2. At quasi-steady state or equilibrium, the net in-flux of the carrier-substrate complex CS is balanced by the net out-flux of the free carrier C.

Transcription

1 Facilitated Transport Instructor: Nam un Wang facilitmcd Process escription In facilitated transport, a carrier molecule C binds to the substrate to form a carrier-substrate complex C at the outer side of the cell membrane The complex diffuses by Fick's diffusion from the outer side of the membrane toward the inner side of the membrane, where the substrate is released into the cell cytoplasm and the free carrier C is regenerated The free carrier shuttles back, again by Fick's diffusion, toward the outer side of the membrane, where it picks up another substrate; and the cycle is repeated The following "reaction mechanism" describes this process C e + C diffusion across cell membrane C C i + C e diffusion across cell membrane C i Alternate Process description with more intermediate steps C e + intermediate C diffusion across cell membrane C intermediate C i + Underlying Ideas Flux of a chemical species across cell membrane via Fick's diffusion flux c e c i in mole/s m 2 where iffusivity of a chemical species thickness of cell membrane c e concentration of the chemical species at the outer cell membrane (extracellular) c i concentration of the chemical species at the inner cell membrane (intracellular) We apply Fick's diffusion to both the complex and the free carrier flux C C C flux of the carrier-substrate complex C flux C C e C i flux of the free carrier At quasi-steady state or equilibrium, the net in-flux of the carrier-substrate complex Cs balanced by the net out-flux of the free carrier C flux C flux C 3 The carrier can take on two forms: complex with the substrate and free We assume that the binding event is reversible and fast; thus, it is described by an equilibrium constant, which is related to the affinity of the carrier molecule for the substrate molecule The extent of complex depends on the substrate concentration C C + C C

2 2 facilitmcd 4 The total number of the carrier molecules is conserved trictly speaking, the concentration of the carrier species is not conserved, and we cannot simply add up concentrations from an arbitrary number of locations, say, at the inner and outer membrane For example, we miss by a factor of 2 if we simply add up concentrations at the inner and outer cell membrane C total Area C( x) Area dx C( x) Area dx Because of Fick's diffusion, the concentration profiles for C and C within the cell membrane are linear C( x) C x e C C C( x) C x e C e C i ubstituting C(x) and C(x) into the integral yields, C total 2 C C C e C i Another way of thinking is that the different forms of carrier (free unassociated form C and associated form C) are conserved C total C average C average where C average C C 2 C average C e erivation Below, we express the above ideas mathematically and let Mathcad crank out the answer Given teady-state in- and out- fluxes flux C C C flux C e C i efinition of dissociation constant C C Conservation of different forms C total 2 C C C e C i The above 5 equations allows us to solve for 5 variables, which are flux (which is what we are really after) and the different forms of carrier species (which are what we do not want to see appearing on the RH of flux=?) C e C i 2 C i

3 3 facilitmcd C C C total i C C 2 C 2 e C e i Find C, C e, C, C i, flux 2 C C C total i C C 2 C 2 e C e C C e total i C C 2 C 2 e C e i C C C e total i C C 2 C 2 e C e i i C C total i C C 2 C 2 e C e flux e C C total C i C C 2 C 2 e i e C C C total e 2 ( α) where α i 2 α C pecial case: identical diffusion coefficient for C and C α C C total flux e e flux max where flux e max i pecial case: Very rapid rate of consumption of the substrate in the cell leads to a minimal intracellular concentration of flux flux e max Michaelis-Menten saturation expression C total

4 4 facilitmcd 4 Alternate a -- correct The total concentration of the carrier molecules is conserved at each side -- a better interpretation of "C totale =C +C e " is not "conservation" but a mere definitino of what "C totale " is Note that conservation of the total number of moles of carrier species is a fact; however, we cannot impose C totali =C totale, because that does not follow from material balance and because doing so will result in an over-specified system of equations with 6 equations and only 5 unknowns to solve for: C, C e, C, C i, flux Indeed, the result below shows that C totali C totale (Think hard why in general the total carrier concentrations may be different between the extracellular and the intracelluar sides) Given teady-state in- and out- fluxes flux C C C flux C e C i C e C i efinition of dissociation constant C C Conservation of different forms C totale C C e C totali C C i The above 6 equations allows us to solve for 6 variables C totale Find C, C e, C, C i, flux, C totali C totale C C e totale e i C 2 i C C C C e totale e i C 2 i C C i C C totale e C C C i e C totale e i C 2 i C C C C C totali C totale in general, C C totali C totale if C C C totali C totale if C =

5 5 facilitmcd 4 Alternate b -- ok Assume the total concentration of the carrier molecules is conserved at each side, and they are equal on both sides "C total =C +C e =C +C i " As mentioned in section "Alternative a" above, we will later examine when this assumption holds Because keeping all the equations will result in an over-specified system of equations with 6 equations and only 5 unknowns to solve for, we take away one of the flux equations Given teady-state in- and out- fluxes flux C C C e C i efinition of dissociation constant C C Conservation of different forms C total C C e C total C C i The above 6 equations allows us to solve for 6 variables C total e C total Find C, C e, C, C i, flux C total C total C C total 2 i e This alternative yields the following expression for flux, which is eqn (42) of huler flux C C i C total J max where J max C total Check: Let's see when the reverse flux equals to the forward flux Basically how the diffusivities of C and C (ie, & C ) need to be related C reverse flux C C e C i C total For the forward and reverse flux to be equal, we require: C total C C total C Thus, the total carrier concentrations are idential on both sides when diffusivities are equal: C =C C Conversely, the total carrier concentrations are different on two sides when diffusivities are unequal

6 6 facilitmcd 4 Alternate b -- repeat of above Instead of assuming the total concentration of the carrier molecules are equal on both sides "C total =C +C e =C +C i ", we state they are (not an assumption) We solve a system of 6 equations and 6 unknowns, by adding as an unknown Given teady-state in- and out- fluxes flux C C C flux C e C i C e C i efinition of dissociation constant C C e Conservation of different forms C total C C e C total C C i The above 6 equations allows us to solve for 6 variables C total C total Find C, C e, C, C i, flux, C total C total C C total 2 i e C As expected, the results are identical to the last section, including the requirement: C

7 7 facilitmcd Graph to show general behavior flux max flux e e flux max e, 5 Facilitated Transport ubstrate Flux Into Cells flux Extracelluar ubstrate Conc Flux as a function of the driving force, which is the difference in the extracellular and intracellular substrate concentrations: - flux flux e max 2 i 2 i ubstrate Flux Into Cells flux Facilitated Transport flux max flux max riving Force (Conc ifference)

8 8 facilitmcd Active Transport -- when the equilibrium values are different at the outer and inner membrane Note that when diffusivities & C are different ( C ), active transport is also possible -- basically we need different/asymetric transport and/or kinetic properties assume, Given teady-state in- and out- fluxes flux C e efinition of dissociation constant e C C C flux i C i C C Conservation of different forms C total 2 C C C e C i The above 5 equations allows us to solve for 5 variables C e total C i C e 2 C e i C i e e 2 C e C i e C total C e 2 C e i C i e e 2 C i Find C, C e, C, C i, flux 2 C e C total C e 2 C e i C i e e 2 i C e C total C e 2 C e i C i e e 2 e i C C total C e 2 C e i C i e e 2 C C C total flux i e C C e 2 i i e C e i 2 C e i C i C C total i e where α e i ( α ) e i i e 2 α C pecial case: identical diffusion coefficient for C and C α C C total flux i e i e flux max where flux e i max i e i pecial case: Very rapid rate of consumption of the substrate in the cell leads to a minimal intracellular concentration of flux flux max e Michaelis-Menten saturation expression C total

9 9 facilitmcd pecial case: symmetric membrane (ie, same properties at both the inner and outer cell membrane e i flux flux e max e For the flux to be positive, we must have > Thus, with a symmetrical membrane it is not possible to achieve a positive flux into the cell against the concentration gradient Active transport, where the substrate moves against the concentration gradient, is possible for an asymmetric membrane ince the denominator of the flux equation is positive, the numerator must be positive For flux> numerator i e e e e > > i Transport against the concentration gradient means the intracellular concentration is higher than than the extracellular concentration For < > Combining the last two inequality relationship yields, e > > i Thus, active transport requires the dissociation constant at the inner cell membrane to be greater than that at the outer cell membrane i > e

10 facilitmcd Graph to show the general behavior flux max k i k e k ie k ii 2 m k ie k e flux flux max, 2 k ii m e k ie k ii k ii k ie k e i k ii k i k i k ie k i e k ii k i k i i k e k ii k ii 4 Active Transport ubstrate Flux Into Cells flux 2 2 e i Facilitated -- Active -- Facilitated Extracelluar ubstrate Conc The above plot has three regions: facilitated transport along the concentration gradient (in out), active transport against the concentration gradient (out in), and facilitated transport along the concentration gradient (out in) pecifically, between =( e / i ) and =, substrate flux is against the concentration gradient