Diffusiophoresis in Cells: a General Non-Equilibrium, Non-Motor Mechanism for the Metabolism-Dependent Transport of Particles in Cells
DDiffusiophoresis in Cells: a General Non-Equilibrium, Non-Motor Mechanism for theMetabolism-Dependent Transport of Particles in Cells
Richard P. Sear
Department of Physics, University of Surrey, Guildford, GU2 7XH, UK ∗ The more we learn about the cytoplasm of cells, the more we realise that the cytoplasm is notuniform but instead is highly inhomogeneous. In any inhomogeneous solution, there are concen-tration gradients, and particles move either up or down these gradients due to a mechanism calleddiffusiophoresis. I estimate that inside metabolically active cells, the dynamics of particles can bestrongly accelerated by diffusiophoresis, provided that they are at least tens of nanometres across.The dynamics of smaller objects, such as single proteins are largely unaffected.
The cytoplasm of cells is far from thermodynamic equi-librium, and far from uniform [1–4]. Here, I consider theeffect of concentration gradients on the motion of largeparticles in the cytoplasm. Large means tens of nanome-tres and above, so an example would be a large proteinassembly. In the cytoplasm, particles and molecules arenot diffusing alone in a dilute solution, but are moving ina concentrated, active and non-uniform mixture of pro-teins, nucleic acids, metabolites such as ATP, small ionssuch as potassium, etc. A schematic of a particle in thecytoplasm, is shown in Fig. 1.It is well known in the fields of colloids [5–18] and ofliquid mixtures [10, 19], that particles of one species willmove in response to a gradient in the concentration ofanother species. In colloids this is called diffusiophore-sis. Diffusiophoresis is typically defined [5–8, 10] as themotion of a larger particle immersed in concentrationgradients of smaller molecules, when both are in a sol-vent (such as water). Although often difficult to mea-sure, there are clearly gradients inside metabolically ac-tive cells. So there must be diffusiophoresis occurring incells, the question is: Does diffusiophoresis make a signifi-cant contribution to the transport of some species? Here,I determine that the answer to this question is proba-bly yes for particles at least tens of nanometres or moreacross, but no for individual protein molecules.I start with the standard Brownian-dynamics approxi-mation for the position of a particle, r ( t ) [20]. With thisapproximation, we can write the change in position overthe time interval t to t + δt , as [20], r ( t + δt ) = r ( t ) + (cid:16) D ( c )P δt (cid:17) / ρ +[ v adv + U ] δt + D ( c )P kT f δt (1)This equation includes four possible transport mecha-nisms for the particle. The second term on the right-handside is the conventional thermal diffusion term. There D ( c )P is the diffusion constant for thermal diffusion in thecytoplasm, and ρ is a vector of random numbers drawnfrom a Gaussian distribution of mean zero, and standarddeviation one. The physics of this term is that the par-ticle is constantly being bombarded by the surrounding molecules, due to their thermal energy. This tends tomove the particle around, but this motion is opposed bythe friction between a moving particle and these samemolecules.The third term on the right-hand side contains the ad-vection and phoresis terms. Advection is motion of a par-ticle because it is carried along by the cytoplasm flowingat a local velocity v adv . U is the diffusiophoretic veloc-ity. If the cytoplasm is inhomogeneous (has gradients) ata point, then locally the stresses on the particle are alsoinhomogeneous, which means that there are unbalancedstresses which will cause the particle to move relative tothe local fluid [5, 6, 8, 12–14]. This local motion is calleda slip velocity, and can be caused by gradients in any-thing. Here we will consider gradients in concentration,and then this slip velocity is a diffusiophoretic velocity, U . Both U and v adv are zero in a system at equilibrium,so in a cell they must come from the cell’s metabolism.The last term on the right is motion due to a force f on the particle, for example due to a motor proteinpushing or pulling on the particle. In eukaryote cells, itis well established that motor proteins pull many cargosaround the cell. Although this is an important process,it is well studied [21] and so here I only consider particlesnot being pulled by motor proteins. FIG. 1. Schematic of a particle (blue), immersed in a cyto-plasm with gradients in the concentrations of two metabolites(red and green). Proteins are magenta. Small ions are notshown. The dotted line indicates the approximate extent ofthe particle/cytoplasm interface, where slip occurs, creatinga gradient in velocity, and hence the diffusiophoretic slip ve-locity U . a r X i v : . [ q - b i o . S C ] M a r To motivate this study, let us consider experimentalevidence for metabolism-dependent mobility of particlesin cells. Parry et al. [22] studied the dynamics of large,around 50 to 150 nm across, particles in the cytoplasmof bacteria (including
E. coli ). The particles includedgranules of an enzyme, a plasmid (of a type withoutan active partitioning system), and particles formed bya self-assembling viral protein. They found that thedynamics of particles in this size range, dramaticallyslowed down when the metabolism was shut off. Themetabolism was shut off by depleting ATP and GTP us-ing 2,4-dinitrophenol (DNP).Parry et al. [22] tracked the displacement of parti-cles ∼
100 nm across over periods of 15 s. When themetabolism was shut down, the particles made manyfewer displacements of order hundreds of nanometres,and this dramatically slowed movement. So we arelooking for a metabolism-dependent mechanism that cantransport assemblies 100 nm across at an effective speedof up to ∼
100 nm/s for periods of 10 s. Here I suggestthat diffusiophoresis is a possible mechanism.It is worth noting that both with and without an activemetabolism, the distribution of displacements was veryfar from the Gaussian distribution expected for thermaldiffusion in a uniform background. This non-Gaussiandistribution implies that the cytoplasm is strongly non-uniform.The results of Parry et al. [22] are for bacteria. Thepresence of motors and the cytoskeleton in eukaryotecells, will make it difficult to unambiguously observe dif-fusiophoresis in eukaryotes. However, I note that Ba-janca et al. [23] studied the motion of the protein dys-trophin in the muscle cells of zebrafish embryos. Thisprotein has been estimated to be 100 nm long. Theyfound effective diffusion constants of order 1 µ m /s, onlyan order of magnitude lower than that of GFP ( ∼ et al. [26], Surovtsev etal. [27], and Walter et al. [28] all modelled what is calledthe ParA/B [29] system of segregating plasmid DNA inbacteria during cell division. The plasmid moves in aconcentration gradient of the ParA protein, and so theirwork [26–28] is an example of diffusiophoresis. However,the molecular interactions and stresses responsible for the plasmid motion were not explicitly modelled in that work[26, 28]. Here I do consider these interactions and stresseshere, and so my work is complementary to that earlierwork [26, 28]. Surovtsev et al. [27] used a Brownian dy-namics model for the interaction, this may overestimatethe strength of diffusiophoresis, as discussed by Sear andWarren [8, 9].There are thousands of species inside cells, many ofwhich may have gradients. To keep things simple, I workwith the gradient in just one example species: the abun-dant metabolite ATP. I select ATP as a test candidateas it is known to interact strongly with proteins at theconcentrations found in cells [30], and to turnover rapidly[21]. The rapid turnover implies large fluxes between thesources and sinks, and the fluxes imply gradients, be-tween these sources and sinks. Thus ATP is my bestcandidate for an abundant species whose concentrationgradients I can estimate. When ATP is consumed ADPis produced, so although here I will refer to an ATP gra-dient for simplicity, in reality it is two gradients, oneof ATP and one of ADP, with the opposite sense. Theeffects of these two opposing gradients may partially can-cel, weakening diffusiophoresis, but as the molecules aredifferent, any cancellation will be partial. Note that smallions such as potassium and chloride are even more abun-dant than ATP inside cells, but as they do not turnoverare expected to have only negligible concentration gra-dients. The numbers needed to characterise cells in mycalculations are gathered together in Table I in the Sup-plemental Material. A particle moving up an ATP gra-dient is shown in Fig. 2.Inside cells, thermal energy and momentum can movemuch more rapidly than even small molecules. So, I ex-pect thermal and pressure gradients to be negligible, seethe Supplemental Material for the justification of this as-sumption.In order to estimate the sizes of the gradients inATP concentration inside cells, I start by estimating thetimescale for ATP to diffuse across a typical bacterial cell1 µ m across. The diffusion constant of ATP both in wa-ter and in cells [21, 31] is of order 100 µ m /s. So an ATPmolecule diffuses across the cell in of order 0.01 s.An active 1 µ m bacterial cell is estimated to have10 ATP molecules and to consume 10 ATP moleculeseach second, see Table I of the Supplemental Material.This gives a time of 1 s between production by ATPsynthase, and consumption. A lifetime 100 times the dif-fusion time implies gradients of order 1% to 10% across acell 1 µ m across. For an ATP concentration of 10 / µ m ,we have gradients of 10 / µ m to 10 / µ m . I will use thegradient value 10 / µ m below. See the SupplementalMaterial for a more detailed calculation that also givesgradients of this size. These are very simple estimatesof steady-state gradients, the gradient will presumablyvary in space and time as particular sources (ATP syn-thase) and sinks (ATP consuming proteins) move. But asATP diffuses much faster than membrane proteins suchas ATP synthase, ATP gradients may often be close to asteady state.The diffusiophoretic velocity U is proportional to thegradient in the concentration c , of a solute U = Λ PH ∇ c (2)There is a standard Derjaguin/Anderson expression [6,8, 12, 32] for the coefficient Λ PH that relates the con-centration gradient to the diffusiophoretic velocity. Thisexpression is valid for a large particle with an interaction φ ( z ) between the particle surface and a smaller speciesthat has a concentration gradient ∇ c . Here z is the dis-tance separating the smaller species from the surface ofthe particle. Between the smaller species and the sur-face is a continuum solvent with viscosity η . The Der-jaguin/Anderson expression isΛ PH = k B Tη (cid:90) ∞ z [exp ( − φ ( z ) /k B T ) −
1] d z (3)Note that as the particle surface is interacting with thesmaller species in water, φ ( z ) is an effective interactionfree energy.From Eq. (3), we see that the diffusiophoretic coeffi-cient Λ PH is approximately k B T divided by the solventviscosity η , and multiplied by the square of the interac-tion range, which we denote by L . So, we obtain theapproximate expressionΛ PH ∼ ± k B T L /η (4)Λ PH is positive for attractive interactions, and then U is directed to higher concentrations of the solute. Forrepulsive interactions the sign is reversed. The integral FIG. 2. Schematic of part of a prokaryote cell, with anATP concentration gradient indicated by shading. Sources ofthe gradient are ATP synthases, in magenta, while our modelassumes that sinks (ATP consuming proteins) are uniformlydistributed in the cytoplasm. We show one particle movingup the concentration at a diffusiophoretic velocity U . in Eq. (3) is of order − L for a repulsive φ ( z ) that is ∼ k B T or stronger over a range L , and is of order + L for an attractive φ ( z ) that is of order k B T over a range L .For a stronger attraction, the integral will be larger, butEq. (3) is an approximation [6, 8, 12, 32], and will breakdown for strong enough attractions. To summarise, theapproximation of Eq. (4) should be the correct order ofmagnitude unless there are attractions (cid:29) k B T in whichcase the Derjaguin/Anderson approximation fails. So Ido need to assume that, for the particles studied by Parry et al. [22], the interactions between the protein and ATPare not strongly ( (cid:29) k B T ) attractive.Here we estimate the diffusiophoretic velocity U of aparticle in a concentration gradient of ATP. The diffu-siophoretic coefficient depends on the free energy of par-ticle/ATP interaction (cid:15) , the range of the surface/ATPinteraction L , and the solvent viscosity η . I approximatethe viscosity by that of water, η ∼ − Pa s. The freeenergy of interaction (cid:15) , I take to be k B T = 4 × − J,and the range L to be 1 nm. From ATP’s diffusion co-efficient of 500 µ m /s, ATP has a Stokes-Einstein radiusof 0 . PH = 4 × − µ m /s, and U ∼ × − |∇ c ATP | [ ∇ c ATP in µ m − ] (5)for c ATP the ATP concentration. We set L = 1 nm, asthat is the order of magnitude of both the size of ATPitself and of the Debye screening length in the cytoplasm.ATP is both highly charged and contains organic groups,so its nature is a little amphiphilic. Therefore, the inter-actions with a protein surface will be complex [30] butwill include electrostatic interactions, with a range of theDebye length. Interactions beyond a few nanometres areexpected to be weak [33].Above, we estimated the gradient in ATP concentra-tion to be 10 / µ m = 10 / m . Putting that gradient inEq. (5), we have a diffusiophoretic speed U ∼
400 nm/s.This is large enough to be consistent with the motion ob-served by Parry et al. [22], so long as the gradient lastsfor of order 10 s or more. Our estimates for the gradi-ents, are steady-state estimates, so they should satisfythis constraint.This is the key result of this work: Physically reason-able concentration gradients of one abundant metabolite,can drive motion of large particles that is fast enough tobe significant for transport inside cells, and fast enoughto be observable. Note that as typical proteins diffuseacross a 1 µ m cell in less than 1 second, an additionalspeed of 100 nm/s has little effect on the dynamics ofsingle proteins, so diffusiophoresis should not affect sig-nificantly affect protein dynamics.My estimate of speeds of hundreds of nanometres persecond is highly approximate, so I would like to commenton sources of uncertainty. It relies on my estimate of thegradients. These could be out by an order of magnitude,and it is difficult to assess how gradients vary in spaceand time. It is also worth noting if the phoretic veloc-ity is directed towards a source of a gradient, there willbe positive feedback as particles will be pulled towardsthe source where the gradient is steepest, an effect thatis magnified when the source itself can move [34]. Thephoretic interaction could pull the particle into contactwith the source, where the concentration gradients arestrongest. This was a theory and simulation study. Ex-periments in vitro by Zhao et al. [35] found that phoreticinteractions can help enzymes move together. Thus ourestimates for U may be underestimates when the phoreticvelocity is towards gradient sources.The estimated speed also relies on our value for Λ PH .The Anderson-Derjaguin expression [6, 8, 12, 32] appliesto dilute systems (the cytoplasm is not dilute), and relieson flow in a fluid interfacial region of width L , drivenby the stresses there. It is uncertain how good theseapproximations are in the cytoplasm.There have been ( in vitro ) experimental studies of pro-teins moving due to active processes. Sen and coworkers[35–38], and Granick and coworkers [39], have both stud-ied enzymes, such as urease, in dilute solution. Bothgroups find that enzymes move faster when they arecatalysing reactions, and Zhao et al. [36] also foundthat active enzymes could speed up the motion of otherspecies. Future work could consider solutions with con-centrations of energy-consuming molecules that are closerto those found in the cytoplasm. Jee et al. [39] have al-ready considered the effect of a crowding agent. Futurework could also use microfluidics to create gradients inATP, in order to look for phoresis.My estimate is for prokayotes. Milo et al. [21] discussthe energy consumption of mammalian cells. The powerconsumption per unit volume of a fibroblast can be com-parable to that of E. coli . Assuming distances of a fewmicrometres between where ATP is consumed, and mito-chondria, the ATP gradients in an active fibroblast willbe comparable to those in growing
E. coli . So diffusio-phoretic speeds should also be comparable.We have only considered a gradient in one of the thou-sands of species in a cell (ATP), and models of theParA/B system of moving plasmids in bacteria [27–29]also only consider ParA gradients. Future work will needto deal with the multicomponent nature of the cytoplasm.Systems that have evolved to localise species such as plas-mids presumably have to work against the forces due toflutuating gradients in the other species present in thecell.Diffusiophoresis is unlikely to be the only non-motor-driven metabolism-dependent transport mechanism incells. See the Supplemental Material for more discus-sion of these other potential transport mechanisms. Ineukaryote cells, there is also transport of particles as thecargos of motor proteins.In conclusion, the more we learn of the cytoplasm ofboth prokaryote and eukaryote cells, the less uniform they appear to us [1–4]. There must be many gradientsin cells, and so phoresis must be occurring in essentiallyall cells. However, quantifying phoretic speeds in cells isdifficult. Cells are complex, and the size of gradients isunknown. In addition the interactions needed to estimatediffusiophoretic coefficients Λ PH are also unknown. HereI estimated Λ PH for ATP, and estimated the size of gra-dients of ATP in an active bacterial cell such as E. coli . Ipredicted that diffusiophoretic speeds of order 100 nm/sare possible. This is large enough to be consistent withthe motions observed by Parry et al. [22], for large (50to 150 nm) particles. However, the complexity of thecytoplasm means that is very difficult to unambiguouslyshow that observed movements are due to one specifictransport mechanism. Experiments on simpler, in vitro ,systems will probably be required to separate out differ-ent non-thermal-diffusion contributions to transport incells.I would like to thank Patrick Warren for teaching memuch of what I know about diffusiophoresis, and DaanFrenkel for many illuminating discussions. I would alsolike to thank the organisers, Julian Shillcock, MikkoHaataja and John Ipsen, and participants of the CE-CAM workshop
Liquid Liquid Phase Separation in Cells ,for helpful questions and feedback. The author confirmsthat no new data were created during this study.
SUPPLEMENTAL MATERIAL
NUMBERS FOR PROPERTIES OF ABACTERIAL CELL
To estimate diffusiophoretic velocities in a bacterialcell, we need estimates for a number of properties of atypical bacterial cell, by which I mean
E. coli . These arecollected in Table I.Many of these numbers are from the excellent refer-ence,
Cell Biology by the Numbers by Milo and Phillips.This is available as both a book [21], and an online re-source [44].
ESTIMATION OF THE SIZE OF GRADIENTS INTHE TEMPERATURE
The relative rates of heat and molecular diffusion, ischaracterised by the Lewis number: Le= α/D . Here α and D are the thermal diffusivity, and the diffusion con-stant of the molecule, respectively. Even for fast diffus-ing species, such as molecules like ATP, the diffusion con-stant D ∼ µ m /s [21, 31], while for water (and hencethe cytoplasm which is mainly water) α ∼ µ m /s [45].Thus for small molecules in the cytoplasm, Le ∼ , andso temperature gradients relax about a thousand timesfaster than gradients in ATP. bacteria cell size 1 µ mtotal protein concentration [21] c P (cid:39) × / µ m in cytoplasm c P (cid:39) × /celltotal metabolite concentration [21, 40] c M (cid:39) / µ m in cytoplasm c M (cid:39)
200 mM10 /cellATP concentration [21, 40] c A TP ∼ / µ m in cytoplasm 10 /cellATP diffusion constant [31, 41] D ATP ∼ × µ m /s(in solution & in vivo)ATP hydrodynamic diameter [21] 1.4 nmATP molecular weight 507 g/molDebye length κ − in 200 mM KCl 0 . η W ∼ − Pa sPower consumption P ∼ − Wof bacterial cell[21, 42] P ∼ ATP/sTABLE I. Table of the values I use for properties of the cy-toplasm. The value for the concentration of ATP is fromBennett et al. [40], who give 10 mM as the ATP concentra-tion in their rapidly growing
E. coli . Other measurements[21] for
E. coli have given similar but typically a little lowervalues, and Traut [43] gives 3 mM ∼ / µ m for examplemammalian cells. The concentration of small cations (eg K + and Na + ) and anions (eg Cl − ) in E. coli will vary with ex-ternal growth conditions, but 200 mM is a typical value [21].N.B., 1 mM (cid:39) × / µ m . Most of the other numbers arefrom Cell Biology by the Numbers [21, 44].
Momentum diffuses with a diffusion constant of thekinematic viscosity, about ν ∼ µ m /s for water. Thusin cells pressure gradients relax even faster than temper-ature gradients, and we therefore expect pressure gradi-ents to be negligible.We can estimate the size of temperature gradients asfollows. A growing bacterial cell of volume 1 µ m hasa power consumption of order 10 − W, see Table I. Ifwe naively assume that the metabolism is concentratedin say, the left-half of the cell, then crossing the mid-point of the cell we have of order 10 − W of heat, or ∼ , for a cell cross-section of 1 µ m . The ther-mal conductivity of water is of order 1 W/K/m [45], soa flux of 1 W/m is driven by a gradient of order 1 K/m.Thus we conclude that active bacterial cells have tem-perature gradients across of them that are of order1 K/m, or that the temperature differences across thecells are no more than 1 µ K. This is a general observation,cells are made of matter with high thermal conductivity,and so cannot support significant temperature gradients.So, presumably, the recent claim [46] that mitochondriaare 10 K hotter than the surrounding cytoplasm is incor-rect.
GRADIENT IN ATP ACROSS THE CYTOPLASM
To obtain a simple estimate of the size of ATP gra-dients I assume a one-dimensional geometry in whichATP synthases are along two parallel flat cell walls at z = ± w/
2. I assume that the system is at steady state.As the time taken for ATP to diffuse across the cell is only0.01 s, steady state will be achieved in much less than asecond. The cell width w = 1 µ m. To get a simple one-dimensional model I then ignore gradients parallel to thewall, and assume the concentration of ATP depends onlyon the distance z from the wall. If the proteins consum-ing ATP (= the ATP sinks) are uniformly distributed,then the concentration of ATP in the cytoplasm obeys D ATP (cid:18) d c ATP ( z )d z (cid:19) − k ATP c ATP ( z ) = 0 (6)Here D ATP is the diffusion constant for ATP, and k ATP isthe rate constant for ATP consumption, assumed uniformin the cytoplasm. If the ATP synthases along the cellwall maintain the ATP concentration at a fixed value c ATP ( z = ± w/ c ATP ( z ) = c ATP ( z = ± w/
2) cosh ( − z/l G )cosh ( w/ (2 l G )) (7)with the lengthscale of the gradient l G =( D ATP /k ATP ) / . The gradients are then of order c ATP ( z = ± w/ /l G . For D ATP = 100 µ m / s and k ATP = 1/s, l G = 10 µ m. Using c ATP ( z = ± w/
2) =10 / µ m , the gradients |∇ c ATP | ∼ / µ m . Thisone-dimensional model neglects both the discrete natureof the ATP source (ATP synthase at the membrane),and fluctuations. GRADIENTS OF METABOLITES NEAR AMETABOLON
In the main part of this paper we considered onemetabolite: ATP. Here we consider a metabolite pro-duced/consumed by a large protein complex — theselarge complexes are sometimes called metabolons [47].Metabolons are physical assemblies of many proteins, in-cluding copies of multiple species of enzyme in the samepathway, i. e. if a synthetic pathway requires enzymes A,B and C, with B catalysing a reaction on a product ofenzyme A, etc, then many of the copies of A, B and Cmay be together in an physical assembly of perhaps hun-dreds or thousands of molecules. This may enhance theefficiency of this pathway [47]. I will show that near thesemetabolons, we should also expect large gradients in theconcentrations of metabolites.For simplicity, I approximate a metabolon by a sphereof radius R MN , producing fluxes of order k = 10 /s, ofa single molecule with diffusion constant D M . A singleurease can catalyse the hydrolysis of urea at a rate of10 /s [39], so this flux could be produced by a hundredcopies of a high turnover enzyme. I assume that just thereactants interact with the particle, including productsjust complicates the expressions a little.Our model is essentially that studied in detail by Reigh et al. [34]. Following Reigh et al. , we estimate the steady-state gradient. At steady state, the concentration c M obeys Laplace’s equation ∇ c M = 0. I use spherical co-ordinates centred on the metabolon, with r the distancefrom the centre of the metabolon. Then the flux is ∇ c M ( r ) = − k πD M r ˆ r (8)as this gives the required total flux k over a surface enclos-ing the metabolon. For a metabolite diffusion coefficient D M ∼ µ m /s, and k = 10 /s, the flux is ∇ c M ( r ) ∼ − r ˆ r (9)where we approximated 4 π by 10, as the expression isapproximate. At a distance of order 100 nm from thecentre of the metabolon, the gradient is of order 10 /m or 10 / µ m .Reigh et al. [34] tested the simple theory above by es-sentially exact computer simulations of a simple model.There was semiquantitative agreement between the the-ory and computer simulations. GRADIENTS OF SMALL IONS SUCH ASPOTASSIUM AND CHLORIDE
Small ions such as potassium, sodium and chloride areabundant in cells, 10 / µ m ∼
100 mM [21], but we ex-pect the gradients in their concentration to be very small.So we do not expect significant phoretic effects due togradients in the concentration of small ions, in cells grow-ing in an environment where the osmotic pressure is con-stant.The timescale for potassium turnover in
E. coli hasbeen measured at of order 10 s, [48]. Potassium is themost abundant cation in cytoplasm, while chloride isthe most abundant anion [21]. Presumably, due to elec-troneutrality, the flux of anions and cations has to be thesame.The diffusion constant of potassium chloride in wateris of order 10 µ m /s [49], so a potassium ion will diffuseacross a bacterial cell in about 1 ms. This is a factor of10 times smaller than the timescale for potassium up-take, and so we expect the gradients in cells, of the con-centration of potassium, and chloride, to be very small.As diffusiophoresis is driven by gradients, this impliesthat diffusiophoresis driven by small ions should typicallybe irrelevant. An exception may be pollen tubes, a very specialised and large type of cell where large gradientsare found, see the work of Lipchinsky [25]. ALTERNATIVE METABOLISM-DEPENDENTMECHANISMS OF TRANSPORT IN CELLS
Diffusiophoresis is unlikely to be the only non-motor-driven metabolism-dependent transport mechanism incells. In this section, I briefly consider two other possiblemechanisms for transport in cells, that rely on the cell’smetabolism. These are advection of a particle due to flowin the cytoplasm, and metabolism-dependent processesaccelerating thermal diffusion by making the cytoplasmless sticky. In eukaryote cells, there is also transport ofparticles as the cargos of motor proteins.
Transport by flow
Cytoplasmic streaming
It is clear that in a number of large cells ( (cid:38) µ m),there is significant flow of the cytoplasm. This corre-sponds to a large v adv term in Eq. (1) in the main text,with a v adv that is relatively uniform over large regions ofthe space, and relatively constant in time, This is some-times called cytoplasmic streaming [50, 51]. Cytoplas-mic streaming is driven by motors and the cytoskeleton,and it clearly contributes to transport in a number ofvery large cells [50, 51]. These large cells include 100 µ m Drosophila oocytes [51], and plant cells that can be cen-timetres long [50]. Speeds of tens of nanometres persecond were measured in the oocytes, while much fasterspeeds are found in larger cells. I am not aware of stud-ies of cytoplasmic streaming in eukaryote cells of moretypical size, ∼ µ m across, or in prokaryote cells. Random stirring of the cytoplasm
Mikhailov and Kapral [52, 53] have considered stirringof the cytoplasm by energy-consuming but non-motorproteins. Here by stir, I mean generate transient flowin more-or-less random directions in the cytoplasm, asopposed to the fast directed flow seen in cytoplasmicstreaming. So, here v adv varies rapidly in space and time.They considered proteins that consume ATP and gener-ate force dipoles, which stir the surrounding cytoplasm,thus accelerating diffusion in this cytoplasm. Note thatproteins free in the cytoplasm generate force dipoles notforces, due to Newton’s Third Law.They studied active proteins at a concentration c FD , that generate force dipoles of root-mean-strengthstrength F δ with a characteristic correlation time τ FD .Mikhailov and Kapral found that these force dipoles in-crease diffusion by an amount (Eq. (10) of Mikhailov andKapral [52]) ∆ = 0 . c FD ( F δ ) τ FD /η l c , for l c a smalllengthscale cutoff, approximately equal to the distanceof closest approach between the active protein, and theprotein whose diffusion is being accelerated. This followsfrom the fact that a force dipole induces flow at speed v ∼ F δ/ηr , a distance r away.Subsequent computer simulations of a simple modelsystem, by Dennison et al. [54] found a relatively smalleffect on diffusion, of order 10% or less. However, the sizeof the increase in diffusion is very sensitive to a number ofparameters so it is hard to estimate how large an affect itcould have in cells, without better data on the cytoplasm. Metabolism-dependent viscosity
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