Intracellular transport mechanisms: a critique of diffusion theory

J. theor. Biol. (1995) 176, 261–272

Intracellular Transport Mechanisms: A Critique of Diffusion Theory P. S. A,† P. C. M‡  D. N. W§> † Department of Biological Sciences, Napier University, Colinton Road, Edinburgh, EH10 5DT, U.K., ‡ 129 Viceroy Close, Birmingham B5 7UY, U.K. and §Cell Pathology, University Medical School, Foresterhill, Aberdeen AB9 2ZD, U.K. (Received on 3 October 1994, Accepted in revised form on 2 May 1995)

It is argued that Brownian motion makes a less significant contribution to the movements of molecules and particles inside cells than is commonly believed, and that the numbers of similar molecules and particles within any near-homogeneous subcompartment of the cell internum are insufficient to justify the statistical assumptions implicit in the derivation of the diffusion equation. For these reasons, it is contended that, contrary to accepted opinion, diffusion theory cannot provide an explanation for intracellular transport at the molecular level. Although attempts have been made to adapt diffusion theory to complex media, the conclusion is that none satisfactorily overcomes the problem of applying the theory to cell biology. However, the heuristic influence of the theory on cellular biophysics and physiology is noted, and possible alternative frameworks for interpreting the valuable experimental data obtained from such studies are outlined. 7 1995 Academic Press Limited

the cell internum. The cell’s liquid phase is visibly compartmentalized by internal membrane systems and other structures. These have a high total surface area; for example a cell with 1000 mm3 of cytoplasm typically contains 100–200 mm3 of cytoskeleton with a total surface area in the order of 106 mm2 (Gershon et al., 1983; Peters, 1986). Even parts of the cytoplasm lacking visible structure may not be monophasic (Fulton, 1982), as evidenced by reports of a ‘‘microtrabecular lattice’’ or ‘‘cytomatrix’’ (Wolosewick & Porter, 1979). Because of this biphasic character of the cytoplasm, intracellular movement of any solute or particle is potentially influenced by the following factors (cf. Wheatley, 1985; Wheatley & Malone, 1993): (i) adsorption and desorption events, (ii) specific binding, (iii) the availability of the cell water as solvent, (iv) the relative dimensions of ‘‘free water’’ channels and intervening solid structures; and (v) cytoplasmic streaming, occasioned by gel-sol transitions in the cytoplasmic actin or by myosin movements along actin fibrils (see e.g. Sheetz & Spudich, 1983), or by syneresis. It is a maxim of modern biology that intracellular transport phenomena must be explained in accordance

Today, thanks to the ingenious work of biologists, mainly geneticists, during the last thirty to forty years, enough is known about the actual material structure of organisms and about their functioning to state that, and to tell precisely why, present-day physics and chemistry could not possibly account for what happens in space and time within a living organism. The arrangements of the atoms in the most vital parts of an organism and the interplay of these arrangements differ in a fundamental way from all those arrangements of atoms which physicists and chemists have hitherto made the object of their experimental and theoretical research. Yet the difference which I have just termed fundamental is of such a kind that it might easily appear slight to anyone except a physicist who is thoroughly imbued with the knowledge that the laws of physics and chemistry are statistical throughout. (Schro¨dinger, 1944: 4)

1. Introduction Intracellular transport of small solutes, macromolecules and organelles is complicated by the structure of

> Author to whom all correspondence and reprint requests should be addressed. 0022–5193/95/180261+12 $12.00/0

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with thermodynamics and kinetic theory. Because the cell internum appears to many as a fluid, diffusion is considered the fundamental kinetic process in intracellular transport (though diffusion cannot be the only kinetic process involved, otherwise all intracellular distributions would tend towards true thermodynamic equilibria rather than steady states; cf. Paine & Horowitz, 1980). Therefore, the tacit consensus assumption is that intracellular transport depends primarily on diffusion, but that the consequences of diffusion are constrained by factors (i)–(v) above, i.e. by the topology, orientation and composition of the solid phase, the composition, concentration and mobility of the fluid phase, and the molecular interactions underlying specific binding, active transport and cytoplasmic streaming. In this paper, we challenge this consensus assumption. We postulate that diffusion is not fundamental to intracellular transport processes and that diffusion theory should be abandoned as the explanatory framework in cell biology. In other words, diffusion is an insufficient explanation for most vital movement/transport processes.† Classical diffusion theory has a macroscopic aspect, relating measurable movement rates of solute molecules and colloidal particles to concentration gradients and physical characteristics of the bulk medium by Fick’s law of diffusion; and a microscopic aspect, consisting of a hydrodynamic model which explains the macroscopic phenomena in molecular terms (the Einstein–Smoluchowski model of Brownian motion). Both Fick’s law and the Einstein–Smoluchowski model yield predictions shown to correspond with experimental data only under certain limiting conditions. The cell internum deviates importantly from these conditions. Although Brownian movement is inevitable in any fluid system, its part in transport through eukaryotic cytoplasm is probably insignificant (and experimentally immeasurable); its effects are quantitatively overshadowed by combinations of factors (i)–(v) above. Moreover, even if Brownian motion did contribute significantly to the movements of a particular class of molecules within an approximately isotropic cellular subcompartment, the numbers of randomly-moving molecules within any

† Note: In this paper, we have chiefly in mind eukaryotic cells, but there seems to be no a priori reason to assume (or presume) that many of our criticisms and suggestions might not equally apply to prokaryotic cells. The fact that intracellular transport mechanisms and movements of aqueous phase may be exceedingly difficult to explore and analyse in prokaryotes does not mean that such directed activity might not exist, as posited in some alternative ideas to diffusion in Section 6. This problem has, however, been addressed in some depth elsewhere (Wheatley & Malone, 1993).

such subcompartment would be too small for the statistical assumptions implicit in diffusion theory to be applicable (Donnan, 1927; Schro¨dinger, 1944; Halling, 1989). This argument is developed in more detail in Sections 2–3 below. Because fundamental criticisms of such a well-established theory as the theory of diffusion might seem to question the principle (to say nothing of the practicability) of seeking physicochemical explanations for biological phenomena, three general caveats need to be mentioned at the outset. (a) Diffusion theory has important scientific characteristics that have to date sustained its application to intracellular phenomena, and the resulting experimental work has generated much valuable knowledge about transport processes, some of these being reviewed in Section 4. We question the explanatory, not the heuristic, value of the theory. (b) Difficulties with the classical theory of diffusion have been encountered in other fields besides cell biology. Physical chemists, engineers and mathematicians have modified the theory to take account of some of these difficulties, and we propose that such modifications must be considered in relation to intracellular transport (see Section 5). (c) If diffusion theory cannot account for intracellular transport processes, then a different physicochemical theory is required. Ideally, any alternative theory should be as well-articulated as classical diffusion theory itself, provide coherent interpretations of the available experimental data, and, most importantly, give rise to novel testable hypotheses. We do not develop alternative theories in detail here, but outline some possibilities in Section 6.

2. The Inapplicability of Fick’s Law to Intracellular Media Fick (1855) derived his law by analogy with Fourier’s (1822) model for heat conduction in homogeneous solids. In the light of the kinetic theory bases of both the Einstein–Smoluchowski model and the mechanism of heat conduction, this analogy has been sustained in the twentieth century (see e.g. Carlslaw & Jaeger, 1959; Jacobs, 1967; Crank, 1975). Let us examine a generalized form of the derivation of Fick’s law (cf. Newman & Searle, 1957) which makes the implicit assumptions readily apparent. Consider an indefinitely thin plane at position X through a medium containing suspended particles (or

    solute molecules or ions). The concentration of particles at that plane at time zero is denoted by C(X, 0). The concentration at a nearby plane, position X ', at time t is C(X ', t); X ' is parallel to X and both planes are perpendicular to an axis x through the medium. We can write C(X', t)=

g

+a

c(X, 0)ft (X'−X) dX,

(1)

−a

where ft (X '−X) is a time-dependent function representing the probability that a particle will arrive within the range X ' to (X '+dX) during time t, having been in the range X to (X+dX) at time zero. Setting x=X '−X, eqn (1) becomes C(x', t)=

g

+a

c(x'+x, 0)ft (x) dx.

(2)

−a

After Taylor expansion of both the right- and the left-hand sides of (2) the equation takes the form a

C(x', 0)+ s t i i=1

g $ +a

=

−a

a ie at i a

%

x dic (x') ft (x) dx . . . i! dx i i=1

C(x', 0)+ s

(3)

which can be simplified (see below) so long as the following conditions are satisfied. (a) The time intervals over which the model is applied are many orders of magnitude greater than the duration of an average Brownian movement of a particle in the medium. (b) Molecular bombardments of a particle by the solvent are equally probable (i.e. equally frequent) from all directions. To satisfy this requirement, (a) must also be satisfied, and in addition the particle must be surrounded by many particle volumes of homogeneous solvent, i.e. a dilute solution/suspension. (c) The system is unstirred, i.e. stationary, showing no bulk solvent movement. (There is inevitably some flux of solvent molecules occasioned by the positional interchange with solute molecules. What is excluded by the assumption is bulk directional flow of solvent, e.g. see Clegg & Wheatley, 1991.) (d) Each suspended particle or solute molecule migrates independently between X and X '; i.e. there are no significant particle-particle interactions.

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Only if these conditions are obtained is ft an even function of x, i.e. ft (x)=ft (−x), and therefore

g

+a

xft (x) dx=0.

(4)

−a

However, conditions (a)–(d) do not obtain inside the cell. Eukaryotic cytoplasm can be considered homogeneous only within very small volumes, the boundaries of which generally have short lifetimes since they are often only transiently formed by macromolecular complexes (see Section 1). The average size of a putatively homogeneous liquid compartment, to judge from the dimensions of the proposed ‘‘microtrabecular lattice’’ (Wolosewick & Porter, 1979), is in the order of 10–20 nm, but protein migration rates through the cytoplasm (cf. Peters, 1986) imply a mean lifetime for structural protein–protein interactions in this ‘‘lattice’’ in the order of 1–100 ms. This implies that the homogeneous volume encompassing any individual particle, and the time intervals between successive changes in the system, are incompatible with conditions (a) and (b). Net solvent flow is commonplace, as indicated by the phenomenon of cytoplasmic streaming (see Section 1) and discussed elsewhere (Wheatley, 1985; with more complicated and refined techniques such as laser-Doppler microscopy, we can begin to quantify of this process; Wheatley et al., 1991). Therefore condition (c) is not met. Finally, ‘‘particles’’ within the cell are not independently mobile because of adsorption and specific binding to surfaces and associations between macromolecules (see Section 1); organelles, of course, are immobilized by cytoskeletal attachments. Even small ‘‘solute’’ molecules such as ATP are to some extent bound in cytoplasm (Ko¨segi et al., 1987; Kellermayer et al., 1994). More generally, Fick’s implicit assumption of continuous resistance to diffusion in the medium (Fick, 1855) is incompatible with the electrostatic and Van der Waal’s interactions between particle and solvent molecules (Hille, 1984), and this assumption is invalid unless the particle is large compared to the solvent molecule and the distance over which migration is measured is much greater than the particle diameter. If, despite these objections, we assume that eqn (4) is at least approximately valid and also note that

g

+a

ft (x) dx=1

(5)

−a

since every particle arriving at (X '+dX) must originate from somewhere else in the system, then

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eqn (3) can be simplified. To simplify it to a statement of Fick’s law, however, it is also necessary that (e) the system is sufficiently homogeneous for the contributions from the terms iq3 in the Taylor expansion on the right-hand side of eqn (3) to be negligible. [If ft (x) is an even function, as assumed, all odd-ordered terms in the expansion vanish after integration; assumption (e) is that even-ordered terms greater than i=2 are negligible.] This can be assumed only if the system is homogeneous, at least with respect to distances within an order of magnitude greater than the particle radius (cf. Section 3), and as already noted, this is not the case in the cell internum. Therefore not all the higher power terms in the expansion are negligible. Importantly, although (5) is invariably true irrespective of the validity of the model’s assumptions, it is only when conditions (a)–(d) are met that (4) and (5) can be applied to the mathematical solution of (3), and only when condition (e) is met that eqn (3) finally simplifies to dc 1 d2c t = dt 2 dt 2

g

+a

−a

x 2 d2c x ft (x) dx= , 2 dx 2 2

where x 2 is the mean square of the x-components of the net displacement, occurring over time t, of all particles in the population, i.e. dc x 2 dc d2c = . 2=D dt 2t dx dx 2

(6)

Equation (6) is Fick’s law of diffusion. Note that the diffusivity, D, is constant only if x 2 is directly proportion to t. This is implicit in conditions (a)–(e) and is discussed further in the following section. What the analysis so far has demonstrated is that D is not constant (has different values over different time intervals) even for a given C(X, 0), and furthermore, since in reality it is not legitimate to ignore all the higher-power terms in (3), that D is a function of c. Fick’s law [eqn (6)] must be replaced by a nonlinear alternative, the exact form of which will depend on the topology of the system (Crank, 1975). This creates two difficulties. First, since the topology is not unusually known in detail and, more importantly, is highly labile, the form of the equation is undecidable. Second, even if a suitable equation could be selected, its nonlinearity would preclude anything but a numerical or graphical solution, and the relevant parameters could not be measured with sufficient accuracy to allow the calculation of D values. The conclusion, which has been implicit since the early arguments of Donnan (1927) and more recently

echoed, for example, by Halling (1989), is that intrinsically statistical models cannot properly be applied to the cell internum. From a quite different perspective, Tyrrell (1961) has drawn attention to general difficulties of interpreting the term ‘‘diffusivity’’ or ‘‘diffusion coefficient’’. For instance, does this parameter measure an intrinsic property of the migrating molecules within the system, or does it measure exchange rates between solute and solvent molecules? Tyrrell argues, as have others, for a replacement of diffusion-theory approaches by irreversible thermodynamic models, and of diffusivities by frictional coefficients. His argument relates to physicochemical applications of the theory, but applies a fortiori to the more complex system presented by the living cell. These arguments show that the case for applying diffusion theory in biology and for purporting to measure ‘‘diffusivities’’ inside the cell is very weak.

3. Limitations of the Einstein–Smoluchowski Model The first coherent mechanistic explanation for Brownian motion, which is held to account for the process of diffusion, was deduced independently by Einstein (1905) and Smoluchowski (1906) from kinetic theory. This work proved scientifically important in that it led to the validation of atomic theory and the evaluation of Avogadro’s number through the work of Perrin and others (e.g. Perrin, 1909), and to the theory of stochastic processes (see e.g. Wiener, 1948). The form of deductive argument presented in this section is essentially Smoluchowski’s. The equation of motion in any given direction (x) of a particle subjected to molecular impacts is m

d 2x dx +v =−Fx dt 2 dt

(7)

where m is the particle mass, v is the damping coefficient due to viscosity and Fx is the x component of the force of molecular bombardment. It follows from (7) that m

d2x v d(x 2 ) + =−Fx (x) dt 2 2 dt

i.e.

0 1

dx v d(x 2 ) m d2(x 2 ) + =−Fx (x). 2 −m dt 2 dt 2 dt 2

(8)

Conditions (a)–(d) (Section 2) must be met before this generalization is valid. Also, it is necessary to assume

   

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(f) The homogeneous medium contains large numbers of identical particles (having the same mass, m)

where a has been introduced to signify

and this, again, is not the case in cytoplasm. Accepting assumptions (a–f), eqn (8) is now applied by averaging it, term by term, over many particles for a given instant of time. Denoting this average by a bar, the equation becomes

Substitution from (11) into the middle term on the left-hand side of (9) yields, after trivial rearrangements,

0 1

m d2(x)2 ) dx v d(x 2 ) + =−Fx (x). 2 −m 2 dt dt 2 dt 2

(9)

Assuming that Fx (x)=0, even though this is almost certainly invalid within a cell, (9) simplifies so that none of the impulse forces F appear. In the first and third terms on the left-hand side of (9), the order of the differentiations with respect to t and the averaging over the particles may be reversed (because the differentiation and averaging are with respect to different variables). For weakly interacting particles at sufficiently low concentration, Maxwell–Boltzmann statistics may be applied. (‘‘Sufficiently low concentration’’ must be less than about {3mkTh−2}3/2 where k=Boltzmann’s constant, h=Planck’s constant.) If, further, successive permitted energy levels are spaced by an amount small compared to kT then the Equipartition of Energy Theorem can be applied, i.e. kT=13 mv 2 . These conditions are met for gases, so it is necessary to assume (g) Equation (9) applies to diffusing particles in a liquid medium as well as those in a gaseous medium, which is at best dubious. From this theorem, the mean kinetic energy

0 0 11 m dx 2 dt

2

(10)

where use has been made of the identity k=R/N, where R=molar gas constant and N=Avogadro’s number. Substituting (10) in (9), we can obtain m dx RT va − + =0, 2 dt N 2

dx va 2RT + = . dt m mN

(12)

Multiplying through (12) by the integrating factor evt/m yields d 2RT vt/m (aevt/m )= e , dt mN

(13)

which on integration gives a=

2RT +Ae−vt/m , vN

(14)

where A is the constant of integration. If, as a result of A being sufficiently small or t being sufficiently large (or both) the second term on the right-hand side of (14) is negligible, then a=

2RT . vN

(15)

Using Stoke’s analysis for the viscous drag acting on a sphere moving through a liquid (i.e. v=6prh where r is the particle radius and h is the coefficient of viscosity) the result for the mean square of the x component of displacement in time t is a=

d(x 2 ) RT = dt 3pNrh

(16)

and by comparison with eqn (6) this gives the well-known ‘‘Stokes–Einstein’’ equation for the diffusivity, D: x 2 RT RT D= = = 2t Nv 6prhN

(17)

The final few steps in this derivation require the following additional assumptions.

attributable to the x components of velocity amongst the particles is given by PV=RT=13 Nmv 2 ,

d(x 2 ) . dt

(11)

(h) Ae−vt/m is negligible This does not apply unequivocally inside the cells because v is locally extremely variable [condition (b) is not met], m may be large, as in the case of macromolecules or complexes of macromolecules, and t is generally short [condition (a) is not met]. If the assumption is made, eqn (15) follows and the graph of particle numbers against distance x is a Gaussian curve with variance (x 2 ) proportional to time of diffusion. If, on the other hand, Ae−vt/m is in the order of (or greater than) 2RT/vN, then the curve is non-Gaussian. In

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principle, it should be possible to measure the relative magnitude of Ae−vt/m by determining the closeness of fit of experimental ‘‘diffusion’’ data to a Gaussian curve. Clearly, the diffusivity is not constant if condition (h) is not met because the mean square particle displacement is not directly proportional to time. This inference was also reached during the discussion in Section 2, above.

(b)

(i) The particles are spherical. The drag force is not particularly sensitive to particle shape so the invalidity of (i) is not serious. (j) The particles are rigid (i.e. all collisions are perfectly elastic). If the particles are not rigid (and in the case of proteins they are not) then momentum transfer during collisions is complicated and ‘‘viscosity’’ is generally increased. Another difficulty arises in relating equation (16) to cells. Both h and T are bulk properties which have significance only when statistical assumptions can be made, and therefore their physical significance in the microcompartments of a heterogeneous system such as cytoplasm is unclear. This difficulty was noted by Donnan (1927) and Wiener (1948), amongst others. To summarize, at least eleven conditions assumed in the derivation of Fick’s law and the Einstein–Smoluchowski model are not met in the cell internum. Attempts to generalize the model to three dimensions are handicapped by the anisotropy of intracellular transport phenomena and the labilities of relevant structural arrangements. In consequence, no possible solution of the diffusion equation (Crank, 1975) can properly be applied to intracellular transport phenomena, and the Einstein–Smoluchowski model does not explain such phenomena. 4. Heuristic Successes of Diffusion Theory Despite these difficulties, explanations based on classical diffusion theory are commonplace in cell biology (discussed in Wheatley, 1993; Wheatley & Malone, 1993). There are several reasons for this. (a) Although it can strictly be applied only under limiting conditions (e.g. homogeneity, high dilution, unstirred medium) the theory makes no special assumptions other than those made in kinetic theory, and kinetic theory is unequivocally well-established. Against this Ockham’s Razor argument, we should note that the Einstein– Smoluchowski model derives from the kinetic theory of gases (more precisely, from Maxwell– Boltzmann statistics for weakly-interacting particles). The kinetic theory of liquids remains

(c)

(d)

(e)

poorly-articulated even today and is conceptually and mathematically much more complicated than the kinetic theory of gases. Fick’s law conforms to one of the general ‘‘patterns of nature’’ (flux rate depends on steepness of gradient) which is also exemplified in Fourier’s analysis of heat conduction and in Ohm’s law (Fick, 1855). This confers upon it an attractive, intuitive explanatory power comparable to those of other familiar ‘‘patterns of nature’’ such as inverse square law relationships and first-order kinetic processes. With few exceptions, large and small molecules injected into a cell distribute themselves visibly over the cell internum in short time scales, typically 1–2 seconds (Wheatley, 1985; Wheatley & Malone, 1987). Experimental observations therefore seem to correspond to everyday experiences, such as the dissipation of smoke in a still atmosphere, which we ascribe to ‘‘diffusion’’. The danger here is of confounding description with explanation; just because an observation seems to merit the description ‘‘diffusion’’ does not necessarily mean that it should be explained in terms of diffusion theory. In the vernacular, ‘‘diffusion’’ is not expected to have the same rigorous dimensions and units which have to be applied if it is to be used as a scientific term. Fick’s law is held to be valid because it is well-supported by accumulated experimental evidence. In fact, the supporting experimental evidence is largely circumstantial and attempts at rigorous testing, including Fick’s own experiments, have been flawed (Tyrrell, 1961; Wheatley & Malone, 1986). The main difficulty is the exclusion of bulk solvent flow, e.g. by convection (Robinson & Stokes, 1965), and in general D is a function of concentration (Tyrrell, 1961). No satisfactory alternative theory has yet been adequately developed, or, more precisely, no theory which has such an amenable mathematical basis as classical diffusion theory. This is not to say that alternatives do not exist. We have published a number of papers indicating that the cell operates, like any good chemical plant, on the regulated flow of reactants over catalytic surfaces (Clegg & Wheatley, 1991), but these are extremely complex vectorial processes which are difficult to formulate mathematically. However, it should be noted that a precondition for the emergence of an alternative theory is the recognition that one is needed; and to date, the need for a fundamental alternative to classical diffusion theory has scarcely begun to be appreciated (see Section 6).

    Classical diffusion theory has been widely applied to the computation of, for example, solute and water diffusivities inside cells and, by substitution of the computed values into eqn (16), of intracellular viscosities. As the discussion in Sections 1–3 suggests, physical interpretation of these results is problematic. However, workers in the field have addressed many theoretical and experimental difficulties, and have developed a variety of techniques for obtaining reliable data. Problems of interpretation notwithstanding, these data have provided insights into the constraints on intracellular dynamics. In short, although classical diffusion theory might not account for intracellular transport, its application to experimental data has led to significant advances in knowledge. The literature has been well reviewed (see e.g. Peters, 1986) and here only an illustrative selection of contributions is considered. Water mobility itself may be important in the transport of many materials inside the cell. This parameter has been measured most commonly by pulse NMR, but this technique is not as versatile as laser-Doppler microscopy (Wheatley et al., 1991). Intracellular water relaxation times are lower than those of pure water (notably T2 is in the order of 100 ms compared with 3 s for pure water; self-diffusivity is usually estimated as two to three times lower, although occasionally higher values have been given (discussed by Clegg, 1991; Luby-Phelps, 1994a,b). These data might be interpreted to mean (i) mobility of most or all intracellular water is reduced; or (ii) some 5% of the cell water is tightly-bound, the other 95% behaving like water in dilute bulk solutions, there being fast exchange between the fractions (the low self-diffusivity could be attributed to compartmentalization or obstruction effects); or (iii) some compromise between these extremes. NMR alone cannot settle the controversy because the limit of spatial resolution of pulse-NMR is greater than the average cell diameter. The questions therefore remain as to whether the vicinal water close to surfaces and macromolecules (Drost-Hansen & Singleton, 1992a,b) possesses substantially different properties from bulk water, and whether a mere 5% of more structured water could contribute significantly to the overall parameters measured (Clegg, 1984). In Artemia cysts, quasi-elastic neutron scattering measurements supported the first interpretation (Trantham et al., 1984). This technique has the advantage of spatial resolutions less than 1 nm and temporal resolutions of 1–10 ps, but it cannot be applied to most biological samples because measurements over several (e.g. 10) days in sealed containers are required. Furthermore, it may not be legitimate to

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generalize from Artemia cysts to cells in general. However, attribution of the intracellular restriction of water mobility to cytoskeletal and other surface binding is supported by ESR studies. Using the spin probe 2,2,5,5-tetramethyl-3-methanol pyrroline-Noxyl, Mastro et al. (1984) found that the intracellular self-diffusivity of water in mouse fibroblasts was only about 50% of the pure water value but was increased by cytochalasin-induced microfilament disassembly. The spatial resolution of ESR using millimolar concentrations of this probe is in the order of 10 nm. Low Mr solutes such as sucrose have intracellular ‘‘diffusivities’’ in the order of two to five times lower than the values in dilute aqueous solution, as measured by ultra low-temperature autoradiography (e.g. Horowitz, 1972). Horowitz and his coworkers have also devised a low-temperature method (cryomicrodissection) for studying ‘‘equilibrium’’ (steadystate) distributions of solutes in large cells such as amphibian oocytes. One extension of the cryodissection approach involves injection of a gelatin gel (internal reference phase) into the cytoplasm, allowing time (20–30 hr) for equilibration even of molecules with slow transport kinetics, freezing the cell in liquid nitrogen, and dissecting the cell in the frozen state to separate reference phase, cytoplasm and nucleus (Paine, 1984). This technique reveals that a significant fraction of intracellular solutes such as ATP is bound in the cytoplasm (unable to equilibrate into the reference phase). Simpler methodological approaches to intracellular ‘‘equilibrium’’ studies (e.g. cell permeabilization) have shown that ions such as K+ can also be bound intracellularly to a substantial extent (Kellermeyer et al., 1984, 1994). The ‘‘diffusivities’’ of exogenous proteins microinjected into cytoplasm have been measured by a variety of techniques including autoradiography, fluorescence microscopy, ultra low-temperature autoradiography and fluorescence microphotolysis (see e.g. Paine et al., 1975; Wang et al., 1982; Jacobson & Wojcieszyn, 1984). Amongst these methods, fluorescence microphotolysis (photobleaching) has the best spatial and temporal resolution (approximately 1 mm and 1 ms; see Peters, 1986). Not only are the measured ‘‘diffusivities’’ of endogenous proteins and RNAs much lower than those in dilute aqueous solution, in some cases by more than two orders of magnitude, but also, in conflict with the prediction of eqn (16), the values show no temperature dependence (Lang et al., 1986). Moreover, for proteins and RNAs, there is no correlation between diffusivity and Mr , as eqn (17) would imply. In the case of mRNA and its precursors, most (effectively all) of the material remains attached

268

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to insoluble cell components during processing and transport; little or none is ‘‘diffusible’’ (reviewed in Agutter, 1988). Microinjected dextrans also show restricted mobility in the cytoplasm but the decrease of ‘‘diffusivity’’ is much less than it is for proteins (approximately six- to eight-fold). Also, the intracellular ‘‘diffusivities’’ of dextrans are inversely proportional to molecular radius, as eqn (16) would imply (Lang et al., 1986). These findings suggest that dextran transport within the cell is not materially affected by adsorption and desorption processes, but by some more general factor such as intracellular inhomogeneity. Studies such as these have several implications for an adequate theory of intracellular transport. (a) Because some or much of the intracellular water appears to have properties different from bulk water, some ‘‘concentration gradients’’ within cells might be attributable to partitioning between aqueous phases with different solvent properties (e.g. different permittivities). In other words, even if the fraction of solute is bound to an intracellular surface is zero, its electrochemical potential might not bear the same relationship to concentration at all points within the cell (Clegg, 1979). (b) Alteration of the solvent properties of water is also likely to alter the binding affinities between molecules; e.g. a receptor-ligand complex formed by ionic bonding will have a lower Kd if the solvent permittivity is decreased. (c) In so far as the non-bulk properties of water result from hydration of the intracellular surfaces (Drost-Hansen & Singleton, 1992b), alterations in, for example, the organization of the cytoskeleton (polymerization/depolymerization) are likely to alter intracellular solute distributions and stabilities of intermolecular complexes because they may change the solvent properties. These changes might be marked when widespread changes in the state of the cytoskeleton occur. (d) The fact that intracellular ‘‘diffusivities’’ are decreased relative to the values measured in dilute aqueous solution means that ‘‘intracellular viscosity’’ computed from the experimentally-determined ‘‘diffusivity’’ by application of eqn (16) is much higher for biological macromolecules than it is for dextrans or for solutes such as sucrose. This difference is generally ascribed to low-affinity adsorption to intracellular surfaces (e.g. Horowitz et al., 1970; Wojcieszyn et al., 1981). In other words, intracellular transport of ‘‘soluble’’ macromolecules involves a process analogous to adsorption chromatography. If a system in which

classical diffusion theory applies is supplemented by an immobile phase containing binding sites for a solute, then the diffusivity of the solute is decreased by a factor of (1+C/Kd )−1 , where C=concentration of binding sites and Kd=dissociation constant of solute-immobile phase complex (cf. Gershon et al., 1983). Many endogenous proteins migrate slowly, if at all; see, for example, Chambers (1940), Feldherr & Ogburn (1980) and Paine (1984). (e) The migration of dextrans and of larger particle within the cytoplasm is impeded not by surface adsorption but by apparent barriers with a spacing in the order 10–20 nm. This spacing decreases, and the channels within the spacing become more tortuous, during osmotic shrinking of the cell because of phase separation. Osmotic shrinkage is therefore accompanied by a marked increase in the ‘‘viscosity’’ of the cytoplasm (e.g. Mansell & Clegg, 1983; Wheatley et al., 1984). Clearly, the cytoskeleton itself does not account for this ‘‘filtering’’ effect because the average geometric radius of channels formed by cytoskeletal fibres is in the order of 100 nm (Gershon et al., 1983). In a medium in which classical diffusion theory applies but there is a gel-like system of barriers, the effective path-lengths of diffusion are increased. Horowitz & Moore (1974) showed that in this situation the diffusivity is decreased to Dl−2 , where D is the diffusivity in dilute aqueous solution and l is the root mean square relative increase in path length attributable to the barriers. Jacobson & Wojcieszyn (1984) showed that in Sepharose beads with mean pore diameters of 50 nm the diffusivity of proteins with Mr 25000 was decreased by some 50% relative to the free-solution value. (f) The rate of movement of a particle through a pore in a ‘‘filtration’’ system of the kind discussed in (e) is given on the basis of classical theory by the permeability coefficient, P=DnAe l−1 , where n is the number of pores per unit area, l is the path length and Ae is the effective pore area. If the geometrical pore area is Ag , then Ae /Ag=f(r/ R)·g(r/R), where r is the solute particle radius and R is the geometrical pore radius, equal to Ag /2p. The functions f and g define, respectively, the effects of steric hindrance at the pore margin and viscous drag within the pore, and they depend on the geometries of both pore and particle. For hard spherical particles and cylindrical pores, f(r/ R)=(1−r/R)2; for other geometries the functions are mathematically more complicated. In general, f, g tend to unity when rR. That is, if

    the particle is much smaller than the pore radius, then the effective pore area is equal to the geometrical pore area. 5. Modifications of Classical Diffusion Theory When a well-established theory fails to account for a particular set of phenomena within its presumed domain, it is usually adapted by more or less ad hoc modifications until the recalcitrant data are assimilated. Only if this approach also fails is it concluded that the problematic phenomena lie outside the domain of the theory (cf. Suppe, 1977). Thus, the failure of classical diffusion theory to account for intracellular transport phenomena has evoked modification such as those discussed in Section 4, which include phenomenological than theory-based mathematical treatments of factors such as adsorption/desorption, channeling of movement, and filtration through pores. Such ad hoc modifications only serve to make the theory more cumbersome and unsatisfactory. However, in other parts of the scientific literature, more theory-derived modifications have been developed. Hartley (1931) ascribed diffusion processes to the driving force of an electrochemical potential gradient rather than a concentration gradient, and suggested that D =constant d ln g I− d ln m

(18)

(where g is the activity coefficient), should be independent of concentration. The principle of this approach can be traced back to the treatment of osmosis by Nernst (1888) and has been strongly supported by thermodynamic analysis (Tyrrell, 1961); see Section 3. Experimental evidence suggests that the ratio shown in eqn (18) is more or less constant at low concentrations but not at higher concentrations, because of attractions between solute molecules in solution (cf. Tyrrell, 1961). When fairly stable complexes form in solution, this ratio is more sensitive to concentration than is D itself (Anderson et al., 1958; Irani & Adamson, 1960). To date, no satisfactory quantitative treatment of a solution with independently diffusing complex species has been achieved. It seems that this kind of modification of classical diffusion theory cannot provide a satisfactory quantitative treatment of intracellular migration because it cannot predict the behaviour of multimolecular complexes in solution. Nevertheless, attempts to measure the activity coefficients of simple cations in oocyte cytoplasms using the reference phase technique

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(see Section 4) have confirmed that water and electrolytes constitute at least two distinct kinetic phases in the cell (Horowitz et al., 1970). Once again, the explanatory inadequacy of the theory has not obstructed its positive heuristic influence. In principle, statistical mechanical analysis of transport processes in liquid media (see e.g. Rice & Frisch, 1962) should replace the Einstein–Smoluchowski model with a more generally applicable hydrodynamic schema. Ultimately, a successful model of this kind would meet Fick’s (1855) criterion: diffusivities should be calculable from the mathematical descriptions of intermolecular forces. To date, however, the statistical mechanical modelling of liquid systems has mainly been limited to homogeneous media of low concentration with little intermolecular complex formation. Such attempts as have been made to extend the theory to cases of higher concentrations and complex formation entail significant mathematical complexity and it becomes very difficult to assimilate experimental measurements to theoretical predictions. Thus, no available model accounts for a system like the cell internum, and there is no immediate prospect of any such model (Polanyi, 1968); and even if one became available, its mathematical complexity would almost certainly preclude rigorous testing against experimental results. In effect, the mathematical complexities of statistical mechanics have failed to generate a treatment of intracellular transport that is significantly superior to the Einstein–Smoluchowski model in respect of either ‘‘realism’’ or practical biological applicability. During the past 20 years there has been revolutionary progress in the provision of computerdependent numerical solutions to problems in non-Fickian diffusion, diffusion in heterogeneous media and moving-boundary problems (see Crank, 1975, for an exposition). All these topics seem prima facia relevant to cell biology because of the phase heterogeneity, phase lability and polymer absorption properties of the cell internum. Briefly, ‘‘non-Fickian diffusion’’ refers to systems comprising a polymer and a solution containing penetrant molecules; the penetrant molecules interact with the polymer and change its surface properties, thus in turn changing the adsorption and migration of the penetrant species. Analytically, non-Fickian diffusion is especially problematic when the polymer relaxation rate is comparable with the diffusivity of the penetrant. ‘‘Heterogeneous media’’ are taken to be either particulate dispersions or laminates. ‘‘Moving boundary problems’’ relate to phase changes such as melting ice in contact with liquid water. Advances in these topics have many applications in physics and

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engineering, but their apparent relevance to the cell internum is chimeric: (i) all treatments assume that molecular movements are primarily random, and this cannot be assumed in biology (Sections 2–3); (ii) not all the relevant parameters can be measured; (iii) the uncertain topological and other characteristics of the system mean that there is no clear basis for constructing a mathematical model. In any case, when water, the ‘‘mother liquor of life’’ (Szent-Gyo¨rgy, 1971), associates with and becomes structured and organized by any surface, whether it be a protein or a steel plate, it changes its properties (Drost-Hansen & Singleton, 1992a). We are only just beginning to understand these phenomena in physical and chemical terms, and they are far from the complexity found within the membranes and polymer surfaces and interfaces of living organisms. Diffusion theory has advanced beyond its simple classical formulation in both macroscopic descriptions and its microscopic explanations, but none of these developments has made it applicable to intracellular media, and there seems no prospect of a development that will succeed in this respect. This conclusion may have been implicitly accepted by an increasing number of biophysicists who seek to explain movements of ions, molecules and particles by combining the relevant approaches to solving the diffusion equation with treatments of obstruction by immobile structures, adsorption, specific binding and flow. Papers by Saxton (1982) and Eckstein & Belgacem (1991) are among the many recent examples. It would be interesting to see whether these and similarly included authors could produce simpler, more general and more testable and applicable mathematical models by replacing the presumption of diffusion altogether with a more credible account of movement. The assertion that little intracellular transport is explicable by diffusion theory, classical or modified, is initially difficult to accept. Diffusion theory is deeply entrenched in modern scientific thought, for at least the reasons discussed in Section 4. There is even a danger that in denying its applicability to the cell internum we might seem to countenance vitalism. The following points therefore need to be re-emphasized. (a) Cells are highly ordered structures, as Peters (1930) so perceptively described. At all levels of analysis from the light-microscopic to the molecular they are high information content (low thermodynamic probability) entities. So far as their internal dynamics is concerned, this means that most physicochemical processes are channeled or ‘‘directed’’ rather than random and suggests that little occurs in the cell on the basis of

chance or as a simple consequence of the law of mass action (Malone & Wheatley, 1991; Wheatley, 1993b). Diffusion theory assumes random molecular events; as the discussion in this paper makes clear, any departure from randomness creates difficulties for the theory that cannot always be overcome even by sophisticated mathematical and computer-assisted approaches. Seen in this way, diffusion theory should not be expected to apply in cell biology, and the conclusion reached in this paper should occasion neither surprise nor scepticism amongst biologists. (b) ‘‘Cytoplasmic diffusivities’’ calculated from experiments of the kinds discussed in Section 4 should not be called ‘‘diffusivities’’ or ‘‘diffusion coefficients’’, because these terms imply the applicability of an inapplicable theory. A term such as ‘‘empirical transport coefficient’’ might be more apt. A significant challenge for alternative theories of intracellular transport is to explain these experimentally-determined parameters. Summing up the developing role of light-scattering techniques in biology, Johnson (1983) drew attention to the problem of self-diffusivity for measuring behaviour at the molecular level: ‘‘This is the region of scale where flow and diffusion are not clearly separated; where the concepts of temperature and molecular movement overlap; where it is not clear whether molecules move or are moved; where the ideas of active and passive lose their meaning’’ (our italics). (c) Any alternative theory must be grounded in accepted physicochemical, principles. Unless this criterion is rigorously applied, the accusation of vitalism will persist and might well become justified. On the other hand, any useful alternative theory must take account of the complexity of intracellular organization.

6. Alternative Theories There is no a priori reason to suppose that a single overarching theory will be adequate to explain all types of intracellular transport. On the contrary, there is a clear intuitive distinction between solute molecules such as glucose that have low probabilities of intracellular immobilization by, for example, adsorption to the cytoskeleton, and entities such as ribonucleoprotein complexes that have comparably low probabilities of being ‘‘in solution’’. For this reason, at least two distinct approaches are needed to alternative theorizing. The perspectives outlined here will be developed in other publications.

    So far as essentially non-adsorbed solutes such as glucose are concerned, at least the following considerations are relevant to the establishment of a quantitative theory. (a) Intracellular transport will depend primarily on solvent fluxes (cytoplasmic streaming) and therefore, minimally, the instantaneous solvent-front velocity needs to be added to the equation of motion (7) of the molecule (Wheatley, 1985). (b) The fraction of cytoplasmic water available as solvent, the mean functional pore sizes of the cytoplasmic gel, and the mean path length of the individual channels, all need to be incorporated into the model (see Section 4 for some possible mathematical approaches). (c) Account must be taken of the multimolecular organization of enzymes that metabolize solutes such as glucose (Srere, 1985; Srivastava & Bernhard, 1986; Clegg, 1979, 1991). This is relevant to the effective dissociation constant of any enzyme–ligand complex. The predominantly adsorbed constituents of the cell are likely to be moved between intracellular locations by a solid-state transport mechanism such as that envisaged for messenger RNA and its precursors (Agutter, 1988, 1991). Aside from the considerable body of evidence relating to mRNA, there is a simple theoretical reason for assuming solid-state transport. If only a small percentage of a substance is freely soluble in the cell, any transport mechanism that relied on moving the freely soluble fraction and replacing it by desorption from the dominant pool would certainly be slow, and probably lethally inefficient. Other processes might also rely on solid-state transport, discussed in Agutter (1994). The point is that all molecules might be directed and escorted in their translocation from one site to another in the cell. The question we are left with is whether this represents a sizeable fraction of the total molecular movements that are necessary for cells to exhibit metabolic activity (to live) over the range from their most quiescent states to working flat out (discussed by Wheatley & Clegg, 1994). I propose to develop first what you might call a ‘naive physicist’s ideas about organisms’, that is, the ideas which might arise in the mind of a physicist who, after having learnt his physics and, more especially, the statistical foundation of his science, begins to think about organisms and about the way they behave and function and who comes to ask himself conscientiously whether he, from what he has learnt, from the point of view of his comparatively simple and clear and humble science, can make any relevant contributions to the question (of what is life?).

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It will turn out that he can. The next step must be to compare his theoretical anticipations with the biological facts. It will then turn out that—though on the whole his ideas seem quite sensible—they need to be appreciably amended. In this way we shall gradually approach the correct view—or, to put it more modestly, the one that I propose as the correct one. (Schrodinger, 1944: 6) We are indebted to Professor B. Goodwin, Dr. P. L. Paine and Dr. K. R. Turvey for constructive criticisms of drafts of this paper. We would also like to thank Dr Kathrine Luby-Phelps for providing us with prepublication information. This work was supported by a grant from the Wellcome Trust, History of Medicine Section (grant no. 035481/Z/92/Z). REFERENCES A, P. S. (1988). Nucleo-cytoplasmic transport of mRNA: its relationship to RNA metabolism, subcellular structures and other nucleocytoplasmic exchanges. In: Progress in Molecular and Subcellular Biochemistry, Vol (10) (Mu¨ller, W. E. G., ed.) pp. 15. Heidelberg: Springer-Verlag. A, P. S. (1991). Between Nucleus and Cytoplasm, London: Chapman and Hall. A, P. S. (1994). Models for solid-state transport: messenger RNA movement from nucleus to cytoplasm. Cell Biol. Internat. 18, 849–858. A, D. K., H, J. R. & B, A. L. (1958). Mutual diffusion in non-ideal binary liquid mixtures. J. phys. Chem. 62, 404–409. C, H. S. & J, J.C. (1959). Conduction of Heat in Solids. Oxford: Clarendon Press. C, R. (1940). The micromanipulation of living cells. In The Cell and Protoplasm (Moulton, F. R., ed.) Washington, DC: AAAS Publication 14. C, J. S. (1979). Metabolism and the intracellular environment: the vicinal water network model. In: Cell Associated Water (Drost-Hansen, W. & Clegg, J. S., eds). New York: Academic Press. C, J. S. (1984). Properties and metabolism of the aqueous cytoplasm and its boundaries. Am. J. Physiol. 246, R133–151. C, J. S. (1991). Metabolic organization and the ultrastructure of animal cells. Biochem. Soc. Trans. 19, 986–991. C, J. S. & W, D. N. (1991). Intracellular organization: evolutionary origins and possible consequences to metabolic rate control in vertebrates. Am. Zool. 31, 504–513. C, J. (1975). The Mathematics of Diffusion, 2nd edn. Oxford: Clarendon Press. D, F. G., (1927). Concerning the applicability of thermodynamics to the phenomena of life. J. gen. Physiol. 8, 685–688. D-H, W. & S, J. L. (1992a). Our aqueous heritage: Evidence for vicinal water in cells. In: Chemistry of the Living Cell (Bittar, E. E., ed.), Chapter 5. Greenwich, CO: JAI Press. D-H, W. & S, J. L. (1992b). Our aqueous heritage: role of vicinal water in cells. In: Chemistry of the Living Cell (Bittar, E. E., ed.), Chapter 6. Greenwich, CO: JAI Press. E, E. C. & B, F. (1991). Model of platelet transport in flowing blood with drift and diffusion terms. Biophys. J. 60, 53–69. E, A. (1905). Von der Molekula¨rkinetischen theorie der Wa¨rme gefordete Bewegung von in ruhenden Flu¨ssigkeiten suspendierten Teilchen. Ann. der Physik 17, 549–554. F, C. M. & O, J. A. (1980). Mechanisms for the selection of nuclear polypeptides in Xenopus oocytes. II: Two-dimensional gel analysis. J. Cell Biol. 87, 589–593. F, J. D., (1936). Statistical evaluation of sieve constants in ultrafiltration. J. gen. Physiol. 20, 95–104.

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