scaled by [ Im ] nu (see text).Parameters as in FIG.2 underlying Markov process, the system has a nonzeroentropy production [22, 37, 38]. This is the energy perunit time required to maintain the NESS, with units ofpower. Following the Schnakenberg description, the EPfor a NESS is given by [39] EP = k B T (cid:88) i,j P SSi W ( i, j ) log W ( i, j ) W ( j, i ) , (1)where P SSi is the steady state probability distribution ofstate i while W ( i, j ) denotes the transition probabilityfrom state i to state j . Concretely, P SSi is the fractionof reactants that participate in the transition reactionstarting from state i while W ( i, j ) can be calculated fromthe relevant reaction fluxes. Note that the sum in Eq.(1)is taken over all links of the reaction network. This isequivalent to summing over the links pertaining to theRan futile cycle. (See Supporting Material for details).This entropy production provides a direct measure ofthe power input to the underlying biochemical circuit.FIG. 4A shows EP for various importin and NTF2 con-centrations. FIG. 4B adds various cargo concentrationsfor a fixed level of [ N T F [Importin] (nM) N L r a t i o [C] tot =20 nM[C] tot =50 nM[C] tot =100 nM [Importin] (nM) EP / k B T [C] tot =20 nM[C] tot =50 nM[C] tot =100 nM
20 40 60 80 100 [C] [I m * ] NL max, [NTF2]=100EP min, [NTF2]=100EP max, [NTF2]=100 X : [Im] nu nM Φ ± [ C ] tot =100 nM, Φ − [ C ] tot =100 nM, Φ +4 BAC D [NTF2] tot = 100nM [C] tot = 100nM
FIG. 4: (Color online) Phase diagram of entropy produc-tion. (A) Entropy production is plotted as a 2D function ofNTF2 and importin, at fixed cargo concentration [ C ] tot = 100nM. Compare with FIG. 2A. (Axes extend to 5 instead of 0nM to avoid numerical divergence.) (B) A family of curvesshows the entropy production for several cargo concentra-tions as a function of importin concentration; NTF2 con-centration fixed at [ NT F tot = 100 nM. The 1D curve for[ C ] tot = 100 nM is a cut across the plot of panel A. Com-pare with FIG. 2B. Peaks and troughs are marked by squaresand circles, respectively. (C) Locations of entropy productionmaximum/minimum (square/circle) and that of nuclear local-ization maximum (diamond). Colors match curves in panel B.The importin concentration at which EP is minimum is closeto but always less than [ Im ∗ ], where NL is maximum. Thus,one strategy for maximizing the efficiency of demixing is tohave the futile cycle operate in regime where its entropy pro-duction is minimized. (D) EP decreases at very high importinconcentration. This reflects a loop around the energetic reac-tion Φ via the reversible reaction Φ . Here [ NT F tot =100nM, as in the panel B. In all panels, [ Ran ] tot = 75 nM. Ki-netics constants as in FIG.2 and 3 (see SM Section II). where importin moves between compartments carryingneither cargo nor RanGTP. As seen in FIG.4D, for suchhigh importin levels the corresponding flux Φ exceedsthat of the RanGTP loading to importin, Φ .To the best of our knowledge the optimal steady-statehas not been observed experimentally. The kinetic rateof nuclear protein uptake was found to be reduced by mi-croinjection of importin receptor to live cells; rate equa-tion simulations done in parallel also pointed to the dualrole of importin (FIG. 3A) [40]. Steady-states were notreported in that study, however. Other possible experi-mental tests include titration of importin protein to nu-clei reconstituted in vitro in Xenopus egg extract andoptical activation of importin receptors, similarly to in-duction of nuclear transport by NLS activation [41]. Animportant point in comparison with literature is that wehave considered a single, collective “cargo” for transport. In reality, multiple cargoes compete for binding to rel-atively few but promiscuous transport receptors. Thiscompetition leads to a partitioning according to equi-librium binding affinities and may lead to vastly differ-ent kinetics. However the steady-state NL ratio (in so-lution) is independent of the affinity, reflecting thermo-dynamic control and equilibration of the chemical po-tentials [24, 25, 34]. Consistent with this paradigm, inwhich a net accumulation occurs together with a bal-anced bidirectional flux, the simulations show that thenuclear and cytoplasmic concentrations of the importin-cargo complex ( X and X , respectively) equilibrate insteady-state. It is also interesting to note that RanGTPloading onto importin (reaction 4) was identified in theearlier analysis as the primary rate-limiting step in accu-mulation kinetics [34].In summary, we have analyzed the biological paradigmfor nuclear transport from a thermodynamic point ofview. Building upon prior understanding that proteincargo demixing is facilitated by hydrolysis of GTP, wedraw the connection between consumption of chemicalenergy and maintenance of the cargo concentration gra-dient at non-equilibrium steady states. We show thatthe efficacy of nuclear localization ratio peaks at inter-mediate importin level, which is not far from the powerconsumption (entropy production) minimal. It is likelythat the cell maintains an importin concentration at anadvantageous level with respect to these operating pointsdefined by the thermodynamic analysis. Interestingly,the system as configured is robust to the quality of thechemical energy source, in the sense that the NL ratiois almost independent of the GTP:GDP ratio θ when θ > ∼
20, FIG. S4. A thermodynamic definition of the sys-tem efficiency remains elusive, however. Whereas con-ventional efficiency of an engine is a dimensionless ratioof mechanical to thermal power, in the NESS a constantfree energetic gradient (chemical potential in the presentcase) is maintained by a constant power input. The ratiohas units of time. This could be renormalized sensiblyby a characteristic remixing time, e.g., the permeabil-ity of the nuclear pores to the cargo-importin complex.There is no guarantee of a bound at unity, however, sothe definition remains ad hoc, a useful figure of merit.It is also interesting to contrast the competitive interac-tions between receptor and RanGTP in nuclear proteinaccumulation (import) with the cooperative interactionsin nuclear protein depletion (export). While these areoften considered as simple inverse processes, they differin this essential aspect [42].This work is part of a larger literature that seeks to ex-amine basic biophysical processes from a thermodynamicperspective. It is now clear that thermodynamics funda-mentally constrains the ability of cells to perform varioustask ranging from detecting external signals [43–45], toadaptation [46], to making fidelity decisions [38], gener-ating oscillatory behavior [47], and of course generatingforces and dynamic structures [48–50]. In all these exam-ples, it is possible to map these tasks to Markov processesand compute the corresponding entropy production rate.This suggests that there may be general theorems aboutthermal efficiency in cells that are independent of theparticular task under consideration [37, 51–53]. It willbe interesting to explore if this is actually the case andto see if these principles can be applied to synthetic bi-ology and ultimately biomimetic engineering [51].
Acknowledgment
PM and CHW were supported by a Simons Investigator in the Mathematical Modeling ofLiving Systems grant, a Sloan Fellowship, and NIH GrantNo. 1R35GM119461 (all to PM). ME acknowledges agrant from the Israel Science Foundation (1369/10), theGerhardt M.J. Schmidt Minerva Research Foundation,and the historical generosity of the Harold Perlman fam-ily. Simulations were carried out on the Shared Comput-ing Cluster (SCC) at BU. [1] M. Dijkstra and D. Frenkel, Physical review letters ,298 (1994).[2] K. H. Mistry, R. K. McGovern, G. P. Thiel, E. K. Sum-mers, S. M. Zubair, and J. H. Lienhard, Entropy ,1829 (2011).[3] P. D. Glynn and E. J. Reardon, American Journal ofScience , 164 (1990).[4] A. Parmeggiani, F. J¨ulicher, A. Ajdari, and J. Prost,Physical Review E , 2127 (1999).[5] J. Parrondo, B. de Cisneros, and R. Brito, StochasticProcesses in Physics, Chemistry, and Biology , 38 (2000).[6] G. G. Maul and L. Deaven, The Journal of cell biology , 748 (1977).[7] B. Talcott and M. S. 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E. Cohen, M. R. Hodel, G. J. Tr-uscott, A. H. Corbett, and A. E. Hodel, Journal of Bio-logical Chemistry , 21361 (2003). Supplemental Material for “A thermodynamic paradigm for solution demixinginspired by nuclear transport in living cells”
I. BIOPHYSICAL MODELS
The 11 basic reactions constituting the whole transport process are depicted in Figure S1. Our model incorporatesthe known (simplified) mechanism of nuclear transport of cargo through binding with importin and the active con-sumption of energy through hydrolysis of GTP. Such process is facilitated by Ran’s intrinsic GTPase activity, which isactivated via interaction with the Ran GTPase activating protein (RanGAP). In addition, we also include the reverseconversion of RanGDP to RanGTP through the action of guanine Exchange Factor RCC1 (known as RanGEF). Inaddition to the standard model of nuclear transport whose biochemistry is summarized below, we also incorporatesthe backward reactions to account for the reversibile nature of this transport process [34]. X X X X X X X X X X X X CompartmentA CompartmentB
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ - X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
NTF2 RanGTP RanGDP cargo Transport receptor cytoplasm nucleoplasm
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ - X X X X X X X X X X X X X X X X nucleoplasmcytoplasm A CB X X X X X X X X X X X X CompartmentA CompartmentB
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ - X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB X X X X X X X X X X X X CompartmentA CompartmentB
COMPARTMENT ACOMPARTMENT B - --- - - ---- + + ++++ ++ +++ -
NTF2 RanGTP RanGDP cargo Transport receptor
COMPARTMENT ACOMPARTMENT B + + ++++ - - ---- X X X X X X X X X X cytoplasmnucleoplasm Importin
FIG. S1:
Molecular reactions involved in nuclear transport . (A) Molecular species in our nuclear transport model. (B)Schematics of the reactions involved in nuclear transport. Note that + and - signs represent self-consistently the start andend points of the reactions, rather than forward or reverse cycle orientations. Thus for example k ”+” and k ” − ” for reaction 9in Eq.(S15) take signs for loss and gain, respectively. (C) The subset of reactions in (B) that forms the futile cycle (i.e. theenergy source for nuclear transport). • (Reaction 10, 9) In the cytoplasm, say, compartment A , the complex formed by importin protein (transportreceptor) and the cargo C interacts with the nuclear pore complex and passes through the channel into thenucleus (compartment B ). • (Reaction 7, 4, 3) In the nucleus, RanGTP competes for binding with the receptor and causes the receptor todissociate from the cargo. The new complex formed by RanGTP and receptor then translocates to the cytoplasmwhile the cargo is left inside the nucleus. • (Reaction 2) Once in the cytoplasm, the GTPase activating protein (RanGAP) then binds to RanGTP, causingthe hydrolysis of GTP to GDP and release of energy. • (Reaction 1, 6) The RanGDP produced in this process then binds the nuclear transport factor NTF2 whichreturns it to the nucleus. • (Reaction 5) Now in the nucleus, RanGDP interacts with a guanine nucleotide exchange factor (GEF) whichreplaces GDP with GTP, resulting again a RanGTP from, and beginning a new cycle. A. Kinetics equations
The whole process can be formulated by a set of kinetics equations involving both cargo protein translocation andRan regulation. The molecular species in the kinetics equations are labelled according to Figure S1.[ X ] + [ X ] k +1 (cid:10) k − [ X ] (S1)[ X ] k +2 (cid:10) k − [ X ] + [ X ] (S2)[ X ] k +3 (cid:10) k − [ X ] (S3)[ X ] + [ X ] k +4 (cid:10) k − [ X ] (S4)[ X ] k +5 (cid:10) k − [ X ] + [ X ] (S5)[ X ] k +6 (cid:10) k − [ X ] (S6)[ X ] k +7 (cid:10) k − [ X ] + [ X ] (S7)[ X ] k +8 (cid:10) k − [ X ] (S8)[ X ] k +9 (cid:10) k − [ X ] (S9)[ X ] + [ X ] k +10 (cid:10) k − [ X ] (S10)[ X ] k +11 (cid:10) k − [ X ] (S11)From this we can write down the following kinetics: d [ X ] dt = − k +1 [ X ][ X ] + k − [ X ] + k +11 [ X ] − k − [ X ] (S12) d [ X ] dt = − k +1 [ X ][ X ] + k − [ X ] + k +2 [ X ] − k − [ X ][ X ] (S13) d [ X ] dt = k +1 [ X ][ X ] − k − [ X ] − k +6 [ X ] + k − [ X ] (S14) d [ X ] dt = − k +9 [ X ] + k − [ X ] + k +10 [ X ][ X ] − k − [ X ] (S15) d [ X ] dt = − k +10 [ X ][ X ] + k − [ X ] (S16) d [ X ] dt = − k +2 [ X ] + k − [ X ][ X ] + k +3 [ X ] − k − [ X ] (S17) d [ X ] dt = k +2 [ X ] − k − [ X ][ X ] − k +8 [ X ] + k − [ X ] − k +10 [ X ][ X ] + k − [ X ] (S18) d [ X ] dt = k +5 [ X ] − k − [ X ][ X ] − k +11 [ X ] + k − [ X ] (S19) d [ X ] dt = k +5 [ X ] − k − [ X ][ X ] − k +4 [ X ][ X ] + k − [ X ] (S20) d [ X ] dt = − k +5 [ X ] + k − [ X ][ X ] + k +6 [ X ] − k − [ X ] (S21) d [ X ] dt = − k +7 [ X ] + k − [ X ][ X ] + k +9 [ X ] − k − [ X ] (S22) d [ X ] dt = k +7 [ X ] − k − [ X ][ X ] (S23) d [ X ] dt = − k +3 [ X ] + k − [ X ] + k +4 [ X ][ X ] − k − [ X ] (S24) d [ X ] dt = − k +4 [ X ][ X ] + k − [ X ] + k +7 [ X ] − k − [ X ][ X ] + k +8 [ X ] − k − [ X ] (S25) II. ESTIMATING THE RATE CONSTANTS
Here we list the kinetics rate constants used in the simulation. Some of them are directly available from literaturewhile others are estimated as described below. In the following, a = 100 µ m s − is the nuclear pore permeabilityand ν N = 100 µ m and ν C = 500 µ m are the nuclear and cytoplasm compartment volumes, respectively. Theexponential free energy difference defined in Eq.(S35)(S43) are set to be: e ∆ F = e ∆ ˜ F = 50. Note that volume factorsmodulate the permeabilities in the usual manner (see Eq.(1) in [34]): Namely, rate constants of cytosolic species (i.e. X , X , X , X , X ) across the nuclear membrane is given by k ± α = a/ν C with α = 3 , , , ,
11 (i.e., reactions thatinvolve crossing the nuclear pores). Rate constants for the nuclear counterparts (i.e. X , X , X , X , X ) are,on the other hand, given by k ± α = a/ν N , with α = 3 , , , ,
11. For example, kinetics equations for X (cytosolicNTF2-RanGDP complex) and X (nuclear NTF2-RanGDP complex) should read (c.f. Eq.(S14) and Eq.(S21)): d [ X ] dt = k +1 [ X ][ X ] − k − [ X ] − aν C ([ X ] − [ X ]) (S26) d [ X ] dt = − k +5 [ X ] + k − [ X ][ X ] + aν N ([ X ] − [ X ]) . (S27)reaction K D or k in /k out k + k − References and Note1 25 nM 0 . − s − ) 2.5 (s − ) From [30]2 ∼ e ∆ ˜ F /θ (s − ) 1 (nM − s − ) estimate using Eq.(S43)3 ∼ − ) 1 or 0.2(s − ) k ± = a/ν N or k ± = a/ν C − s − ) 1 (s − ) From [30]5 ∼ e ∆ F × θ (s − ) 1 (nM − s − ) estimate using Eq.(S35)6 ∼ − ) 1 or 0.2 (s − ) k ± = a/ν N or k ± = a/ν C − ) 1 (nM − s − ) From [54]8 ∼ − ) 1 or 0.2(s − ) k ± = a/ν N or k ± = a/ν C ∼ − ) 1 or 0.2(s − ) k ± = a/ν N or k ± = a/ν C
10 20 nM 1 (nM − s − ) 20 (s − ) From [54]11 ∼ − ) 1 or 0.2(s − ) k ± = a/ν N or k ± = a/ν C Labels SpeciesN NTF2Im Importin (importin)RD RanGDPRT RanGTPN · RD NTF2+RanGDP complexN · RT NTF2+RanGTP complexIm · RD Importin+RanGDP complexIm · RT Importin+RanGTP complexfD (free) GDPfT (free) GTP
A. Reaction 5: Ran exchange mediated by RanGEF
FIG. S2:
Illustration of Ran GDP to GTP exchange reaction mediated by RanGEF
The goal is to estimate the K D for the following reaction:[ N · RD ] k +5 (cid:10) k − [ N ] + [ RT ] , (S28)namely, k +5 k − = [ N ][ RT ][ N · RD ] (S29)Consider the following two constituting reactions[ N · RD ] + [ f T ] k + α (cid:10) k − α [ N · RT ] + [ f D ] (S30)[ N · RT ] k + β (cid:10) k − β [ N ] + [ RT ] (S31)This implies (neglecting labels of steady states SS), k + α k − α = [ N · RT ][ f D ][ N · RD ][ f T ] (S32) k + β k − β = [ N ][ RT ][ N · RT ] (S33)Thus we can reexpress Eq.(S32) using Eq.(S33): k + α k − α = 1[ N · RD ] [ f D ][ f T ] · (cid:32) k − β k + β [ N ][ RT ] (cid:33) = (cid:18) [ N ][ RT ][ N · RD ] (cid:19) · [ f D ][ f T ] k − β k + β = k +5 k − · [ f D ][ f T ] k − β k + β (S34)Thus k +5 k − = k + α k − α · k + β k − β · [ f T ][ f D ] ∼ O (1) · k e ∆ F · exp (cid:18) log [ f T ][ f D ] (cid:19) (S35)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 whichis much larger than 1: ∆ F >>
1. This can also be understood using Eq.(S33) by noting that in the nucleus NTF2seldom binds to RanGTP. Finally, since the free GTP to GDP ratio, [ f T ] / [ f D ], is buffered by cellular metabolism,we simply treat the last term as a free parameter θ . Note that there is far more free GTP than Ran on a molar basis.After rescaling time by τ ← tc k diff , with k diff = 10 sec − nM − and c represent the diffusion-limited reaction rateand the characteristic molar concentration (set to 1nM), respectively, and approximating e ∆ F ≈ ∼
100 , one canestimate ( k +5 /k − ) ∼ (10 ∼ × θ , where θ := [ f T ] / [ f D ] is treated as a free parameter. B. Reaction 2: Ran exchange mediated by RanGAP
FIG. S3:
Illustration of RanGTP to RanGDP exchange reaction mediated by RanGAP
We aim to approximate K D for such reaction:[ Im · RT ] k +2 (cid:10) k − [ Im ] + [ RD ] , (S36) k +2 k − = [ Im ][ RD ][ Im · RT ] (S37)Similarly the estimation is based on the following two steps:[ Im · RT ] + [ f D ] k + γ (cid:10) k − γ [ Im · RD ] + [ f T ] (S38)[ Im · RD ] k + δ (cid:10) k − δ [ Im ] + [ RD ] (S39)This implies (neglecting labels of steady states SS), k + γ k − γ = [ Im · RD ][ f T ][ Im · RT ][ f D ] (S40) k + δ k − δ = [ Im ][ RD ][ Im · RD ] (S41)Thus we can reexpress Eq.(S40) using Eq.(S41): k + γ k − γ = 1[ Im · RT ] [ f T ][ f D ] · (cid:18) k − δ k + δ [ Im ][ RD ] (cid:19) = (cid:18) [ Im ][ RD ][ Im · RT ] (cid:19) · [ f T ][ f D ] k − δ k + δ = k +2 k − · [ f T ][ f D ] k − δ k + δ (S42)Thus k +2 k − = k + γ k − γ · k + δ k − δ · [ f D ][ f T ] ∼ O (1) · k e ∆ ˜ F · exp (cid:18) log [ f D ][ f T ] (cid:19) ∼ k × (10 ∼ × θ (S43) III. STANDARD ESTIMATE OF DIFFUSION-LIMITED REACTION RATE
Considering two type of molecules A and B diffusing in a viscous environment. According the Fick’s law thediffusion flux of one type of molecule assuming the other is at stationary is given as (cid:126)J µ = − D µ ∇ [ µ ] , (S44)where µ = A, B and D µ is the diffusion constant of molecule µ . Assuming spherical symmetry one can integrateFick’s law to get the total number of molecules diffusing through a given surface area: φ tot = 4 πR ( D A + D B )[ A ][ B ] , (S45)where R is the sum of molecular radii of A and B. The factor k a := 4 πR ( D A + D B ) is exactly the reaction rate ofthe overall catalytic reaction under the assumption that the process is diffusion-limited (i.e. upon A and B are incontact, the intermediate complex AB immediately reacts to form the final product P): A + B k a −→ P (S46)Finally, recall Stokes-Einstein relation: D = k B T / (6 πηa ) with molecule (spherical) particle radius a , we have k a = 4 π (2 a ) (cid:18) × k B T πηa (cid:19) (cid:20) m sec (cid:21) → (cid:18) k B T η (cid:19) N A (cid:20) · sec (cid:21) , (S47)where N A is the Avogadro’s constant. The factor 10 appears because we convert the SI unit of volume m to liter.Using η = 10 − (Pa · sec)= 10 − kg/m/sec, we get k diff := k a ∼ × [M − · sec − ] = 10 [nM − · sec − ] (S48) IV. SIMULATION CODES
MATLAB (cid:114) simulation codes are available for download at http://physics.bu.edu/~chinghao/thermo_transport/codes/
V. ENTROPY PRODUCTION
The distinct feature of systems out of thermodynamics equilibrium is the continuous production of entropy. Therate of entropy change (in time) consists of two parts: (i) the internal entropy change and (ii) the exchange of entropywith the environment dSdt = Π − Φ , (S49)where S is the entropy of the system and Π is the rate of entropy production and Φ denotes the rate of entropy flowfrom the system to the outside. Within this context, the 2nd law of thermodynamics dictates Π ≥ • Equilibrium steady states (ESS): Π = Φ = 0 • Nonequilibrium steady states (NESS, i.e. irreversible): Π = Φ > ddt P i ( t ) = (cid:88) j [ P j ( t ) W ji − P i ( t ) W ij ] , (S50)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 appropriatemicroscopic description for the nonequilibrium system amounts to (i) having well-defined entropy for the irreversiblesystems and (ii) the entropy production rate Π should respects the non-negativity and should vanish when systemequilibrates (i.e. when it exhibits reversibility). For systems described by the master equation, thermodynamicsequilibrium is essentially the detailed-balanced condition: P i W ij = P j W ji . The solution for the first is the Boltzmann-Gibbs entropy: S ( t ) = − k B (cid:88) i P i ( t ) log P i ( t ) , (S51)while the entropy production rate is advanced by the Schnakenberg description [39]:Π( t ) = k B (cid:88) ij [ P i ( t ) W ij − P j ( t ) W ji ] log W ij P j ( t ) W ji P i ( t ) (S52)By imposing dS ( t ) /dt = 0 to Eq.(S51) at steady state and using Eq.(S50)(S52) to simply, one gets the steady stateentropy production rate: Π = k B (cid:88) ij W ij P j log W ij W ji , (S53)where P i is the stationary probability distribution. It’s easy to check that Π − Π( t ) = dS/dt → EP := Π × T using (S53) (in the same spirit as F = U − T S , where F is the Helmholtz free energy), we have EP = k B T (cid:88) i,j P SSi W ( i, j ) log W ( i, j ) W ( j, i ) , (S54)where P SSi is the steady state probability distribution of state i while W ( i, j ) denotes the transition probability fromstate i to state j . Concretely, P SSi is the fraction of reactants participating in the transition reaction starting fromstate i while W ( i, j ) can be calculated from the relevant reaction fluxes. For example, P SS is the molar fraction ofcytoplasmic NTF2-RanGDP ( ∼ [ X ]) whereas W (3 ,
10) is the transition probability of of NTF2-RanGDP into thenucleus: W (3 ,
10) = ( k +6 [ X ]) / ( k +6 [ X ] + k − [ X ]) (See FIG.S1C). Note that in principle the summation in Eq.(S54) toobtain the entropy production is taken over all links in FIG.S1B. It can be separated, however, into reactions 1-6 thatrepresent the Ran futile cycle (i.e. FIG. S1C) and the remaining reactions 7-11 that do not explicitly involve Ran.The latter are essentially passive and could be expected to satisfy detailed balance at steady state. We have confirmednumerically that the contributions of reactions 7-11 in Eq.(S54) cancel to zero, so the total entropy production isequal that evaluated in the futile cycle alone. We can also inspect the reactions qualitatively. Reaction 9 is trivially indetailed balance because the concentrations X and X are equal in steady state. These represent the importin-cargocomplex in cytoplasm and nucleus, respectively. Clearly the net cargo binding/unbinding to importin in the cytoplasmmust balance that in the nucleus as well, so the contributions of reactions 7 and 10 cancel. Finally, the free receptorsimportin and NTF2 exchange passively across the nuclear envelope (reactions 8 and 11). Again in steady state theircycle fluxes must balance, so their contributions to the entropy production sum also cancel. VI. WEAK SENSITIVITY TO THE GTP:GDP RATIO θ Ultimately the (chemical) free-energetic fuel driving the transport cycle is the ratio of GTP to GDP, θ , which is heldout of equilibrium by cellular metabolism. We find that the nuclear localization ratio, as well as biased concentrationsof transport receptors, is not strongly dependent on θ . This reflects the counterbalancing effects of RanGEF andRanGAP as described in Section II above. θ [C] nu /[C] cyto [Im] nu /[Im] cyto [Im-C] nu /[Im-C] cyto
20 40 60 80 100 θ : free GTP to free GDP ratio R an G D ( T ) P f r a c t i on n RanGDP n RanGTP θ [C] nu /[C] cyto [Im] nu /[Im] cyto [Im-C] nu /[Im-C] cyto
20 40 60 80 100 θ : free GTP to free GDP ratio R an G D ( T ) P f r a c t i on n RanGDP n RanGTP
A B
FIG. S4:
Effect of free GTP to free GDP ratio θ on nuclear localization . (A) [NTF2] tot =100 nM (B) [NTF2] tot =10 nM.Other parameters are the same for both panels: [C] tot =10 nM, [Ran] tot =75 nM and [Im] tot =100 nM. Kinetics rate constantsused are given in SM Section II. [C] tot = 20 nM [C] tot = 50 nM[C] tot = 100 nM ABC
FIG. S5:
Phase diagram of nuclear localization and entropy production . (A) total cargo concentration [C] tot = 20nM (B) [C] tot = 50 nM and (C) [C] tot = 100 nM. Other parameters used are the same as in FIG.3 and 4: [Ran] tottot