ProcessDB User Guide Wiki    Computational Cell Biology Textbook  

Chapter 7

Writing Rate Laws for Translocation Processes

Translocation is the third broad class of biological processes. Since it includes all process that move chemical species from one physical place to another, we need to develop rate laws for processes as diverse as diffusion, endocytosis, blood flow, and transport of mRNA from the nucleus to the cytoplasm through the nuclear pore complex.

Diffusion is arguably the simplest of these special cases. It is a process that occurs in inanimate systems just as readily as it occurs in living systems. No specialized proteins are required, unless the diffusing molecule must cross a lipid membrane, and even then a nonpolar, lipid soluble molecule can cross without proteinacious assistance.

Imagine a molecule of glucose in the cytoplasm of a mammalian cell. In order for this molecule to begin its trek through glycolysis, it must first be phosphorylated. In light of what you learned in the previous chapter, this phosphate transfer can only take place after the glucose, the phosphoenolpyruvate, and the appropriate enzyme have come together to form the enzyme-substrate complex. This coming together is dependent on diffusion.

Diffusion is a special case of passive movement in a chemical field. The chemical field is the force per mole tending to drive the passive transport of a molecule from one place (phase) to another. This field is conservative in the sense that the free energy associated with a given position is conserved: the work done in transporting a quantity of molecules from one place to another is independent of the path taken. In fact, any arrangement of stationary phases gives rise to a chemical field that is conservative. The mathematical statement of this physical fact is that the chemical field, Fc, can be written as the gradient (derivative with respect to space) of the chemical potential.

Equation 1

If you are familiar with electromagnetic fields, you will see that the chemical field is related to the gradient of chemical potential just as an electric field is related to the gradient of the electrical potential. An electric field has units of N/coul (newtons per coulomb); a chemical field has units of N/mole. For an uncharged molecule, s, the chemical potential is given by

Equation 2

where m 0 is a constant at a given temperature, T, and pressure, P, R is the Gas Constant, Cs is the concentration of s (molar), Vp is the partial molar volume of s, and P is the pressure. Differentiating gives:

Equation 3

In respiratory and cardiovascular physiology and in any experimental system using pumps, you will encounter significant pressure gradients, but animal cell membranes cannot support such gradients. Consequently, for cellular level work, we will set dP/dx = 0 and the chemical field will reduce to the concentration gradient term.

This force, Fc, tending to move mass from an area of high chemical potential to an area of low chemical potential is opposed by a friction force, Ff, proportional to the velocity of the transported mass:

Equation 4

where f is the coefficient of friction. You may have encountered, in Physics, friction forces proportional to v2; these occur at velocities greater than those encountered in biological systems. Particles of the substance, s, will be accelerated by the chemical field until the field is balanced by this friction force, that is until Fc = Ff. Now we're getting close to the result we want, but to make it useful the velocity, v, needs to be related to quantities we actually measure. This is accomplished by recognizing that the flux, Js, of substance, s, (in moles sec-1 cm-2) is equal to the product of concentration and velocity. Thus, the friction force can be written as f Js/Cs. Equating this to our expression for the chemical field gives

Equation 5

With one last piece of information, provided by Albert Einstein in 1905, this derivation is complete. Einstein demonstrated that RT/f is equal to the classical diffusion coefficient, D, used by Adolf Fick to characterize diffusion 50 years earlier. This substitution yields Fick's Law of Diffusion. Fick, however, did not derive his law from first principles as we have done here. He did not even obtain the law as an empirical fit to experimental data. Instead, he argued by analogy to the movement of heat down temperature gradients, which had been analyzed by Fourier in 1822. Thus, by calling equation 6

Equation 6

"Fick's Law of Diffusion", we are honoring the intuition of a great scientist.

Pay attention to the units of flux. In previous chapters we concluded that all of the differential equations that we formulate for biological systems should be written in terms of conservation of mass. You might expect, therefore, that each term on the right hand side of your differential equation would have units of mass/time. Fick's first law, however, produces a flux whose units are, for example, nmol sec-1 cm-2. Indeed many scientists insist that the word flux should only be used when the movement of molecules is expressed in these terms. Rather than enter the fray, I suggest that you listen carefully when "fluxes" are discussed and be certain that you know which units are intended. In many situations you will need to include a diffusive flux term in a model you are constructing, but you will have no estimate of the cross-sectional area through which the flux is travelling. The rate law can then be written to acknowledge that the flux is proportional to the concentration difference,

Equation 7

where D' is related to the classical diffusion coefficient but incorporates both the diffusional cross sectional area and the thickness of the barrier separating the compartments, 1 and 2. In this case, what are the units of D'? It depends, of course, on what units you choose for [S]1 and [S]2; D' will have whatever units are required to convert these concentrations to, say, nmol/sec. Notice that Jdiff changes sign if the concentration difference changes sign. This reflects the passive nature of the process; at one moment the net flux may be moving from 2 to 1 and the next moment from 1 to 2.

Why do we use the word "net" to describe the diffusive flux? The answer is because there are always molecules moving from compartment 2 to compartment 1 and from compartment 1 to compartment 2. In fact, you could write the rate law as which shows the individual forward and reverse contributions to the net flux. You can see here that diffusion is a bi-directional application of the principle of mass action. The individual flux contributions are both proportional to the concentration of S in the originating compartment, the compartment from which the flux flows.

A final point, perhaps obvious by now, is that the concentrations of S are absolutely required on the right hand side of the equation. The mass of S cannot be substituted for the concentration because you could then cause transport up a concentration gradient just by giving the low concentration compartment a much larger volume than the high concentration compartment.

DIFFUSION COEFFICIENTS ARE ALMOST ALWAYS IN THE RANGE FROM 10-6 TO 10-5 cm2/sec. In order to develop some intuition about diffusion, you should learn the absolute values of some typical diffusion coefficients. These are reported values for diffusion in water at 20 oC. Notice that the bigger the molecule, the smaller the diffusion coefficient.

PERMEABILITY CHARACTERIZES DIFFUSION THROUGH MEMBRANES. To apply Fick's law to diffusion through a cell membrane of finite thickness one must satisfy two assumptions: 1) that the membrane thickness is small and 2) that the inside and outside compartments are well mixed. When these assumptions are made, the concentration gradient, dC/dx, can be replaced with (Co-Ci)/b, where the subscripts o and i refer to outside and inside and b is the thickness of the membrane. Then b can be lumped into the diffusion coefficient and a new constant, P = D/b, is formed. P is referred to as a permeability.

DIFFUSION COEFFICIENTS INCREASE WITH INCREASING TEMPERATURE. This empirical fact was given a theoretical basis in the Einstein relation (D = RT/f), mentioned earlier. The temperature appears linearly in this equation, but it is important to remember that this is absolute temperature. Consequently, a change from 20 C to 30 C does not result in a 50% increase in D. Remember that absolute zero is -273 C. Perhaps you anticipated the increase in diffusion coefficient with temperature since you know that increasing temperature causes increased molecular velocity.

A quantity frequently used to characterize the temperature sensitivity of a physical or chemical process is the Q10. This is defined as the rate of the process at T+10 C divided by the rate at temperature T. Certain values of Q10 have sometimes been quoted as "proof" that a process is energy requiring ("active") rather than passive. Such arguments are not supported by the facts. Activation energies for active and passive processes overlap extensively. Another group of membrane proteins is comprised of those that function as channels, carriers or ports. As indicated by their names, these proteins mediate the transmembrane movement of molecules that would otherwise not be able to pass through the phospholipid bilayer. The three names suggest different mechanisms of action, but, in fact, all of them include a specific binding step (or steps) followed by translocation and unbinding. In some cases, there are controls on the translocation step such as gates in ion channels.

Another characteristic of these proteins, that you should always keep track of, is whether or not they carry molecules up or down the chemical potential gradient. If uphill transport is involved, there must be a source of free energy (often an ion gradient produced by a pump) that is being tapped to drive transport. The term pump is generally restricted to those membrane proteins that carry out active transport (that is, transport against a concentration gradient) and which do so at the direct expense of ATP or some other high energy phosphate. The increase in free energy that accompanies transport up the concentration gradient is paid for by coupling the process to the free energy supplied by ATP hydrolysis. The overall free energy change is still negative, as it must be for the pump to work.

All rate laws can be written at any of several levels of detail. The simplest form of a rate law for any process is the linear mass action rate law. This approach is frequently taken for carriers, channels and pumps when electrical forces are irrelevant or can be treated as constant for the duration of the experiment. Ion channels rarely operate in a range in which they saturate, but carriers and pumps saturate almost as regularly as enzymes. It's not known why this is so, but it may depend on the molecular mass of the ligands. If saturation is to be included, formulate your rate law using the principles developed in the chapter on enzyme catalyzed processes. The only difference is that for a transport process, the substrate and product are replaced by the transported species in two different physical places. For example, if glucose (G) is the transported species and it is transported across the cell membrane from extracellular space to cytoplasm, then GECS goes into the diagram in place of S (substrate) and Gcyto replaces P (product). Using the rapid equilibrium approximation, the kinetic constants that appear in the rate law will refer to equilibrium binding of G to the extracellular face of the transporter and equilibrium binding of G to the cytosolic face of the transporter. If the reverse flux is neglected, then only the extracellular binding constant will appear in the final rate law. This equivalence between transport and enzymes is so widely recognized, that biochemists and enzymologists often refer to transporters as "permeases" - enzymes that catalyze permeation.

RATE LAWS FOR ION CHANNELS Ion channels may be treated as carriers if the membrane potential can be treated as constant. Under these conditions, the resulting rate law is simply linear mass action because, as mentioned above, the saturation of ion channels is only observed at very elevated, unphysiological, concentrations of the permeating ion. Two other situations are commonly encountered when modeling ion channels. First, the transmembrane potential often changes. This may be physiologically induced or be part of the protocol in voltage-clamp experiments. In these circumstances, you need a rate law that accounts for the electrical contribution to the chemical potential because the ion flux or current can be dramatically changed by changes in the electrical driving force.

PASSIVE MOVEMENT OF IONS DEPENDS ON BOTH CONCENTRATION AND ELECTRICAL POTENTIAL GRADIENTS. To obtain an expression for the passive flux of any ion, we proceed as we did for diffusion above. The net passive flux of the kth ionic species can be written as where vk is the velocity and Ck is the concentration of the species. In turn, the velocity may be written as where uk is the mobility of the ion, and mk is its chemical potential. This equation gives the velocity as the product of velocity-per-unit-force (mobility) and force (gradient of chemical potential).

In the last lecture we wrote the chemical potential expression for an uncharged molecule; now we need the corresponding equation for an ion. It is where z is the ionic charge, F is the Faraday, V is the electrical potential and the other terms are as before. Differentiating and substituting in the equation for passive flux gives (pressure gradients are again set to zero) or This is the Nernst-Planck flux equation for the passive movement of ions down an electrochemical gradient. Be sure to distinguish the mobility from the chemical potential; uk and mk are similar symbols, but they represent very different quantities. The Nernst-Planck equation is the rate law we require. It expresses the flux of ions as a function of both chemical and electrical driving forces.

One level more in sophistication of ion channel rate laws is often required if the channel is gated by voltage or ligands. This topic is beyond the scope of this small book and the interested reader is referred to excellent books on the topic: Ionic Channels of Excitable Membranes by Bertil Hille and references therein.

Chapter 8
Chapter Guide