rate - equilibrium free energy relationship ( refer ) analysis is a methodology originally developed to characterize the energetics of simple chemical reactions ( marcus , 1968 ; grunwald , 1985 ) . recently , this approach has been successfully applied to glean insight into the dynamics of protein conformational changes , such as the opening conformational transitions of ligand - gated ion channels ( e.g. , grosman et al . , 2000 ; cymes et al . , 2002 ; chakrapani and auerbach , 2005 ) . using this methodology , the relative timing of movements of different structural regions of the channel protein can be deduced ( zhou et al . , 2005 ; so far , refer analysis has been applied to reversible systems that are known to be at thermodynamic equilibrium . for the majority of ion channels , gating is assumed to be an equilibrium process , in contrast to transporters that use an external source of energy to complete their catalytic cycles . as a consequence however , some channel and transporter mechanisms have lately been found to intersect , suggesting that evolutionary divergence from common ancestors has given rise to related families of channels and transporters ( gadsby et al . , 2006 ; miller , 2006 ; chen and hwang , 2008 ; csandy and mindell , 2008 ; gadsby , 2009 ) . channels with transporter - like features and transporters with channel - like features are more abundant than previously envisaged . one such example is cftr , a chloride ion channel that belongs to the abc transporter superfamily ( riordan et al . , transport function of abc proteins is linked to atp hydrolysis and is hence a nonequilibrium process ( higgins , 2001 ; oldham et al . , 2008 ) . likewise , gating of the cftr channel pore has been linked to atp hydrolysis ( gunderson and kopito , 1994 ; hwang et al . , 1994 ) and has been found to violate microscopic reversibility ( gunderson and kopito , 1995 ; ishihara and welsh , 1997 ; zeltwanger et al . , 1999 ) , implying a nonequilibrium mechanism . members of the clc channel / transporter superfamily provide another example ; clc chloride channels appear to have evolved from ancestral transporters , and violations of microscopic reversibility in clc channel gating ( richard and miller , 1990 ; pusch et al . , 1995 ; chen and miller , 1996 ; chen et al . , 2003 ; miller , 2006 ; chen and hwang , 2008 ; lsal and maduke , 2008 ) suggest that these channels have also retained vestiges of their ancestors nonequilibrium conformational cycles . the success of refer studies on ligand - gated ion channels has encouraged its recent application to cftr channel gating ( scott - ward et al . although cftr is indisputably an abc protein , whether cftr gating is a nonequilibrium process is still heavily debated , and some earlier studies were argued to provide evidence for its equilibrium nature ( aleksandrov and riordan , 1998 ; aleksandrov et al . , 2000 ) . most recently , linear refer plots and complementary refer slopes for opening and closure of cftr have been interpreted to provide further support for this notion , and to rule out a nonequilibrium mechanism ( aleksandrov et al . , 2009 ) . at a time when the channel  transporter boundary is broadening , it might be useful to researchers working at the margin between these two families of transport proteins to examine some of the basic thermodynamic features of nonequilibrium schemes . by deriving the analytical form of the brnsted plots for the simplest cyclic gating mechanism , a three - state model , this paper examines whether ( 1 ) a mechanism can be identified as equilibrium or nonequilibrium from the characteristics of refer plots , and ( 2 ) whether information on the transition - state structures can be obtained from refer plots for gating mechanisms that are known to be obligatory cycles . thus , a rate k is obtained as(1)k = ae-grt , where g reflects the height of the free energy barrier ( the transition - state t , for the c to o transition ; see fig . classical refer analysis is based on the assumption that the energetic perturbation of the transition state is a linear combination of the energetic perturbations of the ground states ( marcus , 1968 ; grunwald , 1985 ) . thus , if a structural perturbation changes the free energy of the open ground state ( relative to the closed ground state ) by go , then the free energy of the transition state will change by go ( fig . 2 a , dashed line ) . as a consequence , the logarithms of the forward and backward rate constants ( kco and koc ; fig . 1 a ) will change by go/(rt ) and (1)go/(rt ) , respectively , whereas the logarithm of the equilibrium constant changes by go/(rt ) . refer plots ( brnsted plots ) are log - log plots of forward or backward rates as a function of the equilibrium constant for a series of structural perturbations typically introduced into a single region of a protein . for scheme 1 in fig . 1 a , the slopes of these plots will be  and 1 , respectively , allowing experimental determination of . based on the above simple picture , a high  value ( close to 1 ) indicates that in the transition state , the perturbed region of the protein is already near its open - state conformation , whereas a  value close to 0 is an indication that the region is still closed - like in the transition state . a simple equilibrium and simple nonequilibrium gating scheme . ( a ) simple equilibrium gating scheme ( scheme 1 ) with opening ( ko ) and closing rates ( kc ) reflecting forward and backward passage across the same free energy barrier . ( b ) simple nonequilibrium gating scheme ( scheme 2 ) . because transition o2c is fast compared with transition o1o2 , channel closing rate is determined by the rate of the latter transition ( k2 ) . ( a ) free - energy landscape for the mechanism in scheme 1 ; the reversible , equilibrium transition cto is shown without ( solid line ) and after ( dashed line ) a structural perturbation . ( b ) free - energy landscape for an observable closedopenclosed transition based on the mechanism in scheme 2 ; landscapes without ( solid line ) and after ( dashed line ) a structural perturbation are shown . t1 and t2 denote the transition states for the co1 and the o1o2 transition , respectively . t3 , the transition state for step o2c , reflects a low energetic barrier compared with t2 , and has therefore little effect on closing rate ( dotted lines ) . irreversibility is a consequence of unequal barrier heights for exiting each of the ground states in the forward versus backward direction . notation c * emphasizes that after each closure , the entire system has changed relative to the pre - open situation ( e.g. , 1 atp has been converted to adp+p ) , even though the channel protein itself has adopted its pre - open conformation ( c ) . for simplicity , this study assumes the above simple barrier model and classical interpretation of the  value . in reality , energy landscapes for ion channel gating transitions are likely more complex , but recent work has shown that for such reactions , experimental  values still reflect the temporal sequence in which different regions of the protein move during a closed  open transition ( zhou et al . , 2005 ; auerbach , 2007 ) . to what extent are the results presented here ( eqs . 10 and 11 , although derived here using classical simple energy barriers ( fig . 2 b and eqs . 59 ) , can be equivalently written as lnko=1lnkeq(co1 ) and lnkc=(12)lnkeq(o1o2 ) , where keq(c - o1 ) and keq(o1-o2 ) are the equilibrium constants for steps c - o1 and o1-o2 , respectively ( although the system is not at equilibrium , these constants do exist ) . 1220)really only require that for both steps the logarithm of the forward rate constant be approximately linearly related to the logarithm of the equilibrium constant . the conclusions of this study therefore rely only on this typically observed empirical relationship , not on the actual physical interpretation of the  value . 3 , e and f ) , for go1 values ranging from 2 to + 2 rt ( in 0.2-rt increments ) . 3 , e and f , solid symbols ) of these data points was used to obtain slope factors . calculated refer plots for scheme 2 . unperturbed opening and closing rates were ko = 4.348 s and kc = 4.545 s , and -values were 1 = 0.5 and 2 = 0.5 . ( a  d ) refer plots for opening ( a and c ) and closure ( b and d ) under the assumption of eq . 13 . in a and b , a = 1 , and in panels c and d , a = 0.25 . values for logko and logkc were calculated for go1 ranging from 2 to + 2 rt ( in 0.2-rt increments ) using eqs . 14 and 15 , and plotted as a function of logkapp calculated using eq . 16 . solid lines are linear regressions to the plots , and fitted slope values are printed in each panel . ( e and f ) refer plots for opening ( e ) and closure ( f ) , assuming the arbitrary nonlinear relationship go2=0.25(ego11 ) . values for logko and logkc were calculated for go1 ranging from 2 to + 2 rt ( in 0.2-rt increments ) using eqs . solid symbols correspond to go1 values of 0.6 rt , 0.4 rt , 0.2 rt , 0 , + 0.2 rt , + 0.4 rt , and + 0.6 rt . solid lines are linear regressions through this restricted set of data points , and fitted slope values are printed in each panel . ( in all panels , the more conventionally used log - log plots are shown instead of the natural logarithms used in eqs . this study compares a cyclic mechanism ( fig . 1 b , scheme 2 ) with a two - state equilibrium mechanism ( fig it must be emphasized that , mechanistically , three states are minimally required for a nonequilibrium reaction cycle . although , kinetically , two - state cyclic models can be constructed assuming two states connected by two distinct pathways , such models are necessarily simplifications that , for kinetic reasons , neglect intermediate state(s ) along one or both of those pathways . ( e.g. , kinetically , scheme 2 also behaves as a two - state system because the short - lived o2 state can be lumped into state c. ) here , scheme 1 is assumed to be a true mechanistic two - state model . for the simple reversible scheme 1 in fig . 1 a , the slopes of the brnsted plots for opening ( o ) and closure ( c ) are  complementary  to each other in the sense that c = o  1 ( see materials and methods ) . but let us consider an arbitrary gating scheme that may or may not include irreversible steps . suppose the current of a single channel is recorded for some length of time t , during which the channel produces n openings and closures of duration to1 , tc1,  ton , tcn , respectively . mean open time ( o ) and mean closed time ( c ) and open probability ( po ) are then obtained aso=1nk=1ntok,c=1nk=1ntck , andpo=1tk=1ntok.becauset=k=1ntok+k=1ntck , the previous equation can be rewritten as(2)po=(k=1ntok)/(k=1ntok+k=1ntck)=(1nk=1ntok)/((1nk=1ntok)+(1nk=1ntck))=oo+c . this simple relationship between mean open times , mean closed times , and open probability ( eq . 2 ) is fundamentally true for any gating scheme , reversible or irreversible , and therefore , in contrast to published suggestions ( aleksandrov and riordan , 1998 ; aleksandrov et al . , 2009 ) , provides no evidence concerning the mechanism of cftr channel gating . refer analysis ? using mean opening and closing rates , defined as the inverses of the mean closed and open times ( ko = 1/c , kc = 1/o ) , eq . if the process is reversible , then kapp is the apparent equilibrium constant between the set of open states and the set of closed states ; if the process is far from equilibrium , and hence effectively irreversible , then kapp has no physical meaning . nonetheless , for both cases it follows from eq . 3 that kapp = ko / kc , i.e. , ln kapp = lnko  lnkc . for any functions f1 , f2 , and f = f1  f2 , it necessarily follows that (f1f2)f=1,from whichf2f=f1f1.therefore , by necessity,(4)(lnkc)(lnkapp)=(lnko)(lnkapp)1 . in other words , the slopes of the refer plots for opening ( o , app ) and closure ( c , app ) will always follow the  complementarity  described above for scheme 1 ( i.e. , c , app = o , app  1 ) , whether or not a gating scheme contains irreversible steps . 4 is a reflection of the fact that every ion channel  regardless of the mechanism that operates its gates  spends complementary fractions of time open versus closed . demonstration that this relationship holds therefore can not be interpreted as providing evidence to support an equilibrium mechanism of gating of the cftr channel ( compare with aleksandrov et al . , 2009 ) . experimental refer plots for ligand - gated ion channels have so far been found to be mostly linear ( e.g. , grosman et al . , 2000 ; chakrapani et al . , 2004 , but , compare with mitra et al . , 2005 ) there is certainly no theoretical requirement for refer plots to be linear for equilibrium schemes , and the original theory , which assumed barrier functions to be simple intersecting parabolas ( marcus , 1968 ) , predicts refer plots to be quadratic ( the predicted negative curvature is known as the hammond effect ) . the linearity assumption is a first - order approximation , assumed to be valid for small perturbations . recent studies have concluded that the opening / closing transition of the nicotinic acetylcholine receptor channel reflects sequential passage through a series of short - lived intermediate states , too brief to be detected in patch clamp recordings ( auerbach , 2005 ; zhou et al . , 2005 ) . such transition - state ensembles are best described by broad flat energy barriers that contain multiple shallow wells . passage across such barriers requires first reaching the high - energy plateau , and then leaving the plateau in the forward direction after a random walk across the series of shallow wells . thus , several successful attempts to reach the high - energy plateau are typically required to traverse it . mathematical description of such reversible systems predicts refer plots with substantial curvature , in fact more than has been observed experimentally ( zhou et al . , 2005 ) . thus , linearity of the majority of refer plots observed to date is not a consequence of the reversible , equilibrium nature of the underlying mechanisms . although much recent effort has been devoted to better understanding refer data for ion channel gating , these efforts have focused on reversible , equilibrium mechanisms . as a first step to examine what can be learned about nonequilibrium mechanisms using this methodology , this section derives the refer relationships for the simplest possible cyclic scheme , a three - state model ( fig . 1 b , scheme 2 ) . of note , scheme 2 represents a simplified version of the cyclic gating scheme used by several groups to describe open burst / closed interburst cycles of the cftr chloride channel ( for review see gadsby et al . , 2006 and hwang and sheppard , 2009 ) , in contrast to a scheme 1like equilibrium suggested by others ( aleksandrov and riordan , 1998 ; aleksandrov et al . apart from its unique regulatory domain , cftr conforms to typical abc protein architecture , comprising two transmembrane domains ( which in cftr form the channel pore ) , and two intracellular nucleotide binding domains ( nbds ; nbd1 and nbd2 ) . upon atp binding , abc nbds form tight dimers , rapid disruption of which requires prior hydrolysis of at least one of the two atp molecules buried at the dimer interface ( moody et al . , 2002 ; smith et al . , 2003 ) . for cftr , there is strong evidence that nbd dimer formation and dissociation are tightly coupled to pore opening and closure ( vergani et al . , 2005 ) . thus , in scheme 2 , transition co1 represents formation of the nbd1-nbd2 heterodimer , transition o1o2 represents hydrolysis of atp at the nbd2 catalytic site , and transition o2c represents rapid dissociation of the nbd dimer after atp hydrolysis . although ultimately all three steps are reversible , reverse rates o1c , o2o1 , and co2 are neglected because evidence suggests that they are much slower than the forward rates , at least for rates o1c and co2 ( e.g. , vergani et al . , 2005 ; csandy et al . , the closed interburst states are lumped into a single , fully liganded state ( c ) . one molecule of atp is converted to adp plus phosphate and , therefore , the system does not return to the same g level it experienced before opening ( compare states c * vs. c in fig . 2 b ) . also , opening followed by closure reflects transitions across two distinct g barriers , g1 and g2 ( fig . closure is rate - limited by transition across the barrier o1o2 ( rate k2 ) . therefore , the rates of opening to , and closing from , a burst can be written as(5)ko = a1eg1rt,(6)kc = a2eg2rt , and the apparent equilibrium constant iskapp = a1a2eg1g2rt . suppose a perturbation alters the free energies of the open ground states o1 and o2 ( relative to the free energy of the preceding closed state ) by go1 and go2 , and that the conventional refer assumption holds , i.e. , that the consequent changes in barrier heights are linear combinations of the free energy changes of the ground states that the barriers separate ( fig . correspondingly , the modified opening and closing rates and apparent equilibrium constant will become(7)ko=a1eg1+1go1rt,(8)kc=a2eg2+(21)(go1go2)rt , and(9)kapp=a1a2eg1g2rtego1(12 + 1)+go2(21)rt . thus , for the general case , the logarithms of the above parameters can be written as(10)lnko=lnkogo1rt1,(11)lnkc=lnkc(go1go2)rt(21),and(12)lnkapp=lnkappgo1rt(12 + 1)go2rt(21 ) . to construct refer plots , a series of perturbations are introduced into a single region of the protein ( e.g. , a series of mutations of a given amino acid residue ) , which will result in a series of different go1 and go2 values . the resulting lnko and lnkc values are then plotted as a function of lnkapp. from eqs . 1012 , it is clear that the slopes of these refer plots will depend on the relative extent by which mutations of a given residue destabilize the o1 and o2 ground states . let us first consider the simple ( but not unrealistic ) situation in which the energetic perturbations of the o1 and o2 ground states , caused by mutations of a given residue , are proportional to each other , i.e.,(13)go2=ago1 . in this case , eqs . 1416 , it is clear that in this case the brnsted plots will be linear , with slopes(17)o , app=11(1a)(21)and(18)c , app=(1a)(21)1(1a)(21 ) . from eqs . 17 and 18 , it is easy to see that the obligate complementarity c , app = o , app  1 obviously applies . 3 ( a  d ) show calculated refer plots for scheme 2 under the assumption given in eq . 13 ; in a and b , a = 1 ( i.e. , go1 = go2 ) , and in c and d , a = 0.25 . the plots were constructed for go1 values ranging from 2 to + 2 rt ( in 0.2-rt increments ) using eqs . 13 provides an example of a nonequilibrium scheme that generates linear refer plots . because a single counterexample is sufficient to invalidate a hypothesis , we can conclude that the nature ( equilibrium vs. nonequilibrium ) of the underlying mechanism can not be inferred from linearity of the refer plots ( compare with aleksandrov et al . , 2009 ) . in the more general case , the energetic perturbations of the o1 and o2 ground states , caused by mutations of a given residue , need not be proportional to each other ( eq . however , if at least some kind of correlation exists between go2 and go1 , described by some arbitrary nonlinear function go2=g(go1 ) , then for small perturbations the first - order approximation(19)go2g(0)go1 will still be applicable . thus , the refer plots will be close - to - linear in some range around lnkapp = lnkapp , with slopes given by eqs . whether deviations from linearity can be experimentally observed in such situations will depend on the curvature of the refer plots and on the experimentally testable range of kapp. fig . 3 f ) for scheme 2 , assuming the entirely arbitrary nonlinear relationship go2=0.25(ego11 ) . although these plots are substantially nonlinear , the curvatures are little apparent for an 10-fold range of kapp values ( for logkapp between 0.5 and +0.5 ; solid symbols ) . note , because g(0 ) = 0.25 in this case , linear regressions through the data points of this restricted range returned slope values identical to those in fig . 17 , one might hope that the observable refer slope o , app might carry some information on the two transition - state structures of scheme 2 , i.e. , on 1 and 2 . 17 yields(20)1=o , app1o , app(1a)(12 ) . because neither 2 nor a are known , the refer slopes will provide no information on the true transition states for all but two limiting cases . one limiting situation is if the observed refer slope for opening is unity ( o , app = 1 ) ; this implies ( eq . 17 ) that either 2 = 1 or a = 1 ( in both cases the energetic perturbation of the o1o2 transition state equals that of the o1 ground state ) , whereas the value of 1 remains undefined . the other limiting situation is o , app = 0 ; this implies ( eq . 17 ) that 1 = 0 ( in the co1 transition state , the conformation of the studied residue still resembles its conformation in the c ground state ) , whereas 2 remains undefined . the above simple examples clearly demonstrate that the characteristics of refer plots do not carry information on reversibility or irreversibility of the underlying gating mechanism , given that both reversible and irreversible mechanisms can result in either linear or nonlinear refer plots , whereas the complementarity ( in the sense defined by eq . additionally , only very limited information about the transition - state structures can be obtained by refer analysis for gating schemes that are known to be irreversible . 1 a , the slopes of the brnsted plots for opening ( o ) and closure ( c ) are  complementary  to each other in the sense that c = o  1 ( see materials and methods ) . but let us consider an arbitrary gating scheme that may or may not include irreversible steps . suppose the current of a single channel is recorded for some length of time t , during which the channel produces n openings and closures of duration to1 , tc1,  ton , tcn , respectively . mean open time ( o ) and mean closed time ( c ) and open probability ( po ) are then obtained aso=1nk=1ntok,c=1nk=1ntck , andpo=1tk=1ntok.becauset=k=1ntok+k=1ntck , the previous equation can be rewritten as(2)po=(k=1ntok)/(k=1ntok+k=1ntck)=(1nk=1ntok)/((1nk=1ntok)+(1nk=1ntck))=oo+c . this simple relationship between mean open times , mean closed times , and open probability ( eq . 2 ) is fundamentally true for any gating scheme , reversible or irreversible , and therefore , in contrast to published suggestions ( aleksandrov and riordan , 1998 ; aleksandrov et al . , 2009 ) , provides no evidence concerning the mechanism of cftr channel gating . refer analysis ? using mean opening and closing rates , defined as the inverses of the mean closed and open times ( ko = 1/c , kc = 1/o ) , eq . if the process is reversible , then kapp is the apparent equilibrium constant between the set of open states and the set of closed states ; if the process is far from equilibrium , and hence effectively irreversible , then kapp has no physical meaning . nonetheless , for both cases it follows from eq . 3 that kapp = ko / kc , i.e. , ln kapp = lnko  lnkc . for any functions f1 , f2 , and f = f1  f2 , it necessarily follows that (f1f2)f=1,from whichf2f=f1f1.therefore , by necessity,(4)(lnkc)(lnkapp)=(lnko)(lnkapp)1 . in other words , the slopes of the refer plots for opening ( o , app ) and closure ( c , app ) will always follow the  complementarity  described above for scheme 1 ( i.e. , c , app = o , app  1 ) , whether or not a gating scheme contains irreversible steps . ultimately , the relationship in eq . 4 is a reflection of the fact that every ion channel  regardless of the mechanism that operates its gates  spends complementary fractions of time open versus closed . demonstration that this relationship holds therefore can not be interpreted as providing evidence to support an equilibrium mechanism of gating of the cftr channel ( compare with aleksandrov et al . , 2009 ) . experimental refer plots for ligand - gated ion channels have so far been found to be mostly linear ( e.g. , grosman et al . , 2000 ; chakrapani et al . , 2004 , but , compare with mitra et al . , 2005 ) . there is certainly no theoretical requirement for refer plots to be linear for equilibrium schemes , and the original theory , which assumed barrier functions to be simple intersecting parabolas ( marcus , 1968 ) , predicts refer plots to be quadratic ( the predicted negative curvature is known as the hammond effect ) . the linearity assumption is a first - order approximation , assumed to be valid for small perturbations . recent studies have concluded that the opening / closing transition of the nicotinic acetylcholine receptor channel reflects sequential passage through a series of short - lived intermediate states , too brief to be detected in patch clamp recordings ( auerbach , 2005 ; zhou et al . , 2005 ) . such transition - state ensembles are best described by broad flat energy barriers that contain multiple shallow wells . passage across such barriers requires first reaching the high - energy plateau , and then leaving the plateau in the forward direction after a random walk across the series of shallow wells . thus , several successful attempts to reach the high - energy plateau are typically required to traverse it . mathematical description of such reversible systems predicts refer plots with substantial curvature , in fact more than has been observed experimentally ( zhou et al . , 2005 ) . thus , linearity of the majority of refer plots observed to date is not a consequence of the reversible , equilibrium nature of the underlying mechanisms . although much recent effort has been devoted to better understanding refer data for ion channel gating , these efforts have focused on reversible , equilibrium mechanisms . as a first step to examine what can be learned about nonequilibrium mechanisms using this methodology , this section derives the refer relationships for the simplest possible cyclic scheme , a three - state model ( fig . 1 b , scheme 2 ) . of note , scheme 2 represents a simplified version of the cyclic gating scheme used by several groups to describe open burst / closed interburst cycles of the cftr chloride channel ( for review see gadsby et al . , 2006 and hwang and sheppard , 2009 ) , in contrast to a scheme 1like equilibrium suggested by others ( aleksandrov and riordan , 1998 ; aleksandrov et al . , apart from its unique regulatory domain , cftr conforms to typical abc protein architecture , comprising two transmembrane domains ( which in cftr form the channel pore ) , and two intracellular nucleotide binding domains ( nbds ; nbd1 and nbd2 ) . upon atp binding , abc nbds form tight dimers , rapid disruption of which requires prior hydrolysis of at least one of the two atp molecules buried at the dimer interface ( moody et al . , 2002 ; smith et al . , 2002 ; janas et al . , there is strong evidence that nbd dimer formation and dissociation are tightly coupled to pore opening and closure ( vergani et al . , 2005 ) . thus , in scheme 2 , transition co1 represents formation of the nbd1-nbd2 heterodimer , transition o1o2 represents hydrolysis of atp at the nbd2 catalytic site , and transition o2c represents rapid dissociation of the nbd dimer after atp hydrolysis . although ultimately all three steps are reversible , reverse rates o1c , o2o1 , and co2 are neglected because evidence suggests that they are much slower than the forward rates , at least for rates o1c and co2 ( e.g. , vergani et al . , 2005 ; csandy et al . , the closed interburst states are lumped into a single , fully liganded state ( c ) . is converted to adp plus phosphate and , therefore , the system does not return to the same g level it experienced before opening ( compare states c * vs. c in fig . also , opening followed by closure reflects transitions across two distinct g barriers , g1 and g2 ( fig . 2 b ) . because rate o2c is fast compared with rate o1o2 , closure is rate - limited by transition across the barrier o1o2 ( rate k2 ) . therefore , the rates of opening to , and closing from , a burst can be written as(5)ko = a1eg1rt,(6)kc = a2eg2rt , and the apparent equilibrium constant iskapp = a1a2eg1g2rt . suppose a perturbation alters the free energies of the open ground states o1 and o2 ( relative to the free energy of the preceding closed state ) by go1 and go2 , and that the conventional refer assumption holds , i.e. , that the consequent changes in barrier heights are linear combinations of the free energy changes of the ground states that the barriers separate ( fig . correspondingly , the modified opening and closing rates and apparent equilibrium constant will become(7)ko=a1eg1+1go1rt,(8)kc=a2eg2+(21)(go1go2)rt , and(9)kapp=a1a2eg1g2rtego1(12 + 1)+go2(21)rt . thus , for the general case , the logarithms of the above parameters can be written as(10)lnko=lnkogo1rt1,(11)lnkc=lnkc(go1go2)rt(21),and(12)lnkapp=lnkappgo1rt(12 + 1)go2rt(21 ) . to construct refer plots , a series of perturbations are introduced into a single region of the protein ( e.g. , a series of mutations of a given amino acid residue ) , which will result in a series of different go1 and go2 values . the resulting lnko and lnkc values are then plotted as a function of lnkapp. from eqs . 1012 , it is clear that the slopes of these refer plots will depend on the relative extent by which mutations of a given residue destabilize the o1 and o2 ground states . let us first consider the simple ( but not unrealistic ) situation in which the energetic perturbations of the o1 and o2 ground states , caused by mutations of a given residue , are proportional to each other , i.e.,(13)go2=ago1 . in this case , eqs . 1416 , it is clear that in this case the brnsted plots will be linear , with slopes(17)o , app=11(1a)(21)and(18)c , app=(1a)(21)1(1a)(21 ) . from eqs . 17 and 18 , it is easy to see that the obligate complementarity c , app = o , app  1 obviously applies . 3 ( a  d ) show calculated refer plots for scheme 2 under the assumption given in eq . 13 ; in a and b , a = 1 ( i.e. , go1 = go2 ) , and in c and d , a = 0.25 . the plots were constructed for go1 values ranging from 2 to + 2 rt ( in 0.2-rt increments ) using eqs . 13 provides an example of a nonequilibrium scheme that generates linear refer plots . because a single counterexample is sufficient to invalidate a hypothesis , we can conclude that the nature ( equilibrium vs. nonequilibrium ) of the underlying mechanism can not be inferred from linearity of the refer plots ( compare with aleksandrov et al . , 2009 ) . in the more general case , the energetic perturbations of the o1 and o2 ground states , caused by mutations of a given residue , need not be proportional to each other ( eq . however , if at least some kind of correlation exists between go2 and go1 , described by some arbitrary nonlinear function go2=g(go1 ) , then for small perturbations the first - order approximation(19)go2g(0)go1 will still be applicable . thus , the refer plots will be close - to - linear in some range around lnkapp = lnkapp , with slopes given by eqs . whether deviations from linearity can be experimentally observed in such situations will depend on the curvature of the refer plots and on the experimentally testable range of kapp. fig . 3 ( e and f ) shows calculated refer plots for opening ( fig . 3 e ) and closure ( fig . 3 f ) for scheme 2 , assuming the entirely arbitrary nonlinear relationship go2=0.25(ego11 ) . although these plots are substantially nonlinear , the curvatures are little apparent for an 10-fold range of kapp values ( for logkapp between 0.5 and +0.5 ; solid symbols ) . note , because g(0 ) = 0.25 in this case , linear regressions through the data points of this restricted range returned slope values identical to those in fig . 17 , one might hope that the observable refer slope o , app might carry some information on the two transition - state structures of scheme 2 , i.e. , on 1 and 2 . 17 yields(20)1=o , app1o , app(1a)(12 ) . because neither 2 nor a are known , the refer slopes will provide no information on the true transition states for all but two limiting cases . one limiting situation is if the observed refer slope for opening is unity ( o , app = 1 ) ; this implies ( eq . 17 ) that either 2 = 1 or a = 1 ( in both cases the energetic perturbation of the o1o2 transition state equals that of the o1 ground state ) , whereas the value of 1 remains undefined . the other limiting situation is o , app = 0 ; this implies ( eq . 17 ) that 1 = 0 ( in the co1 transition state , the conformation of the studied residue still resembles its conformation in the c ground state ) , whereas 2 remains undefined . the above simple examples clearly demonstrate that the characteristics of refer plots do not carry information on reversibility or irreversibility of the underlying gating mechanism , given that both reversible and irreversible mechanisms can result in either linear or nonlinear refer plots , whereas the complementarity ( in the sense defined by eq . additionally , only very limited information about the transition - state structures can be obtained by refer analysis for gating schemes that are known to be irreversible .
rate - equilibrium free energy relationship ( refer ) analysis provides information on transition - state structures and has been applied to reveal the temporal sequence in which the different regions of an ion channel protein move during a closed  open conformational transition . to date , the theory used to interpret refer relationships has been developed only for equilibrium mechanisms . 
 gating of most ion channels is an equilibrium process , but recently several ion channels have been identified to have retained nonequilibrium traits in their gating cycles , inherited from transporter - like ancestors . 
 so far it has not been examined to what extent refer analysis is applicable to such systems . by deriving the refer relationships for a simple nonequilibrium mechanism , this paper addresses whether an equilibrium mechanism can be distinguished from a nonequilibrium one by the characteristics of their refer plots , and whether information on the transition - state structures can be obtained from refer plots for gating mechanisms that are known to be nonequilibrium cycles . 
 the results show that refer plots do not carry information on the equilibrium nature of the underlying gating mechanism . 
 both equilibrium and nonequilibrium mechanisms can result in linear or nonlinear refer plots , and complementarity of refer slopes for opening and closing transitions is a trivial feature true for any mechanism . additionally , refer analysis provides limited information about the transition - state structures for gating schemes that are known to be nonequilibrium cycles .