Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models.

Abstract

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.

Full text

Page 1

arXiv:1107.4963v1 [cond-mat.stat-mech] 25 Jul 2011
Stochastic theory of large-scale enzyme-reaction networks: finite copy number
corrections to rate equation models∗
Philipp Thomas,1Arthur V. Straube,1and Ramon Grima2
1Department of Physics, Humboldt University of Berlin, Newtonstr. 15, D-12489 Berlin, Germany
2School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JR, United Kingdom
Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtolitres.
Physiological concentrations realized in such small volumes imply low copy numbers of interacting
molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate
equation models are based on the implicit assumption of infinitely large numbers of interacting
molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concen-
trations. In this article we compute the finite-volume corrections (or equivalently the finite copy
number corrections) to the solutions of the rate equations for chemical reaction networks composed
of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small sub-
cellular compartment. This is achieved by applying a mesoscopic version of the quasi-steady state
assumption to the exact Fokker-Planck equation associated with the Poisson Representation of the
chemical master equation. The procedure yields impressively simple and compact expressions for
the finite-volume corrections. We prove that the predictions of the rate equations will always under-
estimate the actual steady-state substrate concentrations for an enzyme-reaction network confined
in a small volume. In particular we show that the finite-volume corrections increase with decreasing
sub-cellular volume, decreasing Michaelis-Menten constants and increasing enzyme saturation. The
magnitude of the corrections depends sensitively on the topology of the network. The predictions
of the theory are shown to be in excellent agreement with stochastic simulations for two types of
networks typically associated with protein methylation and metabolism.
I. INTRODUCTION
Recent years have seen a distinctive surge in the formu-
lation and application of stochastic models of biochemical
reaction kinetics. This trend has resulted from a deeper,
ongoing appreciation of the conditions characteristic of
the intracellular environment1and of their dissimilarity
from in vitro conditions. Typical in vivo concentrations
are in the range of nanomolar to millimolar; such con-
centrations realized in a macroscopic volume imply very
large copy numbers of interacting molecules whereas the
same concentrations in the small volume of a cell fre-
quently implies copy numbers ranging from few tens to
at most few thousands (for a detailed experimental pro-
tein abundance study see, for example, Ref. 2). Reaction
kinetics is inherently a stochastic process;3this noisiness
is not apparent in macroscopic conditions due to an im-
plicit averaging over a very large number of molecules
but cannot be overlooked when we are studying the ki-
netics of a system in which the copy number of at least
one species is small. This is frequently the case of intra-
cellular kinetics.
The introduction of the stochastic simulation algo-
rithm by Gillespie4has popularized the numerical study
of stochastic reaction kinetics. However to-date the an-
alytical study of the properties of such systems has re-
ceived comparatively very little attention principally be-
∗Paper published in:
J. Chem. Phys. 133, 195101 (2010)
DOI: 10.1063/1.3505552
cause the mathematical formalism of stochastic kinetics
[i.e., chemical master equations, (CMEs)] is very differ-
ent than that of deterministic kinetics [i.e., rate equations
(REs) which are based on ordinary differential equations]
and is less amenable to analysis. This problem is aug-
mented by the fact that many biological networks of in-
terest are considerably large.
One of the main analytical methods for systematically
exploring the stochastic properties of these networks has
been the linear-noise approximation.5,6The advantage of
this method is the relative ease with which one can com-
pute the magnitude of intrinsic noise (i.e., coefficients of
variation and Fano factors). The major drawback is that
the linear-noise approximation gives only meaningful re-
sults provided the copy number of molecules is not small
or to be more precise it is correct in the limit of infinitely
large reaction volumes, i.e., the same limit in which the
REs are valid. Since intracellular reactions occur in the
opposite limit of small volumes, it is highly desirable to
calculate the finite-volume corrections to the concentra-
tions and moments of intrinsic noise.
Developing a theory of finite-volume corrections
presents a considerable analytical challenge. One way
to obtain the latter is via the system-size expansion of
the CME.5The linear-noise approximation comes about
by evaluating the first term (of order V0where V is the
reaction volume) in this expansion. The next term is pro-
portional to V−1/2and hence consideration of this term
will necessarily give a finite-volume correction. Grima7,8
calculated the first such corrections for the mean concen-
trations of species involved in single-substrate enzyme re-
actions and recently also for a general chemical reaction
network of arbitrary complexity.9The general result is

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that the kinetics of a reaction pathway confined in a vol-
ume V can be described by the usual REs plus new terms
which are proportional to V−1. These equations are re-
ferred to as effective mesoscopic rate equations (EMREs).
The differences between the solution of EMREs and the
corresponding REs for finite volumes stems from a cou-
pling between the mean concentrations and the fluctua-
tions about them. EMREs can be explicitly solved for
pathways characterized by a handful of chemical species,
but otherwise one has to resort to numerical solution. In
the latter cases, one obtains a solution at the expense of
losing the insight which typically comes from analytical
results.
In this article we develop an alternative, powerful
method of calculating finite-volume corrections to the
solutions of the REs. Note that in the context of this
article, finite-volume corrections exclusively refer to cor-
rections to the mean concentrations not to the moments
of intrinsic noise.The method is based on the Pois-
son representation of the CME rather than the system-
size expansion used in the derivation of EMREs. This
method unlike the system-size expansion has been ap-
plied to study systems of biochemical or biological rele-
vance in only a handful of cases (see, for example, Refs. 10
and 11) but as we shall show it is a tool with great
potential for this field. We focus on chemical reaction
networks which are composed of enzyme-catalyzed re-
actions, a commonly encountered case in intracellular
biochemistry.12We show that when the timescales of
complex and substrate fluctuations are well-separated,
it is possible to obtain explicit and impressively simple
equations for the finite-volume corrections. For simple
reactions these corrections are shown to be the same as
given by the EMRE. The distinct advantage of the new
method over the EMRE is that it provides analytically
simple results even for complex networks with hundreds
or thousands of species. This is inherently possible be-
cause of the large reduction in the effective dimensional-
ity of the CME when timescales are well separated.
The paper is organized as follows. In Section II, we
derive the Poisson Representation for a general enzyme-
reaction network and use the resulting Fokker-Planck
equation (FPE) to obtain an exact Liouville equation
encoding all information about deviations from the de-
terministic solution of the REs. In Section III we show
that in the limit of well-separated timescales of complex
and substrate species, the Liouville equation simplifies
to a compact approximate form. This is used in Section
IV to compute explicit expressions for the finite-volume
corrections of a general network. In the latter section we
show that the corrections for a simple Michaelis-Menten
type reaction agree with those previously derived using
the EMRE formalism. More importantly we apply the
theoretical results to two common types of large-scale
networks and confirm the predictions using simulations.
We finish by a discussion in section V.
II.THE POISSON REPRESENTATION FOR
THE ENZYME REACTION NETWORK
In this section we use the Poisson Representation to
derive a general FPE for an enzyme reaction network.
The latter while being exactly equivalent to the CME is
much more amenable to analysis and hence is a very con-
venient starting point for detailed calculation purposes.
We consider a generic type of enzyme network com-
posed of two major types of chemical processes: (i) the
input of a substrate species A0 into a subcellular com-
partment; (ii) the transformation of A0 into some final
product AN via N consecutive enzyme-catalyzed reac-
tions of the type
Ei+ Ai
k1
− − ⇀
↽ − −
ki
−1
Ci
ki
2
−→ Ei+ Ai+1, (1)
where Ai, Ci and Ei denote the ithsubstrate, complex
and enzyme species, respectively; the index i takes values
from 0 to N − 1 and the k′s denote the relevant macro-
scopic rate constants. Note that we have assumed here
that the bimolecular reaction rate, k1, is the same for all
substrates. Note also that in such types of networks there
are N distinct substrate species, an equal number of dis-
tinct complex species and a number of enzyme species
varying between 1 and N. The freedom in choosing the
number of enzyme species comes from the fact that an
enzyme can generally bind to more than one type of sub-
strate.
If we assume that we have well-mixed conditions inside
the compartment then the instantaneous description of
the state of a chemical system at time t is simply given
by the vector of the absolute number of molecules of each
species, n = ({nAi},{nEi},{nCi}). Since the mesoscopic
kinetics are stochastic, a full description of the system is
necessarily probabilistic and is achieved by defining the
probability density function, P = P(n,t) and its time-
evolution equation, which is commonly referred to as the
CME (Refs. 5 and 13)
∂tP =
N−1
?
i=0
ΠiP + ΞP, (2)
Πi=k1
V(ΘAiΘ−1
+ ki
+ ki
Ci− 1)nAinEi
−1(ΘCiΘ−1
2(ΘCiΘ−1
Ai− 1)nCi
Ai+1− 1)nCi, (3)
Ξ = kinV (Θ−1
A0− 1), (4)
where V is the compartment volume, Ξ is the contribu-
tion due to the input of substrate species A0 into the
system at a rate kinwhile Πidescribes the ithcatalytic
reaction step, Eq. (1), in the reaction network. The
CME is compactly expressed using van Kampen’s step
operators defined as Θ±1
Xig(nXi) = g(nXi± 1).
Substantially, the CME depends only on the set of vari-
ables {nAi} and {nCi} since an enzyme molecule can be

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either in the free state or in the complexed state and
hence the variables {nEi} are redundant. We can express
this conservation law by writing nEi= Ei
where the matrix Gij is defined by construction to be
T−?
jGijnCj
Gij= (ˆ ei)j=
?1, if enzyme i binds substrate j
otherwise.0,
(5)
The N-dimensional vector, ˆ ei, is associated with the en-
zyme binding substrate in the i-th catalytic reaction. Its
j-th entry is chosen to be equal to one if the enzyme can
form a complex with the j-th substrate and zero other-
wise. Hence the connectivity of the network is explicitly
encoded in the form of these vectors.
We define the moment generating function, param-
eterized by the vector of continuous variables, z =
({zAi},{zCi}), as
G(z) =
?
n
?
i
znAi
AiznCi
CiP(n). (6)
Multiplying the CME, Eq. (2), by?
ing the summation over all values of the variables n and
expressing the resulting equation in terms of the moment
generating function G(z), we obtain the moment gener-
ating function equation
iznAi
AiznCi
Ci, perform-
∂tG(z) =
?
k1
?
iRG
i+ SG?
G,(7)
SG= kinV (zA0− 1), (8)
RG
i= (zCi− zAi)
?ˆEi∂Ai
z
− Ki
M∂Ai
z
?
+ Ki
2(zAi+1− zAi)∂Ci
z, (9)
whereˆEi abbreviates [Ei
Ei
T/V is the total enzyme concentration associated with
the enzyme binding substrate in the ithcatalytic reaction
step. Furthermore we set Ki
Ki
2(commonly referred to as the Michaelis-
Menten constant) and ∂X
z= ∂/∂zX. Note that in Eq. (9)
all derivatives are to the right.
We proceed by making use of Gardiner’s Poisson Rep-
resentation. At the heart of this method is the assump-
tion that the probability density function P(n,t) can be
expanded as a superposition of multivariate uncorrelated
Poissons13,14
T] − V−1?zCi∂i
Cand [Ei
T] =
1= ki
−1/k1, Ki
2= ki
2/k1,
M= Ki
1+Ki
P(n,t) =
?
dα
?
i
×e−αCiV(αCiV )nCi
nCi!
e−αAiV(αAiV )nAi
nAi!
f(α,t), (10)
where the function f(α,t) is usually referred to as the
quasi-probability density function. The vector α is de-
fined to be ({αAi},{αCi}). It has been shown that this
superposition always exists if the range of α is extended
to the complex plane by analytic continuation of the Pois-
son kernel. We have explicitly introduced V in our su-
perposition definition, which is not customarily done in
the Poisson Representation. In doing so, we obtain the
representation in intensive variables, i.e. in units of con-
centrations, as those encountered in the theory of rate
equations. The above expansion is equivalent to writing
the moment generating function as
G(z,t) =
?
dαf(α,t)e
?(zAi−1)αAiVe
?(zCi−1)αCiV
(11)
It follows from Eqs. (7) and (11) (see Appendix A for
details)
−∂tf = (R + S)f,
S = kin∂0
(12)
A, (13)
R =k1
N−1
?
i=0
C− ∂i
+ Ki
Ri,
Ri= (∂i
A)?αAiEi− Ki
2(∂i+1
A
− ∂i
MαCi
?
A)αCi,(14)
where we have utilized the notation ν = V−1/2, ∂i
∂/∂αXi. Note that αAN= αP is the variable for the
product formed after N catalytic steps. Note also that
Eiis not a constant, but is given by the operator
X=
Ei= [Ei
T] −
?
jGij(1 − ν2∂Cj)αCj,(15)
which is essentially the Poisson representation of the con-
servation of total enzyme molecules. The above equa-
tion generally differs from the corresponding determinis-
tic conservation law in terms of concentrations; the latter
is described only by its average, while the former exhibits
finite volume corrections due to the finite copy number
of enzyme molecules. Given Eq. (15), we see that the
Poisson representation, Eq. (12), yields a Fokker-Planck
equation in terms of substrate and complex variables.
Note also that for the case of N = 1 we obtain the repre-
sentation for the single-substrate single-enzyme reaction,
which is usually referred to as the Michaelis-Menten re-
action.
This completes the derivation of the Poisson represen-
tation of our general enzyme-reaction network. Note that
this is not the same FPE as that which arises from the
system-size expansion method of van Kampen.5In the
latter case, the FPE is an approximation to the CME
in the limit of large volumes whereas the FPE obtained
from the Poisson Representation is exactly equivalent to
the CME.
A further boon of the Poisson representation is that
once we have calculated the moments of the continuous
variables, αXi, using the FPE, we can very easily find the
corresponding moments of the copy number of molecules,
nXi, using the following simple relationships:14
?nXi?
V
= ?αXi?, (16)
?nXinXj?
V2
= ?αXiαXj? +1
Vδi,j?αXi?,(17)

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4
where the angled brackets imply the statistical average:
on the left hand side of the above equations these are
given by ?..? =?dn .. P(n,t) while those on the right
hand side imply ?..? =?dα .. f(α,t).
A. The mesoscopic equation
The kinetics becomes deterministic in the macroscopic
limit of infinitely large volumes. This can be easily ver-
ified by noting that all second-order derivatives in the
FPE are multiplied by a factor proportional to the inverse
of the volume. To compute the finite-volume corrections
we will need to separate the mesoscopic and macroscopic
evolution equations. We now show that this can be done
by applying a suitable change of variables to the Pois-
son Representation. The macroscopic corresponds to a
shot noise contribution, which agrees on average with the
mean-field result i.e. with the solution of the REs for the
substrate-enzyme network. The mesoscopic contribution
reflects the non-equilibrium properties of the network due
to the bimolecular character of the substrate-enzyme in-
teraction, and depends parametrically on the mean-field
expectation.
We express the deviation from the deterministic path
by the following change of variables:
(αAi,αCi) → ([Ai](t) + νǫAi,[Ci](t) + νǫCi) (18)
We shall refer to ǫXias the mesoscopic correction to
the deterministic, macroscopic concentration [Xi](t) of
species Xi. Note that the above equation has the same
apparent form as the van Kampen (VK) ansatz,5,6at
the heart of the system-size expansion, but the context
of the application is completely different. The VK ansatz
is applied to the integer number of particles in the CME
leading to an infinite series in powers of the inverse square
root of the volume, which has to be truncated; the first
term of this expansion (the one proportional to V0) gives
a linear FPE which is an approximation to the CME. In
our case the change of variables, Eq. (18), is applied on
the FPE arising from the Poisson representation which
leads to a finite series and allows for exact analytical
treatment; as we shall show now, this divides the exact
FPE into a macroscopic term and a term which captures
all deviations from the macroscopic.
The transformation applied to the FPE transforms the
time derivative into
∂
∂t
????
α
f(α,t) =
∂
∂t
+dǫ
????
dt
ǫ
f([X] + νǫ,t)
????
α
· ∇ǫf([X] + νǫ,t),(19)
where ∂/∂t|xdenotes taking the derivative with x held
constant. It follows from Eq. (18) that dǫ/dt|α =
−ν−1d[X]/dt. Finally by expressing the right hand side
of Eq. (12) in terms of the new variables ǫ and equating
the result to Eq. (19), we find that the FPE takes the
form
−∂
∂tg(ǫ) =k1Lg(ǫ) + ν−1
−∂[A]
?
k1Rmacro+ Smacro
∂t
· ∇ǫA−∂[C]
∂t
· ∇ǫC
?
g(ǫ),(20)
where the relevant operators are
Smacro= kin
∂
∂ǫA0
,(21)
Rmacro=
?
i
([Ei][Ai] − Ki
M[Ci])
∂
∂ǫCi
+
?
i
(Ki
1[Ci] − [Ei][Ai])
∂
∂ǫAi
,(22)
L =
?
i
?(∂i
− Ki
C− ∂i
A)?[Ei]ǫAi+ [Ai]δEi+ νǫAiδEi
MǫCi
?+ Ki
2(∂i+1
A
− ∂i
A)ǫCi
?.(23)
Note that the new probability density function necessar-
ily satisfies g(ǫ)dǫ = f(α)dα. Note also that the nota-
tion ∂i
Xnow denotes the derivative ∂/∂ǫXi. The quantity
[Ei] = [Ei
tion of the free enzyme species associated with binding
substrate in the ithcatalytic reaction step. The operator
δEi= −([Ei]−Ei)/ν = −?
is the contribution of the enzyme-operator Eq. (15).
Note that the resulting form of the FPE is clearly di-
vided into two parts. In the macroscopic limit the terms
proportional to ν−1dominate and their sum must equate
to zero - this leads to the macroscopic equations. It also
then follows that the mesoscopic equation is simply given
by
T] −?
jGij[Cj] is the macroscopic concentra-
jGij(ǫCj−∂j
C[Cj]−ν∂j
CǫCj)
∂τg(ǫ,t) = −Lg(ǫ,t), (24)
where we have renormalized time to τ = k1t. Note also
that due to the Poissonian nature of the substrate in-
put process, it only contributes to the macroscopic part,
a feature which is unique to the Poisson representation.
We emphasize that up till this point, we have made no
approximations and hence the resulting mesoscopic equa-
tion is exact.
III.ADIABATIC ELIMINATION OF THE
COMPLEX SPECIES VARIABLES
In this section we show how to rigorously eliminate the
fast variables from our description. This will be done in
two steps: on the macroscopic contribution of the FPE
and on the mesoscopic contribution including terms up
to order ν (i.e., finite volume corrections). As we shall see
later on, the reduced mesoscopic description does depend
on the reduced macroscopic description and hence the
need to treat the latter first.

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5
A.The reduced macroscopic equations
The macroscopic equations are obtained from Eq. (20)
by taking the limit ν → 0, leading to
d[Ai]
dt
d[Ci]
dt
d[AN]
dt
[Ei] = [Ei
= ki
−1[Ci] − k1[Ei][Ai] + ki−1
2
[Ci−1] + δi,0kin,
= k1[Ei][Ai] − (ki
2+ ki
−1)[Ci],
= kN−1
2
[CN−1],
T] −
?
j
Gij[Cj].(25)
These agree exactly with those that can be obtained from
the RE approach. The elimination of the complex species
from the macroscopic equations is a well-known proce-
dure commonly referred to as the quasi-steady state ap-
proximation (QSSA).15,16Briefly speaking the approx-
imation is tantamount to assuming that the complex
equilibrates on a much shorter timescale than the sub-
strate. This is implemented by imposing the approxima-
tion d[Ci]/dt = 0 on the macroscopic equations. The re-
sulting reduced equations (commonly referred to as the
Briggs-Haldane equations) are then given by replacing
[Ci] in the full time-evolution equations for the substrate
concentrations by
[Ci] =[Ei][Ai]
Ki
M
. (26)
B.The reduced mesoscopic equation
Now we are interested in deriving the reduced meso-
scopic equation corresponding to the reduced macro-
scopic equations that we just considered. Time-scale sep-
aration on the mesoscopic scale is non-trivial because of
the inherent correlations between the mesoscopic fluctu-
ations of the various species. Our presentation shall be
as follows. First we shall show that the mesoscopic Liou-
villian, Eq. (23), can be generally cast into an asymptotic
form of the interaction representation which is typically
encountered in the theory of adiabatic elimination of fast
fluctuating variables.17,18The latter yields a particularly
simple result for the reduced mesoscopic equation.
We will now show that the Liouvillian can be rewritten
in the general form
L(γ) = γL1+ γ1/2L2+ L3,(27)
where γ−1is the characteristic fast timescale of the com-
plex fluctuations which will be specified later on. We
now proceed to derive the operators L1, L2, L3 for the
general enzyme-reaction network under study. We start
by grouping all terms containing only the pair of complex
variables ({ǫCi},{∂i
C}) into L1, terms concerning solely
the substrate ones ({ǫAi},{∂i
the remaining terms as interaction L2. Thus we have
A}) into L3, while treating
L1=
?
i
L(i)
1, L2=
?
i
L(i)
2, L3=
?
i
L(i)
3,(28)
L(i)
1 = ∂i
L(i)
C([Ai]δEi− Ki
MǫCi),(29)
2 = − ∂i
A([Ai]δEi− Ki
+ ν(∂i
MǫCi) + Ki
2(∂i+1
A
− ∂i
A)ǫCi
C− ∂i
A)ǫAiδEi,(30)
L(i)
3 = − ∂i
A[Ei]ǫAi.(31)
It is instructive to rescale all complex variables by their
characteristic timescale γ
zi= γ1/2ǫCi, xi= ǫAi. (32)
The operator L3is trivially obtained
L(i)
3 = − ∂i
x[Ei]xi. (33)
Note that ∂i
stands for the derivative ∂/∂ǫXi. Next we observe that
the enzyme operator transforms as
xdenotes the derivative ∂/∂xi whereas ∂i
X
γ1/2δEi= −
?
j
Gij(zj− γ∂j
z[Cj] − νγ1/2∂j
zzj). (34)
Plugging this into Eq. (29) and putting L(i)
we find that the last term in (34) can be asymptotically
neglected. Hence the dominant contribution in the limit
of large γ is given by
1
→ γL(i)
1
L(i)
1 = − ∂i
z
?
jMijzj+ ∂i
z
?
jDij∂j
z, (35)
where we have utilized the abbreviations
Jij= ([Ai]Gij+ Ki
Mδij), Mij= γ−1Jij,
Dij= Gij[Ai][Cj].
(36)
Thus the asymptotic form of the complex fluctuations
is described by an Ornstein-Uhlenbeck process, centered
on the deterministic expectation Eq. (26). It is clear,
by virtue of the fluctuation-dissipation theorem, that the
matrix J (constant matrices, i.e., those independent of
the ǫAiand ǫCivariables and of the associated partial
derivatives, are underlined throughout the rest of the ar-
ticle) must correspond to the Jacobian of the complex
species while the matrix D determines the strength of
the fluctuations in the Poisson representation. Note that
throughout the article by Jacobian of a species we mean
the negative of the Jacobian matrix as obtained from the
macroscopic rate equations of that species with the time
scaling τ = k1t.
The characteristic timescale γ−1can be inferred from
the explicit form of the Jacobian. We choose γ ≡ tr(J)
giving the relaxation rate of the complex vector z. Con-
sequently M is finite, as required by the stability of L1,
and is given by the Jacobian of unit trace. Equally we