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# Passive and Facilitated Diffusion

So far we have studied molecular aggregates (micelles and bilayers) and macromolecular structure (mostly proteins). Next we studied binding interactions which are the first steps in the expression of biological activity of a macromolecule.

For some proteins, reversible binding is their sole function (consider the binding of dioxygen to myoglobin and hemoglobin). For many others, it is not. For those, what can happen next? The answer to that question clearly depends on the biological function of the macromolecule. We can simplify this process by adding one additional step as reflected in the equilibrium binding expression shown below:

$E + L \rightleftharpoons EL \rightleftharpoons M + X$

This expression indicates that the free ligand has changed in some fashion to x. In the next two chapters, we will consider two kinds of transformations:

• L is a ligand on the outside of a biological membrane (Lout) which binds to a membrane protein receptor, R. This undergoes a conformational change (as we studied in the binding of dioxygen to hemoglobin) which leads to the expulsion of the bound ligand to the inside of the membrane (Lin).This can modeled with the simple equation:
R+Lout ↔ RL↔ R+ Lin. This process is called facilitated diffusion and represents a physical as opposed to chemical process since no covalent bonds are made or broken. This process proceeds down a concentration or chemical potential gradient (Dm < 0) and hence is spontaneous (thermodynamically favored).If the ligand concentration is higher inside the cell, net diffusion moves it to the outside of the cell. Passive (non-facilitated) diffusion is kinetically slow in the absence of a receptor since membranes present formidable barriers to the passage of polar molecules.
• L is a ligand (or substrate S) which binds to a protein enzyme, E. The bound substrate is chemically altered to produce a new product, P, which dissociates from the enzyme. This can be expressed most simply as:

$E + S \rightleftharpoons ES \rightleftharpoons E+ P$

This chapter will discuss diffusion processes. First, diffusion equations will be derived for cases not involving a binding receptor. The equation will show the rate of diffusion of a solute across a membrane from a region of high concentration to a region of low concentration (Δm < 0) is a linear function of [DL] across the membrane. Next we will derive equations for receptor-mediated diffusion across a membrane - facilitated diffusion. We will deal with the situation when the solute must be transported up a concentration gradient (which requires ATP as an exogenous source of energy), a process called active transport

### Case 1: Passive Diffusion with No Receptor

The flux of molecule A (JA) across a membrane of thickness dx, is proportional to the concentration gradient across the membrane, dA/dx., which is expressed as Fick's First Law of Diffusion:

$$J_A \propto \dfrac{dA}{dx}$$ or $$J_A= - D \dfrac{dA}{dx}$$

where D is the diffusion coefficient. The negative sign is necessary since concentration increases in the opposite direction of net flux. For these derivations, we will assume the J_A is the initial flux. That is, the flux is measured for a short enough time that the relative concentrations of A on both sides of the membrane does not change significantly.

It should be clear that eventually the net flux levels off to zero, when the concentrations of A on both sides of the membrane are equal (i.e. the chemical potential of A on both sides is identical). This can also be expressed as JA = -L dmA/dx , which bridges kinetic and thermodynamic aspects of diffusion.

For JA = - D dA/dx, dimensional analysis shows that

J = moles/area/sec = mol/cm2.s = - (cm2/s) mol/cm3/cm.

Hence the units of the diffusion constnat, D, is cm2/s.

Rearranging Fick's Law gives the following:

$$J_A= - D \dfrac{dA}{dx}$$

$$J_A dx = - D dA$$

Integrate both sides with x varying between 0 and x (the thickness of the membrane), and A from Aout to Ain.

JA x = -D [Ain - Aout] = D [Aout - Ain] or

JA = (D/x) [Aout - Ain] or

JA = P [Aout - Ain]

where P is the permeability coefficient, which has units of cm2/s/cm or cm/s. (We discussed permeability coefficients for different solutes traversing model bilayers when we discussed lipids.)

A plot of JA vs (Aout - Ain) is linear, with a slope of D/x

### Case 2: Receptor-Mediated Facilitated Diffusion

Consider the following mechanism:

Let's assume that for this system the initial flux will be measured. We would like to derive equations which show J as a function of Aout (assuming that Ain is negligible over the time course of measuring the initial flux. Also assume that the J facilitated is much greater than J passive. In contrast to passive diffusion, JA is not proportional to Aout but rather to [Abound].

Consider this example to help you understand that proportionality. Pretend that the receptor is a truck which can carry one particle across the membrane at a time (i.e. 1/1 stoichiometry). Also assume that the particle can't get across without being carried by the truck. If there are no trucks in the membrane, no load can be delivered. If there are trucks in the membrane but no particles in them, no load will be delivered. As the number of particles available to be loaded into the truck increase, the truck will have an increased chance to be loaded (depending of course on the affinity of the particle for the truck). If the number of loaded trucks is doubled, the number of particles dumped to the other side will double. Therefore, by analogy,

JA is proportional to [RA], or

Equation 1: JA = const [RA] = k3 [RA]

How can we calculate RA when we know A and R? Let us assume that Atotal (A0) is much greater than R0, as is the likely biological case, and Ain = 0. We can calculate RA using the following equations, and the same procedure we used for the derivation of the binding equation

$ML = \dfrac{ M_oL}{K_d + L}$

Equation 2: Dissociation constant:

$$K_d = \dfrac{ [A]_{eq}[R]_{eq} }{[AR]_{eq}} = \dfrac{[A][R]}{[RA]}$$ (units of molarity)

Equation 3: Mass Balance of R: Ro = R + RA so R = Ro -RA

Since we will assume that A0 is much greater than R0, we will not need the mass balance for A (which is Ao = A + RA)

Substitute 3 into 2:

$(RA)K_d = (Ro)A - (RRA)A$

$(RA)K_d + (RA)A = (Ro)A$

Equation 4: RA = (Ro)A/(Kd + A)

Substitute 4 into 1 gives the final equation,

Equation 5:JA = const [RA] = k3 [RA] = k3 (Ro)A/(Kd + A).

It should be clear to you from this equations that:

• a plot of JA vs A is hyperbolic
• JA = 0 when A = 0.
• JA = Jmax when A is much greater than Kd
• A = Kd when JA = Jmax/2.

These are the same conditions we detailed for our understanding of the binding equation

$ML = \dfrac{ M_oL}{K_d + L}$

### Rapid Equilibrium Assumption

This derivation is based on the assumption that the relative concentrations of A, R, and AR can be determined by the Kd for the interactions and the concentrations of each species during the early part of diffusion (i.e. under initial rate conditions). Remember under these conditions, Aout does not change much with time. Is this a valid assumption? Examine the mechanism shown in the above figure. Aout binds to R with a second order rate constant k1. RA has two fates. It can dissociate with a first order rate constant k2 to Aout + R (to give the original species), or dissociate with a first order rate constant of k3to give Ain + R (as A moves across the membrane). If we assume that k2 >> k3 (i.e. that the complex falls apart much more quickly than A is carried in), then the relative ratios of A, R, and RA can be described by Kd. Alternatively, you can think about it this way. If A binds to R, most of A will dissociate, and a small amount will be carried across the membrane. If this happened, then R is now free, and will quickly bind Aout and reequilibrate. This occurs since the most likely fate of bound A is to dissociate, not to be carried across the membrane, since k3 << k2

### Diffusion of Ions and Molecules Across Membranes

As we saw previously, the permeability coefficients of synthetic membranes (liposomes) to solutes is related to the size and polarity of the solute.

Figure: Permeability coefficientspf molecules through membranes

Smaller, higher charge density ions (like Na+) have a lower permeability coefficient than do larger, lower charge density ions (like K+). What about natural membranes? Look at the chart below.

Permeability coefficient (cm/s) of natural and synthetic membranes to D-Glucose and D-mannitol at 25oC.

 Membrane Preparation D-Glucose D-Mannitol Synthetic Lipid Bilayer 2.4 x 10-10 4.4 x 10-11 Calculated Passive Diffusion 4 x 10-9 3 x 10-9 Intact Human Erythrocyte (red blood cell) 2.0 x 10-4 5 x 10-9

Why do the permeability coefficients differ for D-Glc and D-Mannitol across the red blood cell membrane?

#### "Receptors" in Facilitated Diffusion

Two types of proteins are involved in facilitated diffusion.

1. Carrier proteins (also called permeases or transporters) such as the glucose transporter (GLUT1) move solute molecules across a membrane,
2. Channels facilitate diffusion of ions down a concentration gradient.

In the former cases, the ligand binds the receptor (permease, transport protein) which induces a conformational change in the receptor as illustrated in the animation above and in the figure link below.)

Figure: Models for facilitated diffusion of glucose

In the latter case, a ligand can bind to the receptor (channel protein) which induces a conformational change in the receptor, a "ligand-gated" channel through the membrane. This process would lead to the diffusion of many ions across the membrane (down a concentration gradient) until the channel closes (which can be induced by ligand dissociation or other events). Clearly, the mathematics we derived for the carrier proteins does not apply to the channel proteins. In addition, there are other ways to "gate" open a channel protein, which we will discuss later. Also some transporters can move solute molecules across a membrane against a concentration gradient. These proteins require an external energy source (like ATP or coupling to the favorable collapse of a second transmembrane gradient ) to drive this thermodynamically unfavored process.

Both links above are from the Theoretical and Computational Biophysics group at the Beckman Institute, University of Illinois at Urbana-Champaign. These molecular dynamic simulations were made with VMD/NAMD/BioCoRE/JMV/other software support developed by the Group with NIH support.

Recently, the x-ray structures of two transporters that are powered by the collapse of a second gradient, were reported.

Although it is clear that conformational changes in membrane proteins must occur for activity to be expressed, the exact structural rearrangements of these proteins has been difficult to determine given the lack of structural data for membrane proteins, especially as contrasted to water-soluble enzymes. Transporters allow slow (102to 10-5 s-1) movement of ligands through pores which must be alternately opened and closed to allow extracellular and intracellular access/egress. They need at a minimum an "open-to-out" and "open-to-in" conformational states. The pores must be transient and not continuous. Structural studies have been performed on E. Coli Lac Y Permease, a member of the largest transporter family, Major Facilitator Superfamily (MFS). These proteins have two halves with symmetrical halves each with 6 transmembrane domains. Crystal structures suggest a rocker switch like the figure above for the Glu transport protein.

Structural work by Singh et al on the Leu Transporter (LeuT), a member of the solute carrier 6 or sodium coupled transporters, which is an active transporter requiring movement of Na ion into the cells to power the uptake of Leu, show an "open-to-out" and occluded binding state for ligand (Leu). Tryptophan, a competitive nontransportable inhibitors binds to the open-to-out state, but is too large for the obligate occluded state so it is not transported.

Carriers of ligands need not be proteins. Ionophores are small organic molecules which can bind metal ions and move them down a concentration gradient across a bilayer. Most ionophores are not resident in the cell membrane. Rather they are mobile carriers of ions.They bind ions in solution, ferry them through the membrane, and then release them on the other side. As with receptor carrier proteins, they work in both directions but move ions in a net fashion down the concentration gradient of the ion. One example is valinomycin, a natural product of Streptomyces fulvissimus. It is a cyclic peptide consisting of L and D-Val along with L-lactate and D-hydroxyisovalerate, connected through both ester and amide bonds.

The structure of the valinomycin in its K+ form give clues to its function. The six Val carbonyl oxygens bind the K+ ion. The hydrophilic groups are pointed toward the center, while the hydrophobic groups point to the outside of the structure, allowing the K+ ion to be sequesters in a polar environment as the nonpolar exterior of the complex passes through the membrane. This ionophore is specific for K+ and binds the smaller Na+ ion weakly. This can be accounted for by two factors. The smaller sodium ion doesn't bind as tightly to the chelating carbonyl oxygens. Also, the sodium ion has a higher charge density, so the Na+/water interactions must be more stable and more difficult to break than those to K+. The ion must be desolvated before it binds to the complex. Other ionophores are specific for other ions.

Some ionophores, like gramicidin, from Bacillus brevis, forms a pore in the membrane through which different types of Group I ions may flow. It is a 15-residue peptide consisting of alternating D and L amino acids of significant hydrophobicity. The peptide appears to form a dimer helix (not an alpha helix, however), in a bilayer membrane.

• Jmol: Gramicidin (go to the bottom link for next page in the top frame of this page and continue until you get to the Gramicidin page - an incredible Chime presentation.)

### References

1. Science 322: 1655 (2008)
2. Huang, Y. et al. Structure and mechanism of the Glycerol-3-Phosphate Transporter from E. Coli. Scinec. 301, pg 616-620 (2003)
3. Abramson, J. et al. Structure and mechanism of the Lactose Permase of E. Coli. Science. 301, pg 610-615 (2003)
4. Locher, K. et al. Breaching the Barrier (review of transport proteins LacY and GlpT. Science. 301, pg 603 (2003)
5. Alper, J. Breaching the membrane (new ways to move drugs through membranes).Science.296. pg 838 (2002)
6. Murata et al. Structural Determinants of Water Permeation through Aquaporin I.Nature. 407, pg 599 (2000)
7. Fu et. al. Structure of a glycerol-conducting channel and the basis for its selectivity. Science 290. pg 481 (2000)
8. Sui et al. Structural basis of water-specific transport through the AQP1 water channel.Nature. 414. pg 872 (2001)
9. deGroot and Grubmuller. Water Permeation Across Biological Membranes:Mechanism and Dynamics of Aquaporin-1 and GlpF.Science.294. pg 2304, 2353 (2001)
10. Aqivst and Luzhkov. Ion permeation mechanism of the potassium channel. (dynamics results). Nature. 404 pg 881 (2000)