4278
J. Phys. Chem. 1995, 99, 4278-4285
Thermodynamic and Kinetic Properties of Ions in Narrow, Multistate Channels. 1. Stationary State Distributions and Thermodynamics Michael E. Starzak Chemistry Department, State University of New York at Binghamton, Binghamton, New York 13902-6000 Received: July 18, 1994; In Final Form: November 9, I994@
Ions flowing independently in an ensemble of narrow, multisite channels under a constant chemical or electrochemical potential reach a stationary state distribution that is maintained by the steady dissipation of the driven system. This distribution and the total ion population in an ensemble of channels are proportional to the ratio of the ion injection flux j and the eigenvalues for the multistate system. The macroscopic flux and dissipation, determined by the bathing solution concentrations and potentials, dictate relationships between the forward and reverse transition probabilities in the channel. The character of the stationary state distribution is defined with “thermodynamic” parameters analogous to equilibrium thermodynamic quantities using injection fluxes and transition probabilities. The stationary state distributions generated for these self consistent conditions include the equilibrium Boltzmann distributions as special cases, Le., in the limit of zero net flux through the channels. Various aspects of the model can be tested using both laser Doppler velocimetry and electrochemical measurements.
Introduction A potential difference applied to ionic solution baths separated by a membrane containing ion conducting channels will produce a net ion flux through these channels. The net flux and the applied potential are the experimental parameters that define a dissipation for this nonequilibrium system. However, these parameters also produce a stable population distribution of ions at different locations within the ensemble of channels. For channels with a discrete number of binding sites within narrow channels, a master equation that includes a forcing function to introduce ions from the baths describes both the permeation kinetics and the stationary state intrachannel distribution. The narrow channel ensures nearest neighbor transition probabilities. The homogeneity of binding sites within the channel limits the number of independent transition probabilities. Ion permeation in these membrane channels has been modeled as a sequence of discrete ion translocations between sites within the channel culminating in the net transport of the ion across the membrane.’,* The number of intrachannel sites can be increased to mimic continuous translocation through the channel.* A constant flux through an ensemble of membrane channels produces a stable distribution of populations at the discrete intrachannel sites. These stationary state distributions differ from equilibrium, Le., Boltzmann distributions, because they are maintained by the ion flux through the channels. A model for translocation in a narrow channel with multiple ion binding sites restricts the kinetics to nearest neighbor transitions between a finite number of intrachannel sites. Moreover, channels with site translation symmetry are expected to produce simple relationships between the transition probabilities for state-state transitions. These intrinsic advantages of the membrane channel ensemble make it the ideal vehicle for a study of stationary state distributions. In practice, observed ion currents for a given transmembrane potential are fitted to multistate models where both the number of sites and the transition probabilities are adjustable parameters. Since the number of such parameters may be large relative to the total experimental data, many parameter sets can describe @
Abstract published in Advance ACS Abstracts, March 1, 1995.
these data making it difficult to establish an exact transport mechanism. This difficulty is alleviated with new techniques such as laser Doppler ~ e l o c i m e t r ythat ~ * ~provide direct experimental measurements of ion velocities in the channels. These data, in conjunction with the electrochemical data, provide the means to test the models presented here. Since the stationary state ion distributions within the channels differ from the B o l t ” distributions in homogeneous media, these channel ion distributions can promote reactions or conformational changes that would normally not occur in homogeneous media. For example, an increased ion residence time at a specific site in the channel at stationary state can induce a conformational change in the channel itself. Flow systems have been proposed as “catalysts” for reactions associated with the origin of life.5 Stationary state thermodynamic parameters, analogous to those of equilibrium thermodynamics, are developed to quantitate the potential for such reactions. The population distributions are generated by casting the kinetics in a master equation formalism that includes a forcing function vector.6 The forcing vector contains unidirectional injection fluxes proportional to the concentrations of permeant ions in the bath. The population distribution is then the solution of this modified master The models presented here include transition probabilities from the terminal states to the bathing solutions. In the absence of a forcing function input, the state populations are zero. An alternative f o r m ~ l a t i o n , ~ ~ ~ where a flux input at one terminal channel state is matched by a flux output from the opposite terminal state, conserves the total ion population in an ensemble of channels and is classified as a conservative model. While conservative system master equations have a zero eigenvalue that defines a Boltzmann equilibrium in the absence of injection fluxes, the nonconservative master equations presented here have channel populations only when ions are injected into the channels from either bath. Net fluxes are the difference between this unidirectional injection flux and the loss flux of ions from the corresponding terminal states. For such models, equilibrium is defined by zero net flux. This condition generates relationships between forward and reverse transition probabilities at equilibriumthat may be conserved under nonzero
0022-3654/95/2Q99-4278$09.QQ~Q 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 12, 1995 4219
Properties of Ions in Multistate Channels net flux conditions. The membrane systems and laser Doppler velocimetry allow experimental tests of this hypothesis. The formal solution of the master equation can always be decomposed into a sum of terms with eigenvalues and projection matrices for the transition probability matrix. Although the stationary state population vector can be approximated by selecting the largest terms in this sum, such approximations are often unnecessary for the nearest neighbor models presented here. The translationally symmetric N site channels are described by exact, closed form expressions for the population distributions. Since the total population of ions in an ensemble of channels is proportional to the injection fluxes, the absolute state populations are introduced into an entropy of mixing equation to define thermodynamic parameters for these systems. The approach introduces a natural separation of thermodynamic variables analogous to those of equilibrium thermodynamics and produces two distinct chemical potentials. The fiist results from the distribution of ions within the channels while the second results from the total channel population.
flux is applied, the formal solution for the population vector is
+ + eW’)W-’b)
~p)= eWrlpo) (I
When k f m, the transition probability matrix is not symmetric but can be transformed to an equation with a symmetric matrix by using symmetrizing matrices with diagonal elements
);(k
[Dljj =
The kinetic equation transforms as
General Formulation An ensemble of membrane channels with N equivalent sites arranged sequentially in a narrow channel connecting two bathing solutions is characterized by a set of identical forward transition probabilities k and reverse transition probabilities m k
*m1 = 2m=
k m
...N -
k
1#N+
where W’ is the symmetric transition probability matrix. The “populations” for the symmetric tridiagonal matrix, W’, transform to the distribution for the unsymmetrized matrix as
k
Forward and reverse transition probabilities are defined independently. For nearest neighbor transitions, the transition probability matrix is tridiagonal and the 11 and NN elements include transitions from the membrane terminal sites to the bathing solutions, -(k+m) k
m -(k+m) k
m
.
0
0
.
k
w= 0
IP) = DIP’) The solution for the symmetrized matrix is expanded as a sum of terms in the eigenvalues, Ai, and projection matrices, I?)(?[, as N
N
m -(k+m)/
The kinetic equation for the ith inner state population is
and
Ip) = DIP’) = ze“”Dli’)(i’lpO)
while the terminal states include injection fluxes from their respective baths,
-dpl -j , - (k dt
+ m)pl + mp,
+ z(1- eAif)LY1DIi’)(i’b)
using the known eigenvalues and projection matrices for an N x N symmetric tridiagonal matrix. Stationary State Distributions The populations evolve to stationary values
IP) = w-’li> N
= ZL;’DIi’)(i’V) The transition probabilities are the elements of a transition probability matrix W while the injection inputs to the baths are included as the terminal components (1st and Nth) of a column vector, b). The kinetic equation for the set of site populations IP) is
-dip) - Wlp) dt
+ P)
If Ipo) is the population vector at time t = 0 when the injection
1
dictated by the transition probability matrix inverse. This inverse is determined either by use of cofactors or by expansion in inverse eigenvalues and projection matrices. 1. Microcanonical Ensemble. The chemical potential difference between the ion in a bath and a terminal state produces the injection flux at each terminus. These differences are then local and have no effect on the intrachannel forward and reverse transition probabilities. However, the relationship
4280 J. Phys. Chem., Vol. 99, No. 12, 1995
between the forward and reverse transition probabilities is established by postulating that equal injection fluxes from each bath lead to equal populations for all intrachannel states. Columns 1 and N of the inverse are determined using cofactors since the two nonzero components of the lj) vector multiply these elements. The jth element of the first column (for inputs from bath 1 ) is
while the jth element of the Nth column (for inputs from bath 2 ) is
starzak larger, but equal, injection fluxes and larger equilibrium populations within the channels. Equivalent injection fluxes from baths of equal concentration require k = m to produce zero net flux through the ensemble of channels. This is equivalent to the condition of microscopic reversibility. The condition is established by experimental parameters such as the bath concentrations and net flux in this case and can be tested experimentally by using either fluorescent location probesl0S1lthat establish the populations as a function of transverse position in the membrane or laser Doppler velocimetry to measure the full set of transition probabilities from the velocity distribution of ions in the ~ h a n n e l . ~Other .~ fluorescent probe techniques12 determine the net flux for these systems. The populations for equal injection fluxes are maintained by equal state to state net fluxes everywhere in the system. The net output flux is
J2 = -kp4
with the normalization factor
+j = 0
while the input flux is
J , = j - kp, = 0 For example, the contributing columns for the inverse for a foursite system are
Since the channel populations are equal, the net flux between state i and state i 1 is also
+
Ji
1 k(k2 k2(k
+ km + m2) + m) u = k4
. . . .
m2(k+m) m(k2+km+m2)
+ k3m + m2k2+ km3 + m4
Each element in the first column is multiplied by the injection flux, j1, from bath 1 and each element in the last column is multiplied by the injection flux, j2, from bath 2. A four-state system has a population distribution,
+ + km2 + m3)+ j 2 m 3 ' + km + m2) +j2m2(k + m ) + m ) +j2m(k2 + km + m2) + k2m + km2 + m3) ,
j1(k3 k2m
= kf, - mL+1 = k fk
-j k I=O
Independent injection fluxes produce the dynamic equilibrium. The populations of the states increase as the concentrations in the baths and, therefore, the injectibn fluxes increase, so there is a chemical potential balance between the baths and channel at equilibrium. The equilibrium distribution represents only one possible distribution for the channel system since the concentrations of each bath can be changed independently. Different bath concentrations produce new population distributions and a nonzero net flux. The condition, k = m, is generally assumed valid when the system is not at equilibrium. This hypothesis can be tested with experiments that measure either the channel state populations or the transition probabilities. The nature of the driven, open system distributions is illustrated for situations where a single injection flux is dominant, e.g., j l =j , j 2 = 0. This stationary state distribution,
A dynamic equilibrium is expected when j 1 =j 2 =j since equal bath concentrations must give zero net flux. The stationary state populations for a four-state channel are
j k3 + p m + 2km2 + m2 0
+ + km2 + m3 + + km2 + m3/
k3 2k2m 2k3 k2m
The state populations are proportional to the ratio, jlk. Equivalent concentration increases in each of the baths will produce
is linear (Figure l), and all populations are smaller than those for the equal injection flux condition because ions are added from only one bath. An injection flux j applied only from bath 2 gives the complementary distribution,
so that the equilibrium distribution is the sum of two complementary linear stationary state distributions (Figure 1). The state populations for an ensemble of channels are numbers rather than fractions or probabilities. If the relative
J. Phys. Chem., Vol. 99, No. 12, 1995 4281
Properties of Ions in Multistate Channels
0.10
1
0.24 0
"
a v
0.20
.-
0.18
+.r
0
3
0.12
n 0
a
0.08 0.04
0.00
0.00 1
2
3
4
State Figure 1. State populations for a four state channel system. The sum of single injection flux populations gives the equilibrium population. (a) Equilibrium distribution when jlk =j'lk = 0.1, k = 1. (b) Distribution with injection flux j l k = 0.1, j'lk = 0, k = 1. (c) Distribution with injection flux j'lk = 0.1, jlk = 0, k = 1. populations are summed to give a normalization, 52, the population vector can be converted to an expression proportional to the vector of state probabilities, le), by multiplying and dividing by Q,
1
State
= keeAv&T)
where AY is the difference between local potentials for two adjacent sites, is the proper relationship between the forward and reverse transition probabilities. The decreasing local potentials for the sites with potentials Y and 0 in baths 1 and 2, respectively, require a zero potential drop for the injection flux, and are (N/N)Y, ((N-l)/N)Y, ..., (l/N)Y with Boltzmann factors
8=4+3+2+1 for this four-state model. The modified ratio, I = jQ/k, is a measure of the total ion population in the ensemble of channels while the probability vector, le), gives the relative distribution of these total ions among the channel states. An increase in the j / k ratio increases the slope and the total ion population of the linear state distribution for a single injection flux model (Figure 2). 2. Canonical Ensembles. Although the chemical potential difference between the bath and terminal site affects only the injection flux, an applied electrical potential produces a field that alters the energies of states within the channel. At equilibrium, the ion populations at sites with local electrical potentials, Yi, are proportional to the Boltzmann factors,
The injection fluxes and rate constants are postulated to give populations proportional to these Boltzmann factors when the net flux through the channel is zero. For equally spaced charge sites, the potential drops between adjacent states are equal. The forward transition probabilities for each nearest neighbor step are identical. The reverse transition probabilities are also all equal, but they can be related to the forward transition probabilities in a way that satisfies both the equilibrium (Boltzmann) distribution and zero net flux conditions. The condition of detailed balance at equilibrium,
4
Figure 2. Increasing the ratio j l k increases both the slope and net population of the population distribution. For a four state system and (a) jlk = 0.1,j'lk = 0, k = 1, (b) j l k = 0.2,j'lk = 0, k = 1, (c) jlk = 0.3,j'lk = 0, k = 1.
e-
with
3
2
(ilN)eY/(kT) -
x=e
- xi '
- eY/(N,ksT)
The transition probabilities and the injection fluxes are selected to produce this set of Boltzmann factors at dynamic equilibrium. The general analysis is illustrated with the four-level model. The populations at stationary state for injections of j and j' from baths 1 and 2 are
+ + + \ + + + + + + + + +
/j(k3 k2m km2 m3) +j'm3 1 j(k3 k2m km2)+j'(km2 m 3 ) IP) = j(k3 k2m) +j'(mk2 km2 m3) bk2 +j'(k3 k h km2 m3)
For equilibrium, the net flux between any two adjacent states is
and a net flux of zero requires
For an N state system,
j r = j(Wm)N This general relationship will generate a Boltzmann equilibrium
Starzak
4282 J. Phys. Chem., Vol. 99, No. 12, 1995 :N-n
distribution if the forward and reverse transition probabilities are related by the condition of detailed balance at equilibrium
N-1
2‘
N
a=
The relationship between the injection fluxes for equilibrium is then
c.’ i=O
For a four site system, for example, the populations are IPS?=
[k(l Although the injection fluxes act on the terminal states, a self consistent equilibrium requires that the injection fluxes are related by their absolute potentials, i.e., independent of the number of discrete states in the channel. For a positive potential, j’ must be larger than j to compensate for the increased forward transition rates produced by the applied potential at equilibrium. Since the injection fluxes are proportional to the bath concentrations, this zero-flux relationship gives the electrochemical equilibrium condition for the baths,
l(1 (1 (1
+ x + x2 + x3 + x )3
+ x + x2 + x3)j + x y + x + x2)j + (x2 + x3)j’ + x)j + (x + x2 + x3)j’
When ion injection from bath 1 dominates, i.e., j‘ = 0, the nth site population for an N site channel distribution is :
N
N-n
The four-site system has populations
I1 +x+x2+x3\ where the activities, ai, reflect the ability of the ion to move from the bath to the first channel state. The population vector for a four-level system at equilibrium with x = klm is
\
1 +x+x2+x3+x-l jk3
k4(1
-
+ + x2 + x3 +
1+x+x2+x-’+x-2 x-2 x4) 1
+ + + +
+ x3 + x4 + x5 + x6 1 + x + x2 + x3 + x4 x + x2 + x3 + x4 + x5 (jlk)~-~
x2
\1 + x + x 2 + x 3 + x 4 iX3
x-3
I
\
The relative populations obey Boltzmann statistics, but the partition function for equilibrium probabilities in a homogeneous system is replaced by a function that is linear in the ratio jlk. This ratio gives the total population of ions in the ensemble of channels at dynamic equilibrium. 3. Canonical Flux Distributions. When a potential is applied to the membrane, the transition probabilities are assumed to obey the condition of detailed balance at equilibrium for all stationary states. The actual stationary state population depends on these transition probabilities and the injection fluxes. For injection fluxes j and j’ from baths 1 and 2 respectively, the population of the nth site in an N site channel is
The populations are proportional to sums of Boltzmann factors rather than a single Boltzmann factor defined by the state energy. The relative magnitudes of populations for four-site systems at equilibrium and stationary state systems with single injection fluxes from each bath are compared in Figure 3. Although the Boltzmann distribution for these cations favors the lowest potential state, injection to the high-energy states increases these state populations to produce an “inverted” distribution with greater ion population in the high-energy states. Injection from the opposite bath adds population to the states that are highly populated at equilibrium to produce an enhanced difference in populations for the states (Figure 3c). The total populations for the single injection modes are lower than those for the corresponding double injection flux required for a Boltzmann equilibrium. With an injection flux from bath 1 alone, the Boltvnann factor x for different transmembrane potentials produces a range of distributions (Figure 4). For x < 1, the input flux and the large forward rate constant combine to give a larger net flux and a “flatter” distribution since the injection flux is populating state 1 while the forward transition probabilities populate the remaining states. Transitions to bath 2 lower the population of the final state. The net rate of transport through the channel acts to balance the state populations. For x > 1, the reverse transition probability is larger and both the injection flux from bath 1 and the intrachannel transitions enhance sites 1 and 2 at the expense of the remaining sites to give a Boltzmann-like distribution. The competition between injection flux and larger reverse transition probabilities acts to lower the total ion population in the channel. The same injection flux applied from bath 2 produces very different population distributions (Figure 5). For x < 1, the injection flux and the Boltzmann “tendency” from the applied potential act to increase the population of state 4. The negative applied potentials (x =- 1) by contrast act to move the ions from
J. Phys. Chem., Vol. 99, No. 12, 1995 4283
Properties of Ions in Multistate Channels
-
0.1 0
0.10
0.08
0.08 C
C
.g
2 CI
0.06
-3 E
-3
E
0.06
0
0
0.04
a
0.04
Q
0.02
0.02
0.00
‘
I
I
I
1
1
2
3
4
0.00
2
1
3
State
State Figure 3. Canonical populations of states for a four-state channel. (a) Boltzmann distribution when j / k = 0.1, x = 0.5. (b) Distribution when j / k = 0.1, j‘/k = 0, x = 0.5. (c) Distribution for j‘lk = 0.1, j / k = 0, x = 0.5. 0.10
t 0.06
0.04
n
0.02
0.00
4.0.
Stationary State Thermodynamic Properties
S eJY -=--=dt A d
-3 0
E
Figure 5. Stationary state canonical distributions for an injection flux julk = 0.1 from bath 2. (a) x = 0.25. (b) n = 0.5. (c) x = 2.0. (d) x =
The stationary state distribution for the channel sites is characterized by a flux of ions through the channels. An observer with probes in the baths records a net current density, eJ/A, for a given transmembrane potential across the membrane of thickness d, and the product of these two variables generates the power or dissipation per membrane volume, V,
0.08
w
4
1
5
2
4
State Figure 4. Stationary state canonical distributions for an injection flux jolk = 0.1 from bath 1, x = e-cy/(NLn. (a) x = 0.25. (b) x = 0.5. (c) x = 2.0. (d) x = 4.0.
bath 1 to all the states in the channel to produce a more even distribution of ion populations. At stationary state, the net flux through the channels is constant and can be determined in any region of the channel. The flux determined from the transition from state 4 to bath 2 for the four-state channel, is
J = kp,
- j’
e N V
This dissipation maintains the stationary state population distribution that is characterized by the ratio, j/k, with no time units. By analogy with equilibrium thermodynamics, entropy is used as a bridging parameter to quantitate the order in the stationary population distribution and to correlate this distribution with energies and free energies. The equilibrium entropy for a system characterized by state probabilities ei is
S
N &
kB
i=l
- -=
The state populations for a stationary state are not probabilities, and the equilibrium entropy expression cannot be applied directly. However, the fact that the equilibrium entropy defines the average value of ln(pi),
S
- - = (In@,)) kB
suggests an operational definition for stationary state entropy as
S
- -=
kB where Y is the full potential difference between baths. The difference term appears in both discrete and continuum formulations of ion diffusion and gives the electrochemical potential when the net flux drops. to zero at equilibrium.
N Cpi i
Le., a weighted sum of In@(). For the systems considered, each state population is proportional to the ratio, jlk. A stationary state “sum over states”, 0,
Starzak
4284 J. Phys. Chem., VoE. 99, No. 12, 1995 is generated by summing the vector components. The general population expression for either ensemble,
IP) = C W ) ) is converted into the product of a new constant and a probability vector le) by multiplying and dividing by the sum over state populations,
This rearrangement is the product of the total ion concentration in the channels, r, and a probability distribution for ions within the channel. The correlation between entropy and the thermodynamic energies for an equilibrium system,
is extended to the operational definition for stationary state entropy in an open system,
The expression separates to give
The second term in the state probabilities is the distribution entropy for one ion within the channel multiplied by the total ion population. This distribution entropy defines the intemal energy and free energy for that ion,
in analogy with equilibrium statistical mechanics. r ln(r) defines the chemical potential for the ions in the channels. n ions with chemical potential, p, have a free energy, n(u), so the chemical potential (u) for the population, r, of ions in the ensemble of channels is defined as
&) = k,T
ln(r)
r functions as a concentration. For dimensional consistency, this chemical potential is defined relative to a reference r. The stationary state channel system “stores” free energy in two ways: (1) single ion, nonequilibrium distributions within the channel that produce work during a change to a more random distribution and (2) as a free energy rp that does work by
releasing the driven ions from the channel when the net injection fluxes change.
Discussion A channel with a discrete number of ion-binding sites is ideal for a comparison of equilibrium and stationary state behavior. The nonconservative, open models include equilibria as special limiting cases. These equilibrium systems provide relationships between the system transition probabilities and injection fluxes, e.g., detailed balance at equilibrium, that are assumed valid for the stationary states as well. Since the transition probabilities are related to the local ion velocities that can be measured experimentally using techniques like laser Doppler velocimetry, these fundamental postulates can be tested experimentally. The first-order kinetics of the transition probability matrix are valid when the system is not driven too far from the equilibrium distribution. The states are defined so that there is never more than one ion per channel so that the total ion population for the ensemble, r, can never exceed the total channels in the ensemble. Beyond these limits, multi-ion states must be included and a master equation formulation may be inadequate. The focus on the stationary state population distributions, rather than on the net fluxes they generate, highlights an important difference between these systems and homogeneous, equilibrium systems. The driven systems can have sites with average ion populations above the average ion populations predicted for these sites at equilibrium. This average population is equivalent to a longer average residence time for the permeating ion. The increased residence times can induce reactions and/or conformational changes not normally observed in homogeneous systems. For example, an increased effective ion population might trigger a conformational change in the channel itself. The combination of channel conformation kinetics and ion redistribution kinetics can produce a very fundamental oscillatory system. The operational definition of entropy using the state populations produces a natural separation of the entropies expected for a stationary state. The distribution entropy is the standard thermodynamic entropy expected for a probability distribution of ions in the channel and gives the entropy of a single ion for all distributions, including the equilibrium distribution. This entropy also defines both intemal energy and free energy for the ion in the channel. For the stationary state, nonequilibrium systems, the “sum over states” is more complicated than that for equilibrium system because the state probabilities are defined by sums of Boltzmann factors. The population entropy is associated with the chemical potential for the increased total ion population in the ensemble of channels. This ion excess is stored in the channel system while the net flux through the membrane channels is maintained and might be tapped during a change in the ion population. The time independent ratio, jlk, is a central parameter for both conservative and nonconservative driven systems and serves as the stationary state analog to the law of WaageGuldberg, Ke = kflk,, in homogeneous kinetic systems. Both ratios relate dynamic variables to a “distribution” constant. The rate constants in Waage-Guldberg determine the ratio of products to reactants while jlk determines the stationary state population of ions that remain in the channel following injection, Le., with channel states viewed as “product”.
Acknowledgment. The work was supported by the Office of Naval Research. I thank Dr. E. Arndt for helpful comments.
J. Phys. Chem., Vol. 99,No. 12, 1995 4285
Properties of Ions in Multistate Channels
References and Notes (1) Parlin, R. B.; Eyring, H. In Ion Transport across Membranes; Clarke, H. T., Ed.; Academic Press: London, 1954. (2) Starzak, M. E. In Physical Chemistry of Membranes; Academic Press: Orlando, 1984. (3) Macias, F.; Starzak, M. Biochim. Biophys. Acta 1993, 1153, 331. (4) Macias, F.; Starzak, M. In Membrane Electrochemistry; Vodyanoy, I, Blank, M., Eds.; Advances in Chemistry 235; American Chemical Society: Washington, DC, 1994. (5) Morowitz, H. J. Energy Flow in Biology; Academic Press: New York. 1968.
(6) Starzak, M. E, In Mathematical Methods in Chemistry and Physics; Plenum: New York, 1989. (7) Starzak, M. E. J. Membr. Biol. 1973,13, 37. (8) Starzak, M. E. J. Biol. Phys. 1974,2 , 57. (9) Starzak, M. E. Math. Biosci. 1990,99, 47. (10) Waggoner, A.; Stryer, L. Proc. Nafl. Acad. Sci. U.S.A. 1970,67, 579. (11) Haigh, E. A,; Thulbom, K. R.; Sawyer, W. H. Biochemistry 1979, 18, 3525. (12) Moore, H.-P.; Rdery, M. Proc. Nafl.Acad. Sci. U.S.A. 1980,77, 4509. JP941833C
