Steady-state rate analysis: application to biological transport

J. theor. Biol. (1995) 174, 45–59

Steady-state Rate Analysis: Application to Biological Transport J W  J. B C†

Laboratory of Cardiac and Membrane Physiology, The Rockefeller University, New York, NY, 10021, U.S.A. and †Q.E.D. Unit, Department of Education, Monash University, Clayton, Vic 3168, Australia

(Received on 7 February 1994, Accepted on 4 October 1994)

A novel method for defining the steady-state unidirectional rates of complex reactions has previously been developed (Wagg, 1988 Ph.D. Thesis, Monash University, Australia). This methodology is based upon the method of Wagg (1987, J. theor. Biol. 128, 375–385) for defining the steady-state unidirectional fluxes of chemical species through branched chemical, osmotic and chemiosmotic reactions. It offers a number of distinct advantages over existing approaches to steady-state rate analysis: it is relatively simple to apply to complex reactions and is readily amenable to computer-based application. The method is demonstrated by direct application to a number of hypothetical models for biological transport phenomena.

found to be satisfactory are referred to as elementary processes. Reactions for which this assumption is not satisfactory are referred to as complex reactions and must necessarily comprise more than two transitions of chemical state. The empirically observed kinetic behavior of a complex reaction can only be adequately described on the basis of a mechanism comprising a set of more than two interconnecting transitions of chemical state, because it is only the kinetic behavior of these individual transitions that is defined by the wellestablished principles of chemical rate theory (see Glossary of Terms). Given a kinetic description of each state transition between component states of a complex reaction, the kinetic behavior of the overall reaction is determined by both the absolute and relative rates of transition between all states comprised by the underlying mechanism. The relationship between these rates of transition and the kinetic behavior of the overall reaction is determined according to definable laws of probability. These laws, fundamental to a description of the unidirectional fluxes of chemical species through complex reactions, have been described by Wagg (1987) and are generally applicable to other types of complex kinetic analysis.

Introduction Chemical reactions are initially defined empirically on the basis of an observed stoichiometric net conversion of a set of one or more chemical species (reactants) to another set of one or more chemical species (products). However, the stoichiometric equation of a chemical reaction is of limited value; it conveys no information regarding the time-course of the reaction and the chemical dependencies of the time-course. In order to provide an adequate account of the time-dependent properties of a chemical reaction, one must necessarily develop the theoretical construct of mechanism. A mechanism is a theoretical account of a reaction in terms of a set of transitions of chemical state. A satisfactory mechanistic definition of a chemical reaction is one that accounts for all empirically observable kinetic aspects of the corresponding transformations. The formulation of a mechanism for a reaction begins with the assumption that the stoichiometric equation corresponds to the mechanism, i.e. that the reaction comprises only two transitions of chemical state—one from reactant(s) to product(s), and the other from product(s) to reactant(s). Reactions for which this assumption is 0022–5193/95/090045+15 $08.00/0

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7 1995 Academic Press Limited

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Essentially all biochemical processes are complex. The fundamental difficulty in the elucidation of the mechanism of biochemical processes lies in their inherent complexity. In the present paper, the steady-state rate analysis of complex reactions—together with its relationship to the steady-state flux analysis of some such reactions—is demonstrated by direct application to a number of different mechanisms for complex, biological transport reactions.

of transition from the i-th to the j-th states will be denoted ri, j .

Glossary of Terms

  A complex reaction is one whose mechanism is not given by its stoichiometric equation, i.e. a complex reaction proceeds from an initial set of reactants to a final set of products by two or more elementary processes. In this context, forward direction for an elementary process is defined as proceeding in that direction leading to either (i) binding of reactant, (ii) dissociation of product, or (iii) any change of state proceeding in the direction from reactant binding to product dissociation. Complex reactions may be unbranched or branched. When a complex reaction consists of three or more elementary processes there exists the possibility that the reaction may be branched. For an unbranched (i.e. linear) reaction, each state can undergo no more than two distinct state transitions. For a branched reaction each branch point will represent a chemical state from which three or more state transitions are possible.

This glossary is intended to serve two purposes: (i) to provide the reader in advance with strict definitions of terms used widely in the present paper, and (ii) to provide the reader with a summary of concepts developed and illustrated in later parts of the paper.   An elementary process is one whose empirically observable kinetic behavior conforms exactly with the rate laws predicted directly by its stoichiometric equation. That is to say, an elementary process is one whose stoichiometric equation is a satisfactory description of its reaction mechanism. The stoichiometric equation of such a process defines two transitions of chemical state (state transitions), one in either direction, between two distinct states (see below). In this context, a chemical state refers to the full set of reactants or products participating in a specified elementary process. Unless specified otherwise, individual chemical states of a reaction will be denoted xi , where the subscript i identifies a specific chemical state. Elementary processes will be denoted R(i, j) where iQj, and i and j identify the chemical states of the process. The kinetics of an elementary process are described in terms of the rates of the two state transitions defining the process.   A state transition is a single change from one chemical state to another, the rate of which is governed by chemical rate theory. In particular, the rate of transition from the state xi to the state xj is defined by the chemical activities of all chemical species defining the state xi . In particular, it is directly proportional to the product of the chemical activities of these species, raised to the respective power of each species’ stoichiometric number. The corresponding constant of proportionality is defined as the rate coefficient for the state transition. The rate coefficient for transition from the i-th to the j-th states will be denoted ki, j . The corresponding rate

   The empirically observable change of chemical composition attributable to net reaction across an elementary process is the difference between the forward and reverse rates of state transition. The net rate of reaction across the elementary step R(i, j) in the direction xi to xj is therefore given by ri, j−rj,i .

  A reaction route, R, consists of two distinct state transitions (occurring once only across distinct elementary processes) and any number of interposed elementary processes. The transition leading to the interposed processes is known as the entry transition of the route, and the other transition, leading away from the interposed processes is known as the exit transition of the route. Reaction routes through a mechanism will be denoted Ri, j;k,l , where the subscripts i and j, and k and l, identify the chemical states participating in the entry and exit transitions, respectively. For any given route, Ri, j;k,l , through a reaction mechanism, there exists a complementary route, Rl,k;j,i . The unidirectional rate at which the state i undergoes conversion to the state l via a route Ri, j;k,l through a mechanism will be denoted ri, j;k,l .  ( ‘‘ ’’) A path defines a unique way in which reaction may proceed through a portion (i.e. a route) of a mechanism for a complex reaction. Specifically, a path consists of a sequence of state transitions that

-   completely span a specific route through a mechanism. A path, therefore, consists of a consecutive sequence of n state transitions, beginning and ending at defined entry and exit transitions, the entry and exit transitions (by definition) occurring once only in the path. In the simplest case, where a route contains no elementary processes interposed between the entry and exit transitions, there is only one possible path through the route, consisting of the entry transition followed directly by the exit transition. For complex reactions, where one or more elementary processes are interposed between the entry and exit transitions, any number of transitions may occur repeatedly back and forth across the interposed processes. Whenever a transition is repeated back and forth across an elementary process, a reflection is said to have occurred. Consequently, the length of a given path through a route will depend not only on the absolute number of elementary processes interposed between the entry and exit transitions, but also on the number of reflections that occur between the respective interposed chemical states. Note that there is no finite limit on the number of reflections that may occur across an interposed elementary process. Therefore, reaction through a route, R, will generally involve an infinite set, P, of possible paths through R. This infinite set of paths, deriving from the route, R, and beginning and ending with the same respective initial (entry) and final (exit) transitions (these transitions occurring once only in any given path), defines all ways in which reaction may proceed from the entry transition to the exit transition via the route. In order to account for reaction proceeding through a route, it is necessary to resolve this into an infinite set of components, each component corresponding to the reaction proceeding through the route by each set of paths. Specifically, the infinite set, P, of possible paths must be resolved into an infinite set of subsets, P n, in which each element (path or ‘‘random walk’’) of the respective subset consists of exactly n transitions. Accounting for the reaction proceeding through a route then reduces to the problem of summing an infinite sequence of components of reaction through a route, the n-th component being the sum of all reaction passing through a route via the subset of paths, P n. It is the ability to sum these components of reaction mediated by each such subset of paths (probabilistically determined) according to a Taylor series expansion for a stochastic matrix (defined on the basis of the relative rates of transition between all states of the mechanism), that gives power and generality to the methods outlined previously (Wagg, 1987) and used in the present paper.

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  Complex, branched reactions may involve multiple cyclic routes, Ri,j;k,i , any number of which might be associated with a net change of chemical state. Any cyclic route through a mechanism associated with a net change of chemical state is referred to as a principal cycle. All other cyclic routes are thus referred to as non-principal. Therefore, groups of principal cycles are associated with distinct overall reactions. -   A reaction mechanism is defined as uni-cyclic if all principal cycles through it are associated with one unique chemical, osmotic or chemiosmotic reaction. -   A reaction mechanism is defined as multi-cyclic if there exists more than one set of principal cycles through it, with each set underlying a distinct chemical, osmotic or chemiosmotic reaction. Steady-state Rates and Fluxes Consider the following general reaction involving the stoichiometrically defined net conversion of a set of m reactants, Ri , to a set of n products, Pj : m rf(1) n s Ci Ri JK s Kj Pj , rb(1) j=1 i=1

(1)

where Ci and Kj denote the stoichiometric coefficients of the i-th reactant and the j-th product, respectively, and rf(1) and rb(1) denote the steady-state unidirectional rates (not rate coefficients) of reaction in the forward (left to right) and the reverse (right to left) directions, respectively. In those cases where reaction (1) is elementary, the steady-state unidirectional rates of this reaction (rf(1) and rb(1) ) correspond to the potentially measurable rates of transition from reactants to products and from products to reactants, respectively. Traditionally, these rates are estimated experimentally by measuring the flux of a radioactively labelled chemical group (or chemical species) between a specific reactant/product pair and between a specific product/reactant pair, respectively. This will generally provide a reliable measure of these rates provided there are no significant isotope effects. Thus, for an elementary process the measured unidirectional fluxes of a labelled species will generally provide a suitable experimental definition of rf(1) and rb(1) .

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In cases where reaction (1) is complex, the unidirectional rates, rf(1) and rb(1) , may not be amenable to direct experimental measurement. This is because the measured flux of a chemical group and/or of a chemical species through a complex reaction does not necessarily traverse the entire reaction sequence constituting the overall mechanism of reaction (Chapman, 1982). Thus, individual measurements of the steady-state unidirectional rates of exchange of a labelled chemical species or chemical group between permissible reactant/product pairs do not necessarily yield identical values since different chemical groups or chemical species do not necessarily traverse identical portions of the reaction mechanism. As a consequence, none of these measurable rates necessarily provides an unequivocal definition of rf(1) and rb(1) . An expression developed by Temkin (1971) provides a general method for providing unequivocal definitions of rf(1) and rb(1) for unbranched (linear) complex reactions, irrespective of whether or not these rates are amenable to experimental measurement. The definitions are mechanism dependent and their derivation involves expressing rf(1) and rb(1) (as defined over the entire reaction mechanism) in terms of the unidirectional rates of reaction across each of the elementary steps comprised by the proposed mechanism of the complex reaction. In the case of branched reactions, mediated by either uni-cyclic of multi-cyclic mechanisms, there appears to be only one general approach for unequivocally defining such rates (Hill, 1966, 1968). Hill has modified the diagram methods of King & Altman (1956) and developed a method for defining steady-state unidirectional reaction rates. As the number of discrete chemical states involved in a mechanism becomes greater than about eight, this method becomes too large for practical algebraic purposes (Hill, 1977). In these cases, approaches for obtaining approximate expressions for steady-state reaction rates have been developed (Hill, 1975; Gordon, 1970). Establishing unequivocal kinetic definitions for the unidirectional rates of complex reactions provides a basis for relating experimental flux measurements to these rates and to their ratio (Wagg, 1988). A novel approach for defining these rates, i.e. for steady-state rate analysis, is developed in this paper. The approach is based upon the method of Wagg (1987) for defining the steady-state unidirectional fluxes of chemical species through branched chemical, osmotic and chemiosmotic reactions. The method is demonstrated by direct application to a number of hypothetical models for porter-mediated transport.

Steady-state Flux and Rate Analysis for Reactions Mediated by Uni-cyclic Mechanisms   (a) Flux analysis A general method for flux analysis has been outlined previously (Wagg, 1987). However, in order to emphasize the distinction between steady-state flux analysis and steady-state rate analysis, the pertinent features of flux analysis are demonstrated here for a well-explored uniport mechanism (the simple carrier model). This will provide a basis for comparison with the corresponding methods for rate analysis presented later. Consider the following complex reaction for a transport process, i.e. the movement of a species, S, across a biological membrane from extracellular to intracellular space: rf(2) So JK Si , rb(2)

(2)

where So and Si denote extracellular S and intracellular S, respectively, and rf(2) and rb(2) denote the steady-state unidirectional rates (not rate coefficients) of reaction (2) in the forward and reverse directions, respectively. For the purpose of this discussion it will be assumed that the movements of S across the membrane are mediated solely by the mechanism presented in Fig. 1 (the simple carrier model). The mechanism shown in Fig. 1 comprises an unbranched sequence of elementary steps and the overall reaction is therefore linear. There is only one way in which S may enter the mechanism from any one compartment and, having entered the mechanism, there is only one way in which S may subsequently exit the mechanism and enter the opposite compartment. Hence, there is only one pair of complementary routes through the mechanism via which S may pass across the membrane. For example, for the unidirectional influx of S, there is only one possible entry transition (state 3 to state 4) and only one possible exit transition (state 1 to state 2). Therefore, the mechanism contains only one possible route for influx of S, the route R3,4;1,2 (see Fig. 2); similarly, only one (complementary) efflux route exists, the route R2,1;4,3 . Hence, each of the unidirectional fluxes (efflux and influx) of S mediated by this mechanism consists of a single component corresponding to the steady-state rate of conversion of one naked state of the carrier to the other via those states to which substrate is bound. Thus, the steady-state unidirectional influx of S mediated by this mechanism is equal to the steady-state rate of conversion of E' to E via steps R(3,4) , R(1,4) and R(1,2) (see Fig. 2).

-  

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All state transitions comprising the influx route may be assigned a conditional probability and incorporated into a stochastic matrix. An outline of the procedures involved in the assignment of conditional probabilities is given in Wagg (1987). The resulting stochastic matrix, M, is as follows: State 1 2 3 4 1 2 3 4

K 0 C1,2 G 0 G0 0 G0 kC4,1 0

0

C1,4 L

G

0 0 G. 0 C3,4 G 0

0

l

These probabilities are conditional upon a single state transition having occurred. Hence, the diagonal elements of the matrix are all zero, as are the elements corresponding to net transitions that would require more than one state transition to have taken place. The 3,2 and 2,3 elements of the matrix M are zero because step R(2,3) does not form part of the route for measurable flux of S. Replacement of the 3,4 element of the matrix M (i.e. the element corresponding to the conditional probability for the entry transition) by the

F. 2. The full set of chemical events constituting the influx route R3,4;1,2 through the mechanism of Fig. 1 whereby the chemical species, S, may enter this mechanism from the extracellular compartment by binding to the site on the ‘‘carrier,’’ move randomly back and forth across step R(1,4) , and subsequently dissociate from the ‘‘carrier’’ to enter the intracellular compartment. The thick horizontal lines identify the entry and exit transitions defining this influx route, and the numbers in square brackets denote the different intermediate forms of the ‘‘carrier’’.

rate at which this transition proceeds, r3,4 , yields a matrix, A; State 1 2 3 4 1 2 3 4

F. 1. A simple carrier model involving the binding of a chemical species (S) to a membrane constituent (E) and a subsequent translocation of this species from one side of the membrane to the other. Only one distinct species binding site is assumed to exist on E and two major conformational states of this constituent are identified, E and E', according to whether this site is freely accessible from the intracellular compartment or from the extracellular compartment, respectively. Si and So denote intracellular and extracellular S, respectively. Numbers in square brackets denote the different intermediate states of ‘‘carrier,’’ i.e. the loaded and unloaded forms of E and E'. The vertical arrows denote the translocation steps, R(2,3) and R(1,4) , involving the translocation of the unloaded (‘‘naked’’) and loaded forms of the ‘‘carrier,’’ respectively. All other reaction steps involve and binding and dissociation of S to and from the ‘‘carrier.’’

K 0 C1,2 G 0 G0 0 0 G kC4,1 0

0 0 0 0

C1,4 L G 0 G. r3,4 G 0

l

The matrix A has the property that the 3,2 element of the n-th integral power of A, i.e. An(3,2) , is equal to the unidirectional influx of S mediated by all possible consecutive sequences of individual transitions beginning with the entry transition (state 3 to state 4) and ending with the exit transition (state 1 to state 2), and consisting of exactly n transitions; i.e. it is equal to the component of influx of S mediated by all paths comprised by the subset P n. For example, the 3,2 element of the matrix A3 is equal to the unidirectional influx of S mediated by the path consisting of the following three transitions: E':E'S:ES:E; this is the minimum number of transitions that will completely span this influx route, and corresponds to the simplest path through this route with no reflections occurring across step R(1,4) . In those cases

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where n reflections occur across this step, only paths consisting of 2n+3 transitions will completely span this influx route. For example, the path containing two reflections will consist of the following sequence of seven state transitions: E':E'S:ES:E'S:ES:E'S:ES:E. The foregoing examples illustrate the fundamental distinction between a route and a path (see also the Glossary of Terms). Extracellular S, having entered the mechanism, may move randomly back and forth (reflect) across step R(1,4) any number of times before subsequently dissociating into the intracellular compartment; hence, the influx of S is given by the sum of the 3,2 elements of the matrices An, where n is an odd integer greater than 1. As the 3,2 elements of all other powers of A will be zero, the required flux can be obtained from the Taylor series expansion for the matrix, A (Deif, 1982): (I−A)−1=I+A+A2+A3+· · ·, where I is the identity matrix of A. The required influx is the 3,2 element of (I−A)−1 . Correspondingly, for efflux of S the required value would be the 2,3 element of (I−B)−1 , where the matrix B is given by: State 1 2 3 4

1

K0 Gr G 2,1 G0 kC4,1

2

3

0

0

0

0

0

0

0

C4,3

4 C1,4 L

G G. 0 G 0 l 0

(b) Rate analysis The simple carrier model presented in Fig. 1 comprises an unbranched (linear) sequence of elementary steps for which the steady-state unidirectional rates, rf(2) and rb(2) , are the rates at which any one state of the mechanism undergoes cycling in the clockwise and anticlockwise directions, respectively. These rates are independent of which state one chooses, i.e. it is an inherent property of a cyclic mechanism that all distinct state undergo cycling at identical rates—both unidirectional and, therefore, net. The net rate of cycling of any given state is obviously equal to the net rate of translocation of the substrate S. A fundamental requirement of steady-state rate analysis is the necessity to define these rates of cycling over the entire reaction (Chapman, 1982). For the present example, this means defining these rates over steps R(3,4) , R(1,4) , R(1,2) and R(2,3) . The forward rate of cycling, rf(2) , may be resolved into an infinite sequence of component rates, each

associated with a unique path through the mechanism. Each such path must satisfy the condition that it begin and end with a specific state present within the mechanism. As the mechanism of Fig. 1 is unbranched, any complete set of paths fulfilling the above conditions will comprise a simple cyclic route through the mechanism. Figure 3 presents an example of a cyclic route through the mechanism beginning and ending with state 3, R3,4;2,3 . Note that any valid path through this route begins and ends with state 3 (the entry and exit transitions occurring once only within any such path) and involves any number of reflections across steps R(1,4) and R(1,2) . Therefore rf(2) consists of the sum of the rates mediated by each of the paths through this route. Thus, rf(2) is given by the product of the rate of the entry transition (state 3 to state 4) times the proportion of this rate that passes on to state 2 (via steps R(1,4) , and R(1,2) of this route) and participates in the exit transition (state 2 to state 3). This unidirectional reaction rate cannot be defined by straightforward application of the method of Wagg (1987) since this method only allows one to define the steady-state unidirectional rate of conversion of any given state of this mechanism to any other discrete state. However, a simple modification of the route of Fig. 3 allows definition of rf(2) . The modification involves: (i) the introduction of a dummy intermediate state, state 5, and (ii) removal of all transitions to or from the initial state of the entry transition (in this case, state 3) with the exception of the

F. 3. A cyclic route, R3,4;2,3 , through the mechanism of Fig. 1. The thicker arrows denote the entry (state 3 to state 4) and exit (state 2 to state 3) transitions characteristic of the route. All other notation is as for Fig. 1.

-  

F. 4. The route, R3,4;2,5 , obtained by the following modifications of the route presented in Fig. 3: (i) introduction of a dummy intermediate state, state 5; (ii) removal of all transitions to or from the initial state of the entry transition (state 3) with the exception of the entry transition; (iii) introduction of a new exit transition, state 2 to state 5. All notation is as for Fig. 3.

entry transition. This includes removal of the exit transition (state 2 to state 3); (iii) introduction of a new exit transition, the initial state of which is state 2 (the initial state of the original exit transition) and the final state of which is the dummy state, state 5. The resulting set of transitions defines the route presented in Fig. 4. rf(2) is given by the product of the rate of the entry transition, r3,4 , times the proportion of this rate that passes on to state 2 and participates in the exit transition, state 2 to state 5, i.e. r3,4;2,5 . This reaction rate may be defined by straightfoward application of the method of Wagg (1987). The route of Fig. 4 is resolved into a set of state transitions, conditional probabilities are assigned to the occurrence of each of these transitions, with the new exit transition, state 2 to state 5, being assigned a conditional probability equal to that for the original exit transition, state 2 to state 3. These probabilities are then arranged into a stochastic matrix, M: State

1

2

3

4

1

0

C1,4

2

0 0 0 C3,4 0 0 0 0

0 C1,2 K G C G 2,1 0 3 G 0 0 G 4 C4,1 0 G 5 k 0 0

5 0

L G

C2,5 G 0 G 0 G G 0 l

where Ci,j denotes the conditional probability for the occurrence of the transition from state i to state j. The term for the entry transition, the element in the 3rd row and 4th column of the matrix, M, is then replaced with an expression for the rate at which this transition takes place, r3,4 , to yield a matrix, A.

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Substraction of A from its identity and a subsequent inversion of the resulting matrix yields a matrix (I−A)−1 . The element (I−A)−1 (3,5) is equal to rf(2) . Application of an analagous procedure allows definition of rb(2) . There are four mutually exclusive approaches one may adopt in order to define rf(2) , depending upon the chemical state one chooses as the initial and final state of the original unmodified route. In the above example this state was state 3 (see Fig. 3); this was an arbitrary selection and any one of the other 3 states could have been selected. The above approach is generally applicable to unbranched complex reactions and provides an alternative to the approaches of Temkin (1971) and Hill (1966, 1968). Given a hypothetical mechanism for an unbranched complex reaction, the following general procedure allows definition of the steady-state unidirectional rate of this reaction in the forward direction, rf : 1. select any one of the intermediate states of the mechanism; 2. establish a route through the mechanism, the entry and exit transitions of which involve reaction in a forward direction, with the state selected in (1) above comprising the initial and final state of these transitions, respectively; 3. modify this route by (a) introduction of a dummy intermediate state, (b) removal of all transitions to or from the selected state with the exception of the entry transition (this includes removal of the exit transition), and (c) introduction of a new exit transition, of which the initial state is that of the original exit transition and the final state of which is the dummy state; 4. resolve the modified route into a set of state transitions; 5. assign conditional probabilities to the occurrence of each of these transitions, with the exit transition of the modified route being assigned a conditional probability equal to that for the exit transition of the unmodified route; 6. arrange the above probabilities into a stochastic matrix, M; 7. replace the term for the entry transition with an expression for the rate at which this transition occurs to yield a matrix, A; 8. subtraction of A from its identity and a subsequent inversion of the resulting matrix

.   . . 

52

yields a matrix, a specific element of which is equal to rf . Application of an analogous procedure allows definition of rb , the steady-state unidirectional rate of the overall unbranched reaction in the reverse direction. The only difference is that the entry and exit transitions defining the route through the mechanism (established in procedure 2) should involve reaction in a reverse direction.

or more routes. The number of flux components will equal the product of the number of distinct entry transitions times the number of distinct exit transitions for the transported species. Hence, each flux component can be identified with a unique set of routes through the mechanism. For example, for the mechanism of Fig. 5, there are eight distinct influx

(c) Relations between rates and fluxes Although the net rate of cycling of any given state in the mechanism of Fig. 1 is clearly equal to the measurable net rate of translocation of substrate S, the unidirectional rates of cycling of any given state (rf(2) and rb(2) ) are not generally equal to the unidirectional flux of the species S through this mechanism. The influx and efflux of S, defined as the unidirectional rates at which state 3 undergoes conversion to state 2 and state 3 undergoes conversion to state 2, respectively, via the corresponding flux routes (see Fig. 2 for the influx route) are not equal to the unidirectional rates of cycling as defined in the previous section. This is because the routes subserving these fluxes do not span the full set of elementary processes subserving cycling of any given state of the mechanism (compare Figs 2 and 3 illustrating the respective routes for definition of influx and rf(2) ).   (a) Flux analysis Consider the following complex transport reaction involving the movement of a chemical species, S, across a biological membrane: rf(3) 2S0 JK 2Si , rb(3)

(3)

where So and Si denote extracellular and intracellular S, respectively, and rf(3) and rb(3) denote the steady-state unidirectional rates of the reaction (defined across the entire net catalyzed reaction) in the forward and reverse directions, respectively. For the purpose of this discussion it will be assumed that the movements of S across the membrane are mediated by the branched mechanism shown in Fig. 5. The flux analysis of this mechanism has been fully developed in an earlier publication (Wagg, 1987). In general, the steady-state unidirectional movement of a transported species through a complex transport reaction may be resolved into a number of components, each component corresponding to the steady-state rate of conversion of one intermediate state of the transport protein to another via a set of one

F. 5. A hypothetical mechanism for a second-order porter-mediated transport process involving the binding of a chemical species (S) to a porter (E) and a subsequent translocation of this species from one side of the membrane to the other, two molecules at a time. Numbers in square brackets denote intermediate states of the porter. Two distinct species binding sites are assumed to exist on the porter and four distinct forms of the partially loaded porter are identified, states 2 and 6 corresponding to one of the loaded sites, and states 3 and 7 corresponding to the other. The vertical arrows denote the translocation steps, R(4,5) and R(1,8) , involving the translocation of the unloaded (‘‘naked’’) and loaded forms of the porter, respectively. All other reaction steps involve binding and dissociation of S to and from the porter.

-  

53

routes and eight complementary efflux routes whereby transport of S occurs. (b) Rate analysis Complex, branched reactions generally involve elementary steps across which reaction is an obligatory part of catalysis. For example, in the case of the mechanism of Fig. 5, steps R(4,5) and R(1,8) represent obligatory steps. The initial and final states of the obligatory steps, i.e. states 4, 1, 5, and 8, are referred to as obligatory states. rf(3) and rb(3) may be defined as the unidirectional rates of cycling of any one of these four obligatory states, i.e. definitions of these rates may be established on the bais of a set of cyclic routes derived with respect to any one of these four states. Indeed, having chosen any one of these states, a set of procedures similar to those outlined above for the unbranched reaction allows definition of rf(3) and rb(3) . The only difference is that these rates are defined with respect to more than one route. rf(3) may be resolved into an infinite sequence of component rates, each component rate being associated with a unique path through any one of two distinct cyclic routes through the mechanism. Figure 6 presents a pair of cyclic routes, the initial and final states of which is state 1. One of the routes begins with the state 1 to state 2 entry transition, while the other begins with the state 1 to state 3 entry transition. Both routes end with the same exit transition, state 8 to state 1. Note that any set of paths through this pair of cyclic routes begins and ends with state 1 and involves any number of reflections across steps R(2,4) , R(3,4) , R(4,5) , R(5,6) , R(6,8) , R(5,7) and R(7,8) . The following modifications of the routes of Fig. 6 allows definition of rf(3) : 1. the introduction of a dummy intermediate state, state 9; 2. removal of all transitions to or from the initial state of the entry transitions (in this case state 1) with the exception of the entry transitions. In this case there is only one exit transition (state 8 to state 1); 3. introduction of a new set of exit transitions, one to replace each one removed in step 2 above. The initial state of each new exit transition should correspond to the initial state of the original exit transition it replaces and the final state of all new exit transitions should be the dummy state, state 9. In this case there is only one original exit transition, state 8 to state 1 and, accordingly, only one new exit transition is introduced to replace this, state 8 to state 9. The routes resulting from the above modifications of Fig. 6 are presented in Fig. 7. These routes are resolved

F. 6. A pair of cyclic routes through the mechanism of Fig. 5, R1,2;8,1 and R1,3;8,1 . The thicker lines denote the entry (state 1 to state 2, and state 1 to state 3) and exit (state 8 to state 1) transitions characteristic of these routes. All other notation is as in Fig. 5.

into a set of state transitions and conditional probabilities assigned to the occurrence of each of these transitions, with all new exit transitions being assigned a conditional probability equal to that of the original exit transition they replaced. These probabilities are then arranged into a stochastic matrix, M. The terms for the entry transitions are replaced with expressions for the rate at which the corresponding entry transitions take place, i.e. the element in the 1st row and 2nd column of M is replaced with an expression for r1,2 and the element in the 1st row and 3rd column of this matrix is replaced with an expression for r1,3 . The resulting matrix, A, may then

54

.   . . 

be subtracted from its identity and inverted. The element (I−A)−1 (1,9) is then equal to rf(3) . rb(3) may be defined by application of an analogous procedure.

states of the entry and exit transitions, respectively; 3. modify this route by

The approach presented above for defining rf(3) and rb(3) is generally applicable to branched complex reactions and provides a distinct alternative to the method of Hill (1966, 1988). Given a hypothetical mechanism for a branched complex reaction, the following general procedures allow definition of the steady-state unidirectional rate of the overall reaction in the foward direction, rf :

(a) introduction of a dummy intermediate state; (b) removal of all transitions to or from the selected state with the exception of the entry transitions (this includes removal of all exit transitions); (c) introduction of a new set of exit transitions, one to replace each one removed in step (b) above (the initial state of each new exit transition should correspond to the initial state of the original exit transition it replaces and the final state of all new exit transitions should be the dummy state);

1. select any one of the intermediate states involved in an obligatory step of the mechanism (it is assumed that there will exist at least one such step); 2. establish a route through the mechanism, the entry and exit transitions of which involve reaction in a foward direction, with the state selected in 1 above comprising the initial and final

4. resolve the modified route into a set of state transitions; 5. assign conditional probabilities to the occurrence of each of these transitions, with each exit transition of the modified route being assigned a conditional probability equal to that for the exit transition (of the unmodified route) it replaced; 6. arrange the above probabilities into a stochastic matrix, M; 7. replace each of the terms for the entry transitions with an expression for the rate at which each of these transitions occurs to yield a matrix, A; 8. subtraction of A from its identity and a subsequent inversion of the resulting matrix yields a matrix, a specific element of which is equal to rf . Application of an analogous procedure allows definition of rb , the steady-state unidirectional rate of the overall branched reaction in the reverse direction. The only difference is that the entry and exit transitions defining the route through the mechanism (established in procedure 2 above) should involve reaction in a reverse direction.

F. 7. The pair of routes, R1,2;8,9 and R1,3;8,9 , obtained by the following modifications of the routes presented in Fig. 6: (i) introduction of the dummy intermediate state, state 9; (ii) removal of all transitions to or from the initial state of the entry transitions, state 1, with the exception of the entry transitions; (iii) introduction of a new exit transition, state 8 to state 9. All notation is as for Fig. 6.

(c) Relations between rates and fluxes A comparison of Fig. 6 of the current paper with the corresponding figure in the paper by Wagg (1987) will demonstrate the general distinction between fluxes and rates for branched reaction mechanisms. The unidirectional rates of complex, branched reactions are defined in terms of the unidirectional rates of cycling of obligatory states of the mechanism. By contrast, the unidirectional fluxes of chemical species through such a mechanism are defined in terms of multiple components, each component corresponding to the steady-state rate of conversion of one intermediate state (obligatory or non-obligatory) of the transport

-  

55

protein to another via a set of one or more routes. Thus, the unidirectional rates of complex, branched reactions generally are neither equal nor proportional to the fluxes of chemical species through the underlying mechanism. Steady-state Rate Analysis of Reactions Mediated by Multi-cyclic Mechanisms  By definition, a multi-cyclic mechanism is one associated with the catalysis of more than one distinct complex chemical, osmotic or chemiosmotic reaction. An example would be the mechanism for a porter or osmoenzyme mediating several distinct modes of transport, such as the sodium-potassium pump. Steady-state rate analysis involves establishing unequivocal definitions for the steady-state unidirectional rates of each of these complex reactions, i.e. the unidirectional rates of each distinct mode of operation. Hill (1977) refers to these rates as cycle fluxes. In this context, establishing unequivocal definitions for steady-state rates allows one to resolve the experimentally observable steady-state net and unidirectional fluxes of a chemical species through a multi-cyclic mechanism into components, each component being mediated by a given set of principal cycles within the mechanism. Rate analysis provides a theoretical basis for a more complete understanding of several important biochemical concepts, i.e. it provides a basis for:

F. 8. A hypothetical tri-cyclic mechanism for three distinct osmotic processes: an antiport process and two distinct uniport processes. All three processes are proposed to be mediated by the one porter. Individual intermediate states of the porter are identified by numbers enclosed in square brackets. Horizontal lines represent elementary reactions involving the binding and dissociation of A and B to and from the porter. Vertical lines denote reaction steps involving either species translocation (steps R(1,6) and R(3,4) ) or the translocation of ‘‘naked’’ porter (step R(2,5) ).

Figure 9 presents these three cycles, denoted cycles 1, 2 and 3, each associated with the following complex transport reactions, respectively: rf(4) Ao+Bi JK Ai+Bo (4) rb(4) rf(5) Ao JK Ai rb(5)

(5)

rf(6) Bi JK Bo , rb(5)

(6)

1. a comprehension of the various components of flux present in a steady-state multicyclic mechanism; 2. defining the way in which the chemical and kinetic parameters of a multi-cyclic mechanism determine the distribution of available catalytic activity between the different permissible modes of operation; 3. a comprehension of such properties as thermodynamic ‘‘coupling’’, free energy transfer, etc. (Hill, 1977).

where rf(4) , rf(5) , rf(6) , rb(4) , rb(5) and rb(6) denote the unidirectional rates of reactions (4), (5) and (6) in the forward and reverse directions, respectively, where forward reaction is defined as proceeding from left to right (for the reactions presented above), or as net reaction proceeding in a clockwise direction around any one of the cycles presented in Fig. 9. rf(4) may be resolved into an infinite sequence of components, each associated with a unique path through the mechanism of Fig. 8. Each such path must satisfy the following general conditions:

Unidirectional rate analysis of reactions mediated by multi-cyclic mechanisms is demonstrated by example in the following section.

1. that it begin with a chemical state present within cycle 1 and located at a branch point between this cycle and any number of the other cycles. In addition, this state must participate in a reaction step obligatory for cycle 1; 2. that it finish with this specific state; 3. that all paths beginning and ending with this state transverse only cycle 1, and that cycle 1 be traversed once only.

 -  (a) Rate analysis Consider the mechanism presented in Fig. 8. It consists of three distinct cycles, each of which is associated with a distinct complex transport reaction.

56

.   . . 

There are only two distinct chemical states present within cycle 1 and located at a branch point between two or more cycles, i.e. state 2 and state 5. Figure 10 presents an example of a set of state transitions fulfilling the above conditions and beginning and ending with state 5. Note that (i) any valid path through this route begins and ends with state 5 and

F. 10. A cyclic route R5,6;4,5 through the mechanism of Fig. 8. The thicker horizontal lines denote the entry (state 5 to state 6) and exit (state 4 to state 5) transitions characteristic of the route. All other notation is for Fig. 8.

involves any number of reflections across steps R(1,6) , R(1,2) , R(2,3) and R(3,4) ; (ii) state 5 participates in a reaction step obligatory to cycle 1, i.e. step R(5,6) ; (iii) reflections across step R(2,5) are not permissible since occurrence of the transition from state 2 to state 5 leads to a complete traversal of cycle 2. rf(4) therefore consists of the sum of the rates mediated by each of the paths defined by the above conditions. This set of paths defines a route, the initial and final state of which is intermediate state 5. Thus, rf(4) is given by the product of the rate of the entry transition (state 5 to state 6) times the proportion of this rate that passes on to state 4 (via steps R(1,6) , R(1,2) , R(2,3) and R(3,4) of this route) and participates in the exit transition (state 4 to state 5). This unidirectional rate cannot be defined by straightforward application of the method of Wagg (1987). However, a simple modification of the above approach allows definition of the unidirectional rates, rf(4) and rb(4) . The route of Fig. 10 is modified by:

F. 9. The three distinct cycles constituting the mechanism presented in Fig. 8. (a) Cycle 1, the cycle underlying an antiport process involving the coupled movements of two distinct chemical species, A and B, across a biological membrane in opposite directions and with a 1:1 stoichiometry. (b) Cycle 2, the cycle underlying a uniport process involving the movements of the chemical species A across a membrane independently of the movements of any other chemical species. (c) Cycle 3, the cycle underlying a uniport process involving the movements of the chemical species B across a membrane independently of the movements of any other chemical species. All notation is as for Fig. 8.

1. the introduction of a dummy intermediate state, state 7; 2. removal of all transitions to or from the initial state of the entry transition (in this case state 5) with the exception of the entry transition itself. This includes removal of the exit transition (state 4 to state 5); 3. introduction of a new exit transition, the initial state of which is state 4 (the initial state of the original exit transition) and the final state of which is the dummy state, state 7, i.e. the new exit transition is state 4 to state 7. The resulting set of transitions defines the route presented in Fig. 11. This route may be resolved into a set of state transitions. Conditional probabilities may then be assigned to the occurrence of each of these

-  

57

transitions, with the proviso that the new exit transition, from state 4 to state 7, be assigned a conditional probability equal to that for the original exit transition, state 4 to state 5. These probabilities may then be arranged into a stochastic matrix, M: State 1 2 3 4 5 6 7

1

2

3

4

0 C1,2 0 0 K GC G 2,1 0 C2,3 0 G 0 C3,2 0 C3,4 G0 0 C4,3 0 G0 0 0 0 GC 0 0 0 6,1 G k0 0 0 0

5

6

0

C1,6

0

0

0 0

0 0

0

C5,6

0

0

0

0

7

L G G 0G C4,7G. 0G G 0 G 0l 0

0

The term for the entry transition, the element in the 5th row and 6th column of the matrix, M, is then replaced with an expression for the rate at which this transition takes place, r5,6 , to yield a matrix, A. Subtraction of A from its identity and a subsequent inversion of the resulting matrix yields a matrix (I−A)−1 , the (5, 7) element of which is equal to rf(4) . Application of an identical set of procedures to the route of Fig. 12 allows definition of rb(4) . Figures 13, 14, 15 and 16 present the modified routes, appropriate for the definition of rf(5) , rb(5) , rf(6) and rb(6) , respectively. Each route may be resolved into a set of state transitions and conditional probabilities assigned to the occurrence of each transition. These probabilities may then be arranged into a stochastic matrix, M, and the term for the entry transition replaced by an expression for the rate at

F. 11. The route R5,6;4,7 obtained by the following modifications of the route presented in Fig. 10: (i) introduction of a dummy intermediate state, state 7; (ii) removal of all transitions to or from the initial state of the entry transition (state 5) with the exception of the entry transition; (iii) introduction of a new exit transition, state 4 to state 7. All notation is as for Fig. 10.

F. 12. A route R5,4;6,7 through the mechanism of Fig. 8. The thicker horizontal lines denote the entry (state 5 to state 4) and exit (state 6 to state 7) transitions characteristic of the route. All other notation is as for Fig. 8.

which this transition occurs, to give a matrix, A. Subtraction of this matrix from its identity matrix, I, and a subsequent inversion of the resulting matrix yields a matrix (I−A)−1 , a specific element of which gives the appropriate unidirectional rate. The procedures outlined above are generally applicable to tri-cyclic mechanisms. Nonetheless, the fundamental definition of steady-state rate upon which these procedures were based is generally applicable to multi-cyclic mechanisms irrespective of the number of distinct sets of principal cycles. Consider a multi-cyclic mechanism involving n distinct sets of principal cycles, each set being associated with a characteristic complex reaction. The steady-state rate of the i-th reaction in the forward direction, rf(i ) , may be resolved into an infinite sequence of component rates, each associated with a unique path through any one of the elements of the i-th set of

F. 13. A modified version of a route through the mechanism presented in Fig. 8. All notation is as for Fig. 12.

58

.   . . 

F. 14. A modified version of a route through the mechanism presented in Fig. 8. All notation is as for Fig. 12.

F. 16. A modified version of a route through the mechanism presented in Fig. 8. All notation is as for Fig. 12.

principal cyclic routes through the overall multi-cyclic mechanism. Each such path must satisfy a specific set of conditions:

i-th reaction are permissible provided they do not lead to traversal of these cycles.

1. it must begin with a chemical state common to all principal cycles associated with the net catalysis of the i-th reaction (i.e. it must begin with an obligatory state for the i-th reaction) and located at a branch point between this set of cycles and any number of other cycles; 2. it must end with this specific state; 3. all paths between this state must traverse only those principal cycles associated with the catalysis of the i-th reaction and these cycles must be traversed once only. Reflections into other cycles, not common to the set associated with the

F. 15. A modified version of a route through the mechanism presented in Fig. 8. All notation is as for Fig. 12.

rf(i ) is given by the sum of the rates mediated by each such path. In the case of a tri-cyclic mechanism, the calculation of the steady-state unidirectional rates of the threecomponent complex reactions involves the establishment of three pairs of complementary routes through the mechanism, one pair for each complex reaction. For more complex mechanisms, multiple pairs of complementary routes may need to be established for each component complex reaction. (b) Relation to flux analysis Steady-state flux analysis of a multi-cyclic mechanism involves a straightforward application of the method illustrated earlier in this paper. Although this method was demonstrated within the context of a uni-cyclic mechanism, it is applicable without modification to multi-cyclic mechanisms. This is because flux analysis applies at the level of the overall mechanism without consideration for whether or not it comprises multiple sets of principal cycles. On these grounds, it is apparent that neither the steady-state fluxes of chemical species through a multi-cyclic mechanism nor any proportion thereof, provides a measure of the steady-state rates of any of the component complex reactions. Although analagous mathematical procedures are involved in defining a steady-state rate and steady-state flux, it has been demonstrated that, for complex reactions, flux and rate are fundamentally distinct concepts. This is not to say that flux data contain no kinetic information; on the contrary they can provide

-   a measure of the unidirectional rates of reaction between a restricted set of pairs of intermediate states. Nonetheless, flux in itself can never serve as a general measure of rate for a complex reaction.

59

readily amenable to computer-based implementation at either a numerical or algebraic level. Consequently, there would appear to be no practical limitation on the range of complexity capable of being addressed by these methods.

Concluding Remarks

REFERENCES

A novel method of steady-state rate analysis has been presented and contrasted with the principles of steady-state flux analysis outlined in previous work. This comparison has revealed the essential mechanistic reasons why unidirectional flux measurements do not provide reliable estimates of unidirectional rates for complex reactions; they are fundamentally different concepts. This failure of equivalence between fluxes and rates is conceptually inherent in even the simplest of carrier models and does not require invocation of other complicating factors (e.g. parallel leaks) that undeniably complicate the interpretation of experimental flux data in intact biological systems. The value of the present methods is that they allow rigorous quantitative exploration of the complexities residing in any postulated transport mechanism, totally isolated from other complications. The methods illustrated here offer a distinct advantage over earlier methods in that they are not limited by problems of algebraic intractability. On the contrary, the algorithms defining these methods are

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