.ig
:!cat %|tbl|pic|eqn|troff |dpost >%.ps;ghostview %.ps&
:!cat %|tbl|pic|eqn|troff |dpost >%.ps
:g/omega hat sub size -2 alpha/s//Wa/g
:g/w1/s//w1/g
..
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.EQ
delim $$
define Angstrom '~ up 17 "\(oa" back 60 A'
.EN
.EQ
define SUBqq ' sub {q sub size -4 beta q sub size -4 alpha} '
define SUBvq ' sub {v sub size -4 beta q sub size -4 alpha} '
define SUBqv ' sub {q sub size -4 beta v sub size -4 alpha} '
define SUBvv ' sub {v sub size -4 beta v sub size -4 alpha} '
define SUPN  ' sup down 80 fwd 20 size -4 N '
define Wa    ' omega hat sub size -2 alpha '
define Wb    ' omega hat sub size -2 beta '
define wt    ' omega tilde '
define Gt    ' GAMMA tilde '
define tt    ` t up 55 back 45 "\N'126'" `
define w1    ' omega sub size -2 1 '
define Gp    ` GAMMA ' sub size -2 alpha `
define dag   ' \(dg '
define ddag  ' \(dd '
define ws1   ' {omega sub 1} '
define ws2   ' {omega sub 2} '
define wsx   ' {omega sub x} '
define wsy   ' {omega sub y} '
define ws+   ' {omega sub +} '
define ws-   ' {omega sub -} '
define ws+-   ' {omega sub +-} '
define wxsq  ' {omega sub x} sup 2 '
define wysq  ' {omega sub y} sup 2 '
define w1sq  ' {omega sub 1} sup 2 '
define w2sq  ' {omega sub 2} sup 2 '
define wsq   ' {omega sup 2} '
define wx*   ' {omega sub x back 50 * } '
define w0i   ' { omega sub {i} } '
define W1    ' {- omega sup 2~+~{omega sub 1} sup 2~-~i omega GAMMA sub 11} '
define W2    ' {- omega sup 2~+~{omega sub 2} sup 2~-~i omega GAMMA sub 22} '
define gc12   ' {(- i omega GAMMA sub 12 ~-~ C)} '
define gc21   ' {(- i omega GAMMA sub 21 ~-~ C)} '
define Ga    ' {GAMMA sub a}  '
define Gb    ' {GAMMA sub b}  '
define G11   ' {GAMMA sub 11}  '
define G22   ' {GAMMA sub 22}  '
define G21   ' {GAMMA sub 21}  '
define G12   ' {GAMMA sub 12}  '
define Gxx   ' {GAMMA sub xx}  '
define Gyy   ' {GAMMA sub yy}  '
define Gxy   ' {GAMMA sub xy}  '
define Gyx   ' {GAMMA sub yx}  '
define v2v1  ' sub {v sub 2 v sub 1} '
define v1v1  ' sub {v sub 1 v sub 1} '
define q1v1  ' sub {q sub 1 v sub 1} '
define q2v1  ' sub {q sub 2 v sub 1} '
define v1q1  ' sub {v sub 1 q sub 1} '
define v2q1  ' sub {v sub 2 q sub 1} '
define q2q1  ' sub {q sub 2 q sub 1} '
define q2q2  ' sub {q sub 2 q sub 2} '
define q1q1  ' sub {q sub 1 q sub 1} '
define Wab   ' {- {omega sup 2} ~+~ GAMMA sub a GAMMA sub b}  '
define Dab   ' {- wsq ~+~ Ga Gb ~-~ i omega Ga}  '
define Dabsq ' {(- wsq ~+~ Ga Gb ) sup 2 ~+~ wsq Ga sup 2} '
define Ow2   ' {O ( omega sup 2 )} '
.EN
TIME CORRELATION FUNCTIONS.
.SP 3
We think of variables as correlated if when one variable has a large value,
the other also has a large value,
and if this parallel behavior persists for some period of time.
Intuitively, we understand that 
after some longer period of time the correlation will vanish.
A formal expression of the correlation between two variables
for a system at equilibrium is the time correlation function,
.DS I
.EQ (A.1)
C sub {beta alpha} (t) ~=~ {< beta (t) alpha (0) >} 
.EN
.DE
where $alpha (t) ~and~ beta (t)$ are time-dependent values for the 
two variables given as deviations from their means,
and the angle brackets indicate ensemble average.
The extremum of $C sub {beta alpha} (t)$ is the limit value as $t -> 0$, 
$C sub {beta alpha} (0)$.
Over a sufficiently long time $C sub {beta alpha} (t)$ decays to zero.
.SP
Correlation functions are sometimes normalized as
.DS I
.EQ (A.2)
C sub {beta alpha} SUPN (t) ~=~ {< beta (t) alpha (0) >}  over
{< alpha (0) alpha (0) > sup 1/2 < beta (0) beta (0) > sup 1/2}
.EN
.DE
.SP 2
Correlations within  sets of variables are described by a
matrix of correlation functions, 
$bold C (t) ~=~ < bold alpha (t) bold alpha sup T (0) >$, 
where $bold alpha$ is a column vector of variables
and $bold alpha sup T $ its transpose.
.SP 
The time correlation function describes averaged properties of
a system at equilibrium.
.in +1
.SP
(a) \fI$C (0)$ describes fluctuations at equilibrium.\fP
For a single variable,
the zero-time autocorrelation function $C sub {alpha alpha} (0)$ 
is equal to the mean squared deviation of the
variable from its average value, equal to the variance for a
Gaussian distribution of values of the variable.
For a set of two variables, 
the off-diagonal elements of the zero-time correlation matrix, 
$C sub {beta alpha} (0)$, are equal to
the covariance for the variable pair. 
If $C sub {beta alpha} (0)$ = 0, there is no
correlation between the variables
(they are statistically independent).
The joint probabilities for values of correlated variables at equilibrium
can be calculated from the zero-time correlation matrix.
.SP
(b) \fI$C (t)$ describes relaxation of the fluctuations.\fP
For a single variable, $C sub {alpha alpha} (t)$ 
defines a characteristic time ($tau sub alpha$), 
under the often good assumption 
that relaxation (decay) of a fluctuation back to the equilibrium (time-average)
value is a first order process.
For two variables, $C sub {beta alpha} (t)$ defines
a characteristic time, $tau sub {beta alpha}$, for the loss of correlation.
.in
.SP
The time correlation function $C (t)$,
although it is determined by the behavior of a system at equilibrium,
contains information on the dynamics 
and properties of similarly-constituted nonequilibrium systems.
The connection between equilibrium and nonequilibrium systems
is given by the fluctuation-dissipation theorem:
variables of a nonequilibrium system not far from equilibrium
relax and respond on average
in the same way as fluctuations in the same variables
relax on average in the equilibrium system.
.in +1
.SP
(a) \fI$C (0)$ describes static response:\fP
the response of one variable after long time to a constant perturbation
of the same or another variable;
the response observed
after the system perturbed from its original equilibrium 
has reached its new equilibrium.
For one variable, $C sub {alpha alpha} (0) ~=~ k sub B T / K$,
where $K$ is the force constant describing a region of the potential surface, 
generally a well bottom, 
within which the correlation function was measured.
Thus $C (0)$ relates the response of a variable to a static external force
that was applied to the same or another variable 
and that perturbed the system from equilibrium.
.SP
(b) \fI$C (t)$ describes dynamical properties of the nonequilibrium system:\fP
the relaxation of a nonequilibrium system to equilibrium.
$C sub {alpha alpha} (t)$ defines $tau sub alpha$ 
for relaxation of a nonequilibrium state 
just as it does for relaxation of equilibrium fluctuations.
$C sub {alpha alpha} (t)$ also describes the opposite of relaxation
to equilibrium, the time response to an external force that displaces
the variable $alpha$ and drives the system away from equilibrium.
Interpretation of $C sub {beta alpha} (t)$ is similar: it
describes the time response of the variable $beta$ with
external force applied to displace the variable $alpha$.
.SP
(c) \fIThe response may lag the force.\fP
The lag, due to friction and inertial effects,
is greater for high rate of change of a variable and
is absent for the zero-frequency response, $C (0)$, 
which reflects only the potential function.
Energy is dissipated when work is done by an external force.
Dissipation is related to the lag in response 
and to the contributions of friction and inertial effects.
.in
.SP
As an example of the correspondence between equilibrium and nonequilibrium
systems, consider a chemical reaction.
Let $alpha (t)$ be the concentration of one of the two species in 
an isomerization equilibrium.
Then $C sub {alpha alpha} (t)$ is the time correlation function for
concentration fluctuations at equilibrium.
The characteristic time for the relaxation of fluctuations described by
$C sub {alpha alpha} (t)$ in the equilibrium system 
is equal, by the fluctuation-dissipation theorem,
to the relaxation time for a system
prepared to have a nonequilibrium concentration of the $alpha$ species.
This relaxation time for the nonequilibrium system
is a simple function of the forward and reverse
rate constants for the isomerization reaction.
Thus the rate constant for a nonequilibrium process is directly related to
the concentration fluctuations at equilibrium and to $C (t)$.
Why is this important?
The central point is the correspondence between reaction rate and
$C (t)$.  
This is easy to make plausible as done above.
It can also be shown in other ways, such as by use of
the Mori-Zwanzig formalism.
Once one knows that $C (t)$ defines the rate constant,
effective molecular dynamics algorithms for equilibrium systems
can be used to compute $C (t)$ and thus the reaction rate. 
The modeling of the molecular dynamics gives a deep
understanding of the microscopic
events that determine the rate constant and underlay the macroscopic process
described by the rate constant: 
friction;
barrier and well shapes;
collective motions, especially of the environment
(protein and solvent for an enzyme active site); 
time scales and coupling to the environment; 
cross correlations and dependence of the reaction rate on other variables; 
etc.
.SK
CORRELATION -- TIME SCALE SEPARATION, RELAXATION, CROSS CORRELATION, DYNAMIC AND STATIC RESPONSE.
.SP 3
We want to make general statements about the time dependence of $C(t)$,
the mathematical form of the cross-correlation $C sub {beta alpha} (t)$,
and the relation of the cross-correlation characteristic time 
$tau sub {beta alpha}$
to the characteristic times of the separate motions of the
$alpha$ and $beta$ variables.
The intent is to understand coupling between protein motions
of the same or different time scale as it applies to 
protein processes, such as enzyme rate processes.
.SP 2
A system at equilibrium but transiently in a fluctuant state
relatively far from equilibrium
will evolve by relaxing back toward equilibrium (decay of the fluctuation).
For example, 
a fluctuant value for a coordinate of a system in a potential well
will evolve toward the value at the well bottom.
Considered within the framework of the Generalized Langevin equation,
the characteristic time for relaxation of some particular system variable 
is determined by interactions (i) with other variables of the same set $bold A$
and (ii) with variables that are sensed only as part of the environment;
the effects of these interactions are described, 
respectively, by the factors $i omega hat$ and $K (t)$ of
the Generalized Langevin equation.
.SP
An external force, acting upon some variable to displace it from
its equilibrium value, elicits a response in the variable displaced and
also in the other variables of the set $bold A$.
By the fluctuation-dissipation theorem,
the time course of this response is on average exactly that for
relaxation of the corresponding fluctuant quantity of the equilibrium
system.
.SP
For some dynamical variable 
that relaxes and responds on a particular time scale, 
we may expect there to be certain other variables that relax and respond on 
approximately the same time scale,
and there to be yet others with dynamics on much faster 
or slower time scales.
Considering, for example,s
a particular torsional libration of a protein 
that relaxes on the 0.1-1 ps time scale,
we would expect there to be other torsions relaxing and responding
on the same time scale,
with the hard degrees of freedom of bond vibration, bond bending, etc.,
being on faster time scales,
and on slower time scales, 
flips between torsional minima, charge redistribution, 
cooperative structural events, etc.
Some of the motions, such as torsional libration and bond vibration,
are motions within wells of a high-dimensional potential-energy 
surface;
others, like flips between torsional minima, involve crossing
a barrier between potential-energy wells.
The event of barrier crossing is typically on the 0.1-1 ps time scale, with
the slowness of a process such as torsional flip, charge movement, etc.,
being due to the low probability of a crossing event (a high barrier).
.SP
We can group dynamical variables as fast or slow
according to time scale of relaxation and response.
If we pick a particular time-scale group, 
such as torsional librations for a protein,
and we observe the system on the time scale of this group (0.1-1 ps),
then we can call "fast" 
any variables relaxing on a much shorter time scale (<0.1 ps),
and "slow" those variables relaxing on the observation time scale.
It is proper as well as convenient to
ignore, for the present, variables with much longer characteristic times, 
because these are invariant on the observation time scale.
The fast variables, insofar as they influence the slow, comprise the random
force driving fluctuations of slow variables about their average trajectories, 
the term $f(t)$ of the Generalized Langevin equation, 
and cause damping of the motion, 
the factor $K(t)$ or $GAMMA (0)$ of the Generalized Langevin equation.
.SP
For a particular process, such as catalysis of a reaction by some enzyme,
one can identify,
at least in principle,
a group of slow variables, the set $bold A(t)$,
that fully describe the process.
Different processes are described, of course,
by different sets $bold A$ of slow variables.
.SP
If we specify some one slow dynamical variable $beta$ of the set $bold A$, 
say a particular torsional libration of a protein,
a time correlation function $C (t)$ describes its relaxation.
The mathematical form of $C (t)$ is a sum of exponentials in the time:
.DS I
.EQ (D.1)
C(t) ~=~ size +6 sum from i=1 to n ~ R sub i ~ e sup {gamma sub i t}
.EN
.DE
.SP
where the $gamma sub i$ are the poles (singularities) of the Laplace
transform of $C (t)$ and the $R sub i$ are the residues at the $n$ poles.
The $gamma sub i$ can be complex valued.  The real part of 
$gamma sub i$ if negative defines a relaxation on the time scale
$tau sub i = 1/Re( gamma sub i )$.
There must be least one pole, corresponding to the slow variable $beta$,
for which $tau  >> tau sub c$, the correlation time of
the random force (the fast variables).
If there are other poles corresponding to a value of $tau >> tau sub c$,
these represent other slow variables potentially 
coupled with the slow variable $beta$.
Cross correlations $C sub {beta alpha} (t)$
as well as single-variable autocorrelatons $C sub {alpha alpha} (t)$
have the mathematical form of Eq. (D.1).
For a particular set of slow variables $bold A$, 
every correlation function is described by the same set of exponentials in
the time, one exponential term for each variable,
with the summations differing in the coefficients $R sub i$ of Eq. (D.1).
We want to understand the rules for selecting which exponential term
dominates for a particular correlation function.
The following section is a summary of some relevant points.
.SK
COMMENTS -- TIME SCALE SEPARATION, RELAXATION, CROSS CORRELATION, DYNAMIC AND STATIC RESPONSE.
.SP 3
(1) An analysis based on Eq. (D.1) can be used
as a guide to understanding the structure of time correlation functions,
particularly two-variable cross-correlation functions, 
$C sub {beta alpha} (t)$.
.SP
.in +1
(a) The correlation function is a sum of exponentials in the time.
.SP
(b) Each exponential term contributes to the time dependence of the 
correlation either as a relaxation (a decay toward zero), 
or as a divergence if the exponent is positive.
.SP
(c) If the exponent has an imaginary part,
the term has a periodic factor 
(an oscillation superimposed on the relaxation).
.SP
(d) The number of terms is less than or equal to the number of slow
variables that are coupled.
Two slow variables,
such as a coordinate and a velocity,
may be associated with even an apparently simple, single process.
.SP
(e) The relative weight of each term and its relaxation time $tau$ 
are functions of parameters of all the coupled slow variables.
However, one variable
or a small number of variables can dominate,
and the sum of exponentials can reduce to one or two terms,
or equivalently,
the relaxation can be determined by one 
or a few variables of similar $tau$.
.in
.SP
(2) Protein motions cover a range of time scales.
It is important to consider coupling
between processes of widely different $tau$.
For the correlation function $C sub {beta alpha} (t)$,
let $beta$ be the slower process and $alpha$ the faster.
.SP
.in
(a) For wide time-scale separation,
the relaxation of $C sub {beta alpha}$ will be controlled by
$tau sub beta$, the slower characteristic time.
If observation is on the time scale of the faster $alpha$ process,
then no relaxation of the $beta$ variable or of $C sub {beta alpha}$
will be seen.
.SP
(b) This can be stated also 
in terms of response of the $beta$ variable to
displacement of the $alpha$ variable by an external force.
If displacement is carried out on the $alpha$ process time scale, 
there can be no response of the $beta$ variable.
If displacement of the $alpha$ variable is carried out
slowly and on the $beta$ process time scale, 
the $beta$ variable will respond.
.SP
(c) If the external force is applied over a very long time, 
the $beta$ variable will respond fully.
This long-time response is the static, or equilibrium, response.
The size of the static response is proportional to $C sub {beta alpha} (0)$,
the covariance of the equilibrium fluctuations.
.SP
(d) The symmetries of the generalized Langevin equation require that
$C sub {beta alpha} (t) = +- C sub {alpha beta} (t)$,
with the sign negative if the $alpha$ and $beta$ processes differ in
time reversal symmetry.
.SP
(e) Consequently, $tau sub {beta alpha} = tau sub {alpha beta}$.
The $alpha$ variable when driven by displacement of the $beta$ variable
responds on the time scale of the $beta$ variable.
This can come as no surprise, since the $beta$ variable cannot be displaced
in times shorter than $tau sub beta$.
Because $tau sub alpha << tau sub beta$, the faster $alpha$ variable shows
instantaneous response (static or equilibrium response) to change in the
$beta$ variable.
.in
.SP
(3) The statements of paragraph (2) have several consequences 
with regard to analysis of an enzyme rate process.
.SP
.in +1
(a) A slow enzyme process cannot be coupled to a fast process
on the time scale of the fast process.
For example, a pK shift cannot be coupled to barrier crossing,
the time scale of the latter being 0.1-1 ps and of the former, 
$\(*m back 10 s$ or longer.
.SP
(b) For a long-lifetime ES complex, 
there is the possibility for static coupling (correlation) 
among all processes with characteristic
times shorter than the lifetime of the ES complex.
.SP
(c) During barrier crossing, those variables with characteristic
times greater than that of barrier crossing maintain the 
quasi-equilibrium values of the ES complex
from which barrier crossing was initiated.
The distribution of these values is described by
the static correlation function $C sub {beta alpha} (0)$ for those variables
with characteristic time less than the lifetime of the ES complex.
.SP
(d) The nature of the 
distribution of quasi-equilibrium correlations carried into
the transition state is a factor in determining height of the barrier.
This contribution should be viewed as an equilibrium solvent effect.
Such correlations are almost certainly important for enzyme catalysis:
consider the distribution of protons among charged groups in the
transition state for an enzyme RDS to which acid-base catalysis contributes.
.in
.SP
