3-0Chapter 4STEADY-STATE CHEMICALLYMEDIATED TRANSPORT4-14-2CHAPTER 4. STEADY-STATE CHEMICALLY MEDIATED TRANSPORTContents4 STEADY-STATE CHEMICALLY MEDIATED TRANSPORT 4-14.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-74.2 MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-74.2.1 Single solute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-74.2.2 Two solutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-104.2.3 Choice of numerical parameters . . . . . . . . . . . . . . . . . . . 4-124.3 GUIDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-124.3.1 Interactive environment . . . . . . . . . . . . . . . . . . . . . . . 4-124.3.2 Graph environment . . . . . . . . . . . . . . . . . . . . . . . . . . 4-164.4 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-204-34-4CONTENTSList of Figures4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-84.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-104.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-134.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-154.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-174.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-184-54-6LIST OF FIGURES4.1. INTRODUCTION4-74.1 INTRODUCTIONTransport of many solutes including important metabolites (e.g. simple sugars or aminoacids) through cellular membranes is accomplished by membrane-bound carrier molecules(transporters) that combine with the solute molecule on one face of the membrane, thentranslocate in the membrane and uncombine at the other face. Thus transport involvesbinding and unbinding chemical reactions at a site on the transporter. Different moleculescompete for this site; for example, glucose and sorbose (two sugars) compete for the sugartransporter site. Hence, transport of one sugar can inhibit transport of another simply by oc-cupying a site to which both can bind. This type of transport is called chemically-mediatedtransport.There are canonical models of chemically-mediated transport that capture importantproperties of the transport of metabolites. It is important to understand these canonicalmodels in order to understand how metabolites are transported across membranes. Deriva-tions of predictions of these models are not particularly difficult to follow; the individualsteps are simple. However, the models typically result in messy algebraic expressions thatrelate flux to concentration and transport parameters. Thus it is easy to get lost in algebraicmanipulation as well as in a sea of parameters so that an intuitive grasp of the models canbe missed. The simulation of these equations is intended to develop intuition for thesemodels.4.2 DESCRIPTION OF THE MODELDescriptions of chemically-mediated transport as well as models of such transport pro-cesses can be found elsewhere [Weiss, 1996]. Here we consider two models, one a specialcase of the other, and list both the assumptions and important results. First we consider atransporter that binds only one solute; then we consider a transporter that binds two soluteswith different affinities. Because the resulting equations for equilibrium of the transporterwith solute are analogous to those of the binding of an enzyme to its substrate we refer tothe transporter as an enzyme.4.2.1 Single soluteWe assume that the membrane contains moles of enzyme per of membrane.Each of these enzymes exist in one of four states which we label , , , and(Figure 4.1). In the states, the solute is bound to the enzyme ; in thestate the enzyme is unbound. In the and states, the enzyme, bound and unbound,communicates with the solution on the inner side of the membrane. In the andstates, the enzyme, bound and unbound, communicates with the solution on the outer sideof the membrane. The concentrations of enzyme in the four states are , , , andmoles/cm . The fluxes of bound and unbound enzyme are and and the flux ofsolute is . The flux is defined as positive when the flux is in the outward direction; theunits are in moles/cm -sec. The model is defined by the following assumptions:4-8LIST OF FIGURESFigure 4.1: Kinetic diagram of a chemically-mediated transport model in which the carrier binds asingle solute.The total amount of enzyme, bound and unbound, is constant, i.e. the sum of theconcentration of enzyme overall of its states equals the total concentration of enzyme(4.1)Since the enzyme resides permanently in the membrane, the total flux of enzymemust be zero, i.e.(4.2)The only way the solute can cross the membrane is when it is bound to the enzyme;is assumed to undergo a reversible change in conformation to the form .We assume that the unbound enzyme undergoes a similar reaction between the twoconformations and . These two reactions are assumed to be first-order reactionswith forward and reverse rate constants, , , , and . Therefore,(4.3)and(4.4)The fluxes equal the rates of change of enzyme concentration, so that(4.5)Similarly,(4.6)4.2. MODEL4-9The binding reactions at the membrane interfaces are assumed to be rapid comparedto the rate of transport of solute across the membrane so that the membrane interfacereactions are assumed to be at equilibrium, i.e.,and (4.7)where and are the dissociation constants on the two membrane interfaces.We wish to sove these equations to find the ’s and the ’s as a function of the concen-trations of solute and of the transport parameters. One way to obtain these solutions isto solve for the ’s in terms of the …