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Chapter 3: Thermodynamics of Biochemical Reactions
3.1 Introduction
3.2 Representation of a biochemical reaction
3.3 Maximal available work that can be obtained from a biochemical reaction
3.4 The spontaneous direction of the reaction and the equilibrium constant
3.5 The standard state activities (concentrations) used for biochemical reactions
3.5.1 Standard states of reaction components
3.5.2 The activity of the solvent, water, is 1 and remains unchanged for
dilute solutions.
3.6 Biochemical reactions in buffered solution at constant pH: the transformed
Gibbs free energy of reaction.
3.6.1 A simple approach to dealing with thermodynamics at constant pH.
3.6.2 The pH-dependence of the transformed Gibbs reaction free energy.
3.6.3 One can also modify the expressions for the Gibbs free energy of reaction
under conditions of constant Mg2+ or any other chemical species.
Box 3.1: The importance of transformed Gibbs free energy in biochemistry
3.7 Transformed enthalpy of reaction
3.7.1 Determining 'or H by calorimetry.
3.7.2: Example: Enthalpy of reaction catalyzed by tryptophan synthase
3.7.3 Determining 'or H by the temperature dependence of 'o
rG
3.8 Transformed entropy of reaction
3.9 Example: Temperature-dependence of the reaction catalyzed by lysine 2,3
aminomutase.
1
3.10 Transformed Gibbs free energy, enthalpy and entropy of formation
3.11 Reactions coupled by enzyme mechanism: example: Acetate-Co ligase
3.12 Diagrammatic representations of free energy relationships of coupled reactions 3.13 Metabolic networks 3.13.1 Examples from hepatocyte metabolism
3.14 Summary
2
Chapter 3: Thermodynamics of Biochemical Reactions
3.1 Introduction
Having established the meaning of the chemical potential in the formal context of
thermodynamics, we will now see how to apply thermodynamics to biochemical
reactions. In this Chapter we will consider biochemical reactions taking place in a single,
homogeneous phase, such as in a test tube, or the cytoplasm of the cell or within the
mitochondrial matrix. In Chapter 4 we will extend the same concepts to include redox
reactions, and in Chapter 5, we will include biochemical reactions that involve the
transport of components between separate compartments, i.e., transport across a
membrane. In any metabolic system, many of these reactions, which can be examined as
separate processes in vitro, are coupled to other processes within the cell. Several
enzymes might utilize the same substrate, or result in the formation of common products
(Figure 3.1).
Figure 3.1: An isolated biochemical reaction carried out in vitro will reach an
3
equilibrium point which can be defined by thermodynamic parameters. In the
context of a metabolic system in a cell in the steady state, the concentrations of the
reactants and products are determined by kinetics of formation and breakdown of
each biochemical component. Thermodynamics can define how far away these
steady state concentrations are from equilibrium.
The product of one reaction is likely to be the substrate for other reactions within the cell.
Substrates may need to be imported into and products exported out of a biological
compartment. Studying the global metabolism of an organism involves recognizing these
interconnected processes and modeling both the thermodynamics as well as the kinetics
of the system under defined circumstances. Examining the entire set of coupled processes
constitutes systems biology. The ground rules for treating simple systems will be our
focus, but the extension to more complex systems is a growing endeavor of biologists.
Thermodynamics provides essential information about the conditions that constitute
equilibrium, given the composition and knowing which enzyme-catalyzed reactions are
occurring. Biological systems, however, are always out of equilibrium and are better
considered to be in a "steady state". In the steady state condition, substrates enter the cell
at a defined rate and products are produced. These products would include biomass (new
proteins, for example) as well as excreted materials such as water and CO2. If mass
balance is maintained, i.e., the amount of substrate being consumed and the amount of
product being produced are equal, then the concentration of each of the biochemical
intermediates inside the cell attains a constant value as the system is running. The steady
state concentrations of the cellular components, i.e., the substrates and products of each
individual biochemical reaction, are determined by the rates of all of the reactions. In the
4
context of a metabolic network, thermodynamics provides the information for each
enzyme-catalyzed reaction of how far the steady state concentrations are from
equilibrium, the direction of the spontaneous change towards equilibrium (i.e., in which
direction the reaction will go) and the magnitude of the driving force towards
equilibrium.
3.2 Representation of a biochemical reaction
Figure 3.2 shows a biochemical reaction as typically written. The reactions are on
the left, the products are on the right and the presumed direction of the reaction is from
left to right. The stoichiometries of each reaction component are also indicated.
Thermodynamics can address several questions about biochemical reactions such as the
one in Figure 3.2.
Figure 3.2: Representation of a biochemical reaction. The direction of the reaction is presumed to be from left to right (reactants to products), and the stoichiometry of each component is indicated by the stoichiometry number, usually denoted i. The reaction is written in terms of biochemical components (e.g., pyruvate) rather than distinct chemical species such as protonated pyruvic acid and deprotonated pyruvate anion. The structures are indicated, with labile protons in parentheses.
5
a) Given the initial concentrations of each component of the reaction, in which
direction will the reaction proceed?
b) What will the final concentration of each component be at equilibrium?
c) What is the work available per mole of component from the reaction under any
set of defined conditions, considering the concentration (activity) of each component, pH,
ionic strength and temperature. Work can be done by coupling to other reactions, i.e.,
chemical work to generate biomass or various metabolites, or transport between cellular
compartments, mechanical work, etc.
3.3 Maximal available work that can be obtained from a biochemical reaction
Let�’s consider a schematic reaction:
aA + bB cC + dD where a, b etc. are stoichiometry numbers, which are positive for chemical species on
the right side of the reaction (presumed products), and negative for components on the
left side (presumed reactants). We will start with initial concentrations of each species
[A]i, [B]i, [C]i and [D]i and then allow the reaction to progress. We are now using the
more common notation of brackets to indicate concentration (Molar unless otherwise
indicated) rather than "ci" as in the previous Chapter. If we are examining the isolated
reaction in vitro (in a test tube), the reaction will proceed until equilibrium is attained.
The concentrations of the species at equilibrium are designated [A]eq, [B]eq, [C]eq and
[D]eq. In a biological context, the substrates and products are often maintained at steady
state concentrations as other reactions generate the substrate being utilized and use the
products. What is most useful is to be able to determine the molar free energy change as
the reaction proceeds under conditions of defined concentrations of each component. This
6
will tell us the maximal (nonPV) work that can be done when the reaction is coupled to
other processes.
Recall that the chemical potential of any component, i , is equal to the partial
molar Gibbs free energy of that component, i.e., the change in free energy per mole of
that component due to an incremental change in its concentration, under conditions where
the concentrations of all the other components are held constant and defined. For
convenience, we can define the extent of reaction parameter, , which has the units of
moles, and quantifies how much reactant has been converted to product. Allowing the
reaction to progress by an infinitesimal amount, converting ad moles of A plus bd
moles of B to form cd moles of C and dd moles of D, we can write the change in the
molar free energy of the system by simply adding up the partial molar free energy of each
component. The chemical potential of each component is dependent on its concentration,
as in equation 2.64.
ln[ ]
ln[ ]
ln[ ]
ln[ ]
oa a
ob b
oc c
od d
RT ART BRT CRT D
(3.1)
The change in the free energy of each component for the differential progress of the
reaction, d , is given by multiplying the change in the number of moles of each
component by the partial molar free energy (which is the free energy per mole) of that
component and adding them together.
7
( )( )( )( )
a a
b a
c c
d d
dG ddG b ddG ddG d
a
b
c
d
(3.2)
( ) ( ) ( ) ( )
( )
c c d d a a b
c c d d a a b b
dG d d d d
dG d
b
(3.3)
We can now define the change in molar free energy for the reaction progress, ,
under the defined conditions.
rG
,
(r c c d d a aT P
GG )b b (3.4)
This tells us that if we know the value of the chemical potential of each component
under the conditions of the reaction, we can easily obtain the maximal nonPV work per
mole that can be obtained from this reaction, rG . Note that this is not the amount of
work that can be obtained by the reaction running to the equilibrium point. Rather, this is
the amount of work obtained (per mole of reaction progress) by the reaction running
under the starting conditions. In a biological context, this is usually more meaningful
because the products and reactants are often maintained at the same concentration due to
other reactions, i.e., a steady state (Figure 3.1). Figure 3.3 illustrates the meaning of
graphically as the slope of the plot of the total free energy of the system as a
function of the extent of reaction. The minimum value of the total free energy is by
definition the equilibrium condition, and the reaction will proceed spontaneously towards
this state.
rG
8
Figure 3.3: A plot of the total Gibbs free energy of a system undergoing a chemical reaction. The free energy of reaction, rG , is the change in the total Gibbs free energy per mole of reaction progress and is determined by the slope of the plot of G versus the extent of reaction parameter . The minimum of the total Gibbs free energy defines the equilibrium condition where 0rG . 3.4 The spontaneous direction of the reaction and the equilibrium constant
Note that if then the reaction will proceed spontaneously from left to
right since the Gibbs free energy will decrease. However, if
0rG
0rG as the extent of
reaction gets larger, then the reaction will proceed in the reverse direction, from right to
left (Figure 3.3). The system will evolve to a point where the Gibbs free energy reaches
its minimal value, which is the equilibrium condition. This can be expressed as the
condition where
,
0 or
0 ( )d a
rT P
eq eq eq eqc c d a b
G G
b
(3.5)
where the chemical potential of each component at equilibrium depends on the
concentration of that component at equilibrium.
9
ln[ ]
ln[ ]
ln[ ]
ln[ ]
a
b
d
eq oa e
eq ob e
eq oc c e
eq od e
q
q
q
q
RT A
RT B
RT C
RT D
(3.6)
If we insert the expressions (3.6) into the equilibrium condition (3.5) then we get the
following equation.
(3.7) 0 00 ( ) ( ln[ ] ln[ ] ln[ ] ln[ ] )o oc c d d a a b b c eq d eq a eq b eqRT C D A B
The first term in this equation is a constant, and is equal to the change in the partial molar
free energy for the reaction when the components are present under standard state
conditions (usually 1 M, but see the next Section).
(o o o or c c d d a a bG )o
b (3.8)
We can now express equation (3.7) as follows
( ln[ ] ln[ ] ln[ ] ln[ ] )
or
[ ] [ ]ln
[ ] [ ]
c d
a b
or c eq d eq a eq b eq
eq eqor
eq eq
G RT C D A B
C DG RT
A B
(3.9)
Since is a constant, we can conclude that the expression on the right side of
equation
orG
(3.9) must also be a constant, and will not depend on the initial concentrations
of the components in the reaction mixture. The term within the logarithm is defined as the
equilbrium constant for the reaction, Keq. The equilibrium constant will be a function
of the temperature, pressure, pH, ionic strength, etc.
10
[ ] [ ][ ] [ ]
c d
a b
eq eqeq
eq eq
C DK
A B
eq
(3.10)
We can also write
(3.11) lnorG RT K
It is important to realize that we have chosen as a matter of convenience to
express the equilibrium constant (3.10) and the chemical potentials (3.6) in terms of the
concentration of each component, but this is standing in place of the ratio of the
concentration divided by the standard state concentration, e.g., in equation (3.10), in
place of [ ]
[ ] we should really write [ ]
c
c eqeq o
CC
C, etc., which is dimensionless. Under
conditions far from ideality, activity must be used in place of concentration, and the
difference between activity and concentration (i.e., the activity coefficient) needs to be
experimentally determined.
We can now rewrite the expression for the Gibbs free energy of reaction under the
initial choice of component concentrations by substituting the expressions for the
chemical potential (3.1) into equation (3.4)
[ ] [ ]ln[ ] [ ]
c d
a b
or r
C DG G RTA B
(3.12)
where the expression inside the logarithm term is sometimes referred to as the reaction
quotient, Q. The concentrations in the reaction quotient are simply those that apply to
the conditions being studied. The maximal work (per mole of reaction progress) that can
be obtained from the reaction, , can be calculated if one knows the concentration of
each of the components and the value of , which is readily obtained by measuring
rG
orG
11
of looking up the value of the equilibrium constant and using equation (3.11). If the
components are at equilibrium, note that expression (3.12) shows that the reaction free
energy is zero, (see Figure 3.3). Under equilibrium conditions, there is no
change in the Gibbs free energy of the system by a small change in the progress of the
reaction,
0rG
d (Figure 3.3). By the same token, being far away from equilibrium
conditions results in a larger amount of work that can be obtained per mole of reactants
converted to products. The progress of the reaction will be towards equilibrium, since
only in this direction is the change in Gibbs free energy negative, the condition for a
spontaneous process at constant T and P, shown in the differential form for the change in
Gibbs free energy of the system. At constant temperature and pressure, the change in the
total free energy depends only on the change in the extent of reaction parameter under the
specified conditions.
rdG SdT VdP Gd (3.13)
The amount of work (per mole) that can be derived from the reaction will
decrease as the concentrations approach the equilibrium condition, at which point, no
work can be obtained since will be zero at equilibrium. In a living cell, the
concentrations are determined by the kinetics of each of the coupled enzyme reactions,
but the direction of each reaction must be consistent with the sign of
rG
rG for that
reaction. As we will see when we discuss enzyme kinetics, the rate of each reaction is
dependent on the concentrations of the products and reactants, but cannot be determined
simply by knowing the value of rG for the reaction.
3.5 The standard state activities (concentrations) used for biochemical reactions
12
The selection of standard states is purely a matter of making our calculations easy
and convenient. Biochemical systems present several particular issues due to the fact that
the reactions are always carried out in aqueous solutions that are buffered to maintain
constant pH. Generally, in addition to temperature and pressure, the reaction conditions
must specify the pH, ionic strength, and in many cases the concentration of divalent
cations which might interact with the components of the reaction.
3.5.1 Standard states of reaction components
1) The standard state of any biochemical component in solution is defined as an
activity of 1, which is normally taken to mean 1 M concentration, assuming ideal solution
conditions, with T = 25oC (298.15K), P = 1 bar. Ionic strength must also be specified but
there is no single value that is an agreed standard.
2) The standard state of a gas, such as CO2(g) or O2(g) is defined as the pure gas
at 1 bar pressure at 25oC. The �“(g)�” defines that the compound is in the gas state.
However, usually it is the concentration of the dissolved gas in solution that is
biochemically relevant, in which case the standard state is 1 M concentration, with the
assumption that the standard state behaves as an ideal solution. We can therefore, write
CO2(aq) and O2(aq) to indicate the dissolved compound in aqueous solution. In the case
of CO2(aq), there is an additional complication, which is that CO2 in the aqueous phase
can exist as CO2, HCO3- or H2CO3, which must all be taken into account. This is
designated CO2(tot).
Unless specified, for biochemical reactions, it is implicit that all components are
dissolved in the aqueous phase.
13
3) The standard state of an element is the natural state (gas, liquid, solid) at 25oC
at 1 bar pressure.
3.5.2 The activity of the solvent, water, is 1 and remains unchanged for dilute
solutions.
The standard state of water is taken to be that of pure water at 298.15K and 1 bar
pressure, which corresponds to a concentration of 55.35 M. Since the concentrations of
most biochemical components are in the millimolar range (10-3 M), any water formed or
consumed by a reaction in which water is a direct participant will be negligible compared
to the ~55 M concentration. Hence, in the equilibrium expression the activity of water
remains a constant 1: 2
2
2
( )H OH O o
H O
ca
c1. As a result, it is the convention to omit the term
for water in the equilibrium constant.
As an example, consider the conversion of malate to fumarate, catalyzed by the
enzyme fumarase (structures in Figure 3.4).
Figure 3.4: The reaction catalyzed by fumarase, interconverting fumate and malate.
(3.14) 2fumarate+H O L-malate
The equilibrium constant can be written as
2
[[ ][
L malateK ]]fumarate H O
(3.15)
14
and the Gibbs reaction free energy
2
[ln[ ][
or r
L malateG G RT ]]fumarate H O
(3.16)
Since the activity of water is 1, this is simply written as
[ ] [ and ln[ ] [ ]
or r
L malate L malateK G G RT ]fumarate fumarate
(3.17)
The fact that water is a participant in the reaction is taken into account in the value of
, but the contribution of water toorG rG does not change as a function of the extent of
the reaction.
Note that the assumption that the activity of water remains constant with an
activity of 1 is not valid when we have a high concentration of solutes, such as in the
cytoplasm of a cell or under any reaction conditions where the activity or effective
concentration change of water is not negligible. For example, protein can be present in
the cell at several hundred mg/ml, which is far from "ideal" solution conditions. The
change in the activity of water due to solutes is responsible for the phenomenon of
osmotic pressure, which will be discussed in Chapter XX.
3.6 Biochemical reactions in buffered solution at constant pH: the transformed
Gibbs free energy of reaction.
Biochemical reactions are virtually always performed under conditions of constant
pH, both in the cell and in vitro. Hence, even if protons are consumed or released by the
biochemical reaction itself, the bulk concentration of free protons does not change
because of the presence of the buffer. We will discuss how buffers do this in the Chapter
on ligand binding, but for now just accept that the concentration of protons is maintained
constant due to the presence of buffers. Because of this, it is necessary to modify the way
15
we express the equilibrium constant and the Gibbs free energy of reaction in reactions in
which protons are involved, which is most biochemical reactions.
This is handled by using a transformed Gibbs free energy, in place of the Gibbs
free energy used traditionally by chemists. This is done as a convenience to make matters
more simple, though it may not look that way at first. Just as we modified the internal
energy to the Helmholz free energy for systems constrained at constant temperature, and
to the Gibbs free energy for systems at constant temperature and pressure, we now define
the transformed Gibbs free energy for systems constrained to constant temperature,
pressure and pH. The transformed Gibbs free energy is denoted by using a superscript
�“prime�”, , is 'G
' ( ) ( )CG G n H H (3.18)
where nC(H) is the amount of moles of hydrogen component bound to all the chemical
species and as protons in solution), and µ(H+) is the chemical potential of protons in
solution under the conditions of the reaction. Table 3.1 illustrates the definitions of the
most useful thermodynamic free energy functions, depending on which environmental
parameters are held constant. To temperature and pressure (see Chapter 2), we are now
adding pH as an environmental parameter which can be held constant.
Table 3.1: Transformed thermodynamic free energy functions that are useful when the system is constrained to constant temperature, pressure and pH. PARAMETERS HELD
CONSTANT
MOST USEFUL THERMODYNAMIC
FUNCTION
EQUILIBRIUM
CONDITION Temperature Helmholz free energy: A = U + TS Minimize A
Temperature
Pressure
Gibbs free energy:
G = U + TS �–PV
G = A - PV
Minimize G
16
Temperature
Pressure µ(H+) or pH
Transformed Gibbs free energy:
G' = U + TS �– PV - nC(H)µ(H+)
G' = G �– nC(H)µ(H+)
Minimize G
Note the pattern of subtracting the product of the intensive variable being constrained (T,
P or µ(H+) ) and the conjugate extensive variable (-S, V or nC(H)). This mathematical
operation is called a Legendre transform. It is not obvious why this should end up
making matters more simple in the end, but it does, as we will see. Note we are adding
nC(H), an extensive variable, and µ(H+), an intensive variable to the list of conjugate
variables (Table 2.1).
It is the transformed Gibbs free energy that is used in dealing with biochemical
systems, and it is this function that is minimized under conditions of constant T, P and
pH. This has very significant consequences, with the result that problems that might
appear daunting become relatively simple. The reason for this is that the use of the
transformed Gibbs free energy allows us to group together species that differ only by the
state of protonation (called a pseudo-isomer group), and treat these as simple reaction
components, rather than a set of distinct chemical species. To illustrate this, let�’s take a
look at the hydrolysis of ATP (structure in Figure 3.5).
17
Figure 3.5: Structure of fully deprotonated ATP. The hydrolysis reaction generates ADP and phosphate. As biochemists, it is natural to express biochemical reactions, such as the
hydrolysis of ATP in the following manner.
2ATP H O ADP Pi (3.19)
ATP, ADP and inorganic phosphate (Pi) are components of this chemical reaction, but
each of these terms represents several distinct chemical species. ATP, for example,
Table 3.1: Relevant protonation reactions involving ATP, ADP or Pi between pH 3 and pH 9. Also tabulated is the equilibrium constant for each reaction, expressed as pK (= -log(K)). When the pH is equal to the pK for a protonation reaction, half of the species is protonated and half is deprotonated. protonation reaction pK
(298.15K, 0.25 M ionic strength) 2- 3- +HADP ADP +H 6.32
18
- 2-2H ADP HADP +H+ 3.79
3- 4- +HATP ATP +H 6.46 2- 3- +
2H ATP HATP +H 3.83 - 2-
2 4 4H PO HPO +H+ 6.65 can exist in three protonation states in the pH range 3 �– 9: ATP4-, HATPP
3- and H ATP22-.
Similarly, inorganic phosphate can exist as HPO42- or HPO4
- and ADP can be ADP3-,
HADP2- or H ADP2-. Treating each of these individually would make dealing with
biochemical equilibria very unwieldy, to say the least. Fortunately, the situation can be
simplified substantially because of the specification that the pH is maintained constant.
At constant pH, we can define an apparent equilibrium constant and transformed Gibbs
free energy of reaction by
'
' '
'
'
[ ][ ] ln[ ]
where
ln
and
[ ] [ ][ ]
o ir r
or
eq i eq
eq
ADP PG G RTATP
G RT K
ADP PK
ATP
(3.20)
in which the term for water has been omitted from the equilibrium constant, as already
discussed in Section 3.52. The parameters are referred to as the
transformed Gibbs free energy of reaction and the transformed standard state Gibbs free
energy of reaction, and these are defined at a specified pH. is an �“apparent�”
equilibrium constant since the biochemical components are not simple chemical species
'
and o
r rG G
'K
19
but each represents a sum of species with different protonation states. The �“prime�” marks
in the upper right indicate the parameters are all defined at a specified pH. Usually, such
expressions as in (3.20) are simply taken for granted as being valid but, in fact, it is not a
trivial matter to demonstrate their validity. When we denote a biochemical component
such as "ATP" in such an expression, "ATP" is not a chemical species but is, rather, a
group of chemical species (ATP4-, HATP3- and H2ATP2-). Justification of these
expressions is given in Box 3.1, which is a lengthy derivation, but not complicated. A
more intuitive, though perhaps not as rigorous, way to get to the same place is described
in the section below.
3.6.1 A relatively simple approach to dealing with thermodynamics at constant pH.
We will start by simply writing out the biochemical reaction as one normally
would, but to recognize that there will be a protons released or taken up as reactant is
converted to product. The stoichiometry number for protons H
is non-integer since only
a fraction of each biochemical reactant (e.g., ATP, ADP, Pi) will be protonated or
deprotonated, depending on the pH and on the pK values (Table 3.2). For example,
Figure 3.6 shows the pH titration of ATP, showing the number of bound protons as a
function of pH.
20
Figure 3.6: The number protons bound to ATP as a function of pH at 298.15K and an ionic strength of 0.25 M. At pH 3 about 1.8 protons are bound but as the pH is increased, this drops to zero at pH 9. The two pK values are pH 3.46 and pH 6.46 for the two protonation sites (Table 3.2)
Let's define rH H
N as the number of protons released or picked up from
solution per mole at the selected pH as the reaction progresses. The value of r HN will
depend on the pH as well as all of the pKa values for the protonation reactions (Table
3.1). The reaction can be written as
2 i r HATP H O ADP P N H (3.21)
Ignoring that ATP, etc. are not unique chemical species, we will proceed to define the
equilibrium condition in terms of chemical potentials. The more rigorous derivation in
Box 3.1 demonstrates that this kind of representation is justified.
21
Figure 3.7: The biochemical components represented in the hydrolysis reaction eacg represent groups of chemical species that differ only by the extent of protonation. The number of protons bound by ATP, ADP and Pi vary depending on the pH and the pK values of the protonation reactions (Table 3.2). For this reason the number of protons gained or lost during the hydrolysis depends on pH and is non-integer. We will now put "primes" over each chemical potential term to denote that we
are assuming constant pH and grouping chemical species into pseudo-isomer groups that
are in rapid equilibrium (see Figure 3.7).
2
2
' ' ' ' '
' ' '
' '
'
0
( ln[ ] ) ( ln[ ] ) ( ln[ ])
( ln[ ] ) 0
ADP i ATP
ADP i H
ATP
ADP
P r H OH H
o o oeq P i eq r H
o oeq H O
oP
N
RT ADP RT P N RT H
RT ATP
2
' ' ' ' [ ] [ ]( ) ln[ ] ln
[ ]i ATPH
eq i eqo o o or H O rH H
eq
ADP PN N RT H RT
ATP
(3.22)
By definition, the standard state chemical potential for the proton, i.e., a solution of
protons at 1 M concentration at 298.15K, is 0, so the term drops out of '(H
or HN )
(3.22). Since we are holding the pH constant, we will include the term ln[ ]r HN RT H
in the definition of the standard state for convenience. 'orG
22
2
' ' ' ' ' ln[ ]ADP i ATP
o o o o or P H O r HG N RT H (3.23)
We will also recognize that the change in the number of free protons in solution
r HN is equal to the negative of the change of the number of bound protons r H
N .
Since ln[ ]log[ ]2.303
HpH H , we get
2
' ' ' ' '
'
' '
[ ] 2.303 ( )
[ ] [ ]ln at equilibrium
[ ]
[ ][ ]ln under any reaction conditions[ ]
ADP i ATP
o o o o or P H O r H
eq i eqor
eq
o ir r
G N
and
ADP PG RT
ATP
and
ADP PG G RTATP
RT pH
(3.24)
These expressions can be generalized to any biochemical reaction in which pH is
held constant. The beauty of using this approach is that all one needs to do is to
determine the concentrations of each biochemical component at equilibrium at the
desired pH, and then use equation (3.24) to determine the value of the standard state
transformed reaction free energy, . One need not be concerned about all the terms
in the messy expression in
'orG
(3.23) of how much ATP is present in the form of ATP4-,
HATP3- or H2ATP2-.
3.6.2 The pH-dependence of the transformed Gibbs reaction free energy.
23
Using the expressions in (3.24), we can now readily obtain the pH-dependence of
either by taking the derivative of the expressions in ' or or rG 'G (3.24) with respect to
pH.
' '
'
' '
, , , ,
'
, ,
( ) ( ) 2.303( ) ( )
or
log
or r
r HT P T P
r HT P
G G RT NpH pH
K NpH
(3.25)
This is identical to equation (3.26) in Box 3.1.
Equation (3.25) says that the slope of a plot of the transformed Gibbs reaction free
energy as a function of pH gives the change in the number of bound protons as one
converts reactants to products at any particular pH value. If one is interested in the
thermodynamics of ATP hydrolysis under the same conditions used for the tabulated
data, these data can be used directly. However, if the tabulated data refer to a standard pH
of 7.0 but you are interested in the equilbrium value at pH 9, then additional information
is required. Figure 3.8 shows how the value of the standard state transformed Gibbs
reaction free energy for ATP hydrolysis, varies as a function of pH. 'o
rG
Figure 3.8: Standard state transformed Gibbs reaction free energy (kJ/mol) plotted as a function of pH for the hydrolysis of ATP at 2998.15K and ionic strength of 0.25 M.
24
As the pH becomes more alkaline, the value of becomes more negative, meaning
that the driving force towards products is more favored at higher pH than at lower pH.
The slope of the plot in Figure 3.8 is negative since the value of gets more
negative as the pH is increased. According to equation
'o
rG
'o
rG
(3.25), this means that the value
of is negative throughout this pH range. As ATP is hydrolyzed, fewer protons are
bound to the biochemical components (ATP, ADP and P
r HN
i), which means that protons are
released during the hydrolysis of ATP to ADP plus Pi. As the pH is increased, the
products, which bind fewer protons that ATP, are favored over the reactant, ATP.
Conversely, making the solution more acidic changes the equilibrium to favor the
reaction in the direction to favor the more protonated components. Figure 3.9 shows the
value of as a function of pH. r HN
Figure 3.9: Plot of the change in the number of bound protons, r HN , as a function of pH for the hydrolysis of ATP at 298.15K and 0.25 M ionic strength. As the pH is increased, the number of bound protons decreases by about 1 proton. The total number of protons bound by ADP + Pi is about 1 less than bound to ATP at pH 9. The equilibrium shifts to favor the more protonated side of the reaction (ATP) at acid pH and the less protonated components (ADP + Pi) at alkaline pH.
25
Expressing the pH-dependence of the reaction in terms of the apparent equilibrium
constant (Figure 3.10) shows that hydrolysis is more favored at alkaline pH by about
a factor of 100, comparing pH 9 to pH 5. Using equation
'K
(3.20), the value of is
easily converted to the apparent equilibrium constant, which varies from about 10
'o
rG
6 at pH
5 to 108 at pH 9, increasingly favoring the products at more alkaline pH.
Figure 3.10: pH-dependence of the apparent equilibrium constant for the hydrolysis of ATP at 298.15K and ionic strength of 0.25 M.
3.6.3 One can also modify the expressions for the Gibbs free energy of reaction
under conditions of constant Mg2+ or any other chemical species.
A similar situation exists for metal chelation as for protons under physiological
conditions. Mg2+ in particular, binds to nucleotides as well as to polynucleotides such as
RNA and DNA. Many metal-chelated species that constitute a single biochemical
reactant are in equilibrium and can be treated as a single component of the reaction. To
the three protonation states of ATP (ATP4-, HATP3-, H2ATP2-), for example, we can add
two metal-chelated species: MgATP2- and MgHATP1- and two additional equilibria to
those listed in Table 3.1.
2 2
3 2
MgATP ATP Mg 2
MgHATP HATP Mg
26
If the concentration of free [Mg2+] is held constant along with [H+], all five of the ATP
species (ATP4-, HATPP
3-, H ATP22-, MgATP2- and MgHATP1-) can be grouped together
and simply written as "ATP". The implicit assumption is that the equilibration of the
various ATP species is rapid compared to the rate of the biochemical reaction where ATP
is hydrolyzed. Using the same procedure to derive a transformed Gibbs free energy to
use under conditions where the Mg2+ concentration is constant, one obtains a set of
equations relating the change in the equilibrium constant for the reaction as a function of
the Mg2+ concentration. Defining pMg = -log[Mg2+]
'
'
'
, ,
'
, ,
( ) 2.303( )
or
log
rr Mg
T P
r MgT P
G RT NpMg
K NpMg
(3.26)
where is the net gain or loss of bound Mgr MgN 2+ per mole of as reactants are converted
to products. These expressions are useful also in examining the thermodynamics of
protein binding to DNA or RNA, where the formation of the complex involves displacing
bound ions, resulting in a strong dependence of the equilibrium constant on the
concentration of the ion being released, as defined by equations analogous to those in
(3.26).
One can use the same procedure to modify the thermodynamic parameters under
conditions in which any biochemical component is held constant, such as maintaining
constant O2 or constant NAD concentration.
27
One aspect to note is that the mechanism of the enzymes catalyzing the reactions
of interest is not relevant to the thermodynamics. If, for example, an enzyme reacts only
with MgHATP1- in the hydrolysis reaction, the equilibrium is still determined by the total
ATP concentration, which is rapidly equilibrated with MgHATP1-. We will see in later
chapters, however, that although the rate of the reaction can be very dependent on the
concentration of the reacting species (e.g., [MgHATP-] ) , the equilbrium end point will
not be influenced.
Values of o
and the corresponding transformed equilibrium constants, rG K ,
have been tabulated for many enzyme-catalyzed reactions at different solution conditions
(pH, [Mg2+], ionic strength) by the National Institute of Standards and Technology
(NIST) and are available on the internet. These are usually established experimentally by
determining the equilibrium constant for each reaction under specified conditions.
Box 3.1: The importance of transformed Gibbs free energy in biochemistry
In this section, a more rigorous derivation of the transformed Gibbs free energy or
reaction will be demonstrated, again using the hydrolysis of ATP as an example. The
justification for using biochemical components in place of chemical species in the
equilibrium expressions is shown along the way.
2 iATP H O ADP P
Each of these reaction components represents a number of protonated species in
equilibrium. Hence, for ATP, for example, the following equilibrium equations can be
written.
28
3 4HATP ATP H (3.27)
2 32H ATP HATP H (3.28)
2 42 2H ATP ATP H (3.29)
These species will remain in equilibrium as ATP is hydrolyzed and its concentration is
diminished. The equilibrium condition defines the following relationships, corresponding
to the three equilibria above. The chemical potentials, or molar Gibbs free energies are
defined for each of the specific chemical species, e.g. HATP3-. For the three reactions
above, we obtain the following.
4 0ATP H HATP3
2
(3.30)
32
0HATP H H ATP
2 0
(3.31)
42
2ATP H H ATP (3.32)
We will rearrange these expressions. From (3.30)
4 3ATP HATP H (3.33)
and from (3.31)
4 22
2ATP H ATP H (3.34)
We now define a set of transformed chemical potentials for each species, where we
subtract out the chemical potential term for protons, which is multiplied by the number of
bound protons in each species. For example, for the species HATP3-, one proton is bound,
or NH(HATP3-) = 1, . Note that when we do this, the transformed chemical potential of
the protons since . ' 0H
4' '
4ATP ATP (3.35)
29
3 3'HATP HATP H
(3.36)
2 22 2
' 2H ATP H ATP H (3.37)
' 0H H H
(3.38)
At equilibrium we can now write, for the ATP component 4 2
2
' ' 'ATP H ATP HATP3 (3.39)
We can define the chemical potentials in equation (3.39) simply as 'ATP .
The differential form of , as defined in equation 'G (3.18) is
' ( ) ( ) ( ) ( )C CdG dG dn H H n H d H (3.40)
Now we can obtain an expression for dG. Assuming constant T and P, the small changes
in the number of moles of species j, dnj, results in a small change in the Gibbs free
energy, dG. The sum extends over all chemical species present, including protons.
1
SN
j jj
dG dn (3.41)
Now substitute into equation (3.41) the transformed chemical potentials from equations
(3.35) -(3.38).
1
'
1 1( )
S SN N
j j Hj j
dG dn N j dnj (3.42)
The first term is summed over one less species, since the transformed chemical potential
of protons . The second term is the change in the total amount of bound
hydrogen (protons) in the system, summed over all species. This is not conserved since
the pH is constant, so protons gained or lost by the chemical species are not accounted for
by changes in the free protons present. We define the total amount of bound protons,
n
' 0H
C(H) and its differential as follows.
30
(3.43) 1
1
( ) ( )
( ) ( )
S
S
N
C Hj
N
C Hj
n H N j n
dn H N j dn
j
j
Therefore, equation (3.42) becomes
(3.44) 1
'
1( )
SN
j j CHj
dG dn dn H
Substitute expression (3.44) into (3.40)
1
' '
1( )
SN
j j C Hj
dG dn n H d (3.45)
To see why this is useful, let�’s go back to the hydrolysis reaction (3.19). The sum in
equation (3.45) contains a term for each of the eight chemical species present: ATP4-,
HATP3-, H2ATP2-, HPO42-, H2PO4
1-, ADP3-, HADP2- and H2ADP-. However, the
transformed chemical potentials of each of the species making up a reaction component
are equal, (3.39).
4 4 3 3 2 22 2
4 3 22
8' ' ' '
1
'
{terms for ADP and P }
= ( ) {terms for ADP and P }
j j iATP ATP HATP HATP H ATP H ATPj
ATP iATP HATP H ATP
dn dn dn dn
dn dn dn(3.46)
But the sum of the changes in each of the chemical species comprising ATP is just equal
to the change in the total amount of ATP.
31
(3.47)
4 3 22
3 22
2 14 2 4
'
'
'
( )
and similarly
(
and
( )i
ATP ATP HATP H ATP
ADP ADP HADP H ADP
P HPO H PO
dn dn dn dn
dn dn dn dn
dn dn dn
1 )
We can now express the differential change in the transformed Gibbs free energy,
equation (3.45) as
' ' ' ' ' ' ' ( )i iATP ATP ADP ADP P P C H
dG dn dn dn n H d (3.48)
We can now introduce the transformed extent of reaction parameter, ' , and the
transformed Gibbs free energy of reaction, , just as we did in Section 3.2. The prime
indicates that the changes are in the number of moles of the reaction components, not
chemical species, and that the pH is constant. For the hydrolysis of ATP, for example, the
stoichiometry number for ATP is
'rG
' 1ATP and '' ' 'ATP ATPdn d d
'
. Similarly,
' ' ' and iADP Pdn d dn d .
' ' '
' ' ' '
''
', ,
( )
where ( )
and
i
r C H
r ADP P A
rT P pH
dG G d n H d
G
dG Gd
TP
(3.49)
The transformed chemical potentials are
32
''
' '
' '
ln[ ]
ln[ ]
ln[ ]
o
i i
ATP ATP
oADP ADP
oP P i
RT ATP
RT ADP
RT P
(3.50)
So we have
' '
' ' ' '
[ ][ ]ln[ ]
where i
o ir r
o o o or ADP P
ADP PG G RTATP
G ATP
(3.51)
The equilibrium condition is that the transformed Gibbs free energy of reaction .
Hence, at equilibrium
' 0rG
' '
'
ln
where
[ ] [ ][ ]
or
eq i eq
eq
G RT K
ADP PK
ATP
(3.52)
We have arrived at a simple expression, which tells us that we can measure the
total concentrations of ATP, ADP and Pi ,without regard to protonation state, and define
an apparent equilibrium constant ( ) and standard state reaction Gibbs Free
Energy( ) at any specified pH.
'K
' orG
Dependence of on pH. ' orG
The value of must be dependent on pH, because changing the pH will alter
the apparent equilibrium constant, favoring either the products or reactants. To see how
' orG
33
to quantify this dependence, we will first rewrite the expression for , equation 'dG (3.49),
in terms of pH. The chemical potential for the proton is
ln[ ] 2.303 log[ ] (2.303 )o o oH H H H
RT H RT H RT pH (3.53)
Hence, (2.303 ) ( )Hd RT d pH . Substitute into equation (3.49) to yield
(3.54) ' ' ' 2.303 ( )r CdG G d n H dpH
Thermodynamic functions are exact differentials, and one mathematical property of exact
differentials is that cross derivatives are equal. This means that if we have a function
f(x,y), and its differential form is
( , )
then
yx
df x y Adx Bdy
A By x
We can apply this to all the differential terms of the thermodynamic state functions,
leading to a bewildering array of equations, known as Maxwell equations. We are only
interested in applying this to equation (3.54).
'
'
'
', ,, ,
'
, ,
( )( ) 2.303 2.303( )
or
log
Crr H
T P pHT P
r HT P
n HG RT RpH
K NpH
T N
(3.55)
34
r HN is the net gain or loss of bound protons per mole as reactants are converted to
products. If we determine at several different pH values, from equation 'rG (3.55) the
slope of the plot of pH vs is 2.303RT'rG r HN . Alternatively, the pH dependence of
log K' can also yield . r HN
END BOX 3.1
3.7 Transformed enthalpy of reaction
We know from integrating equation (2.16) that G H T dS , from which it
follows that we can write a similar expression for the standard state transformed reaction
Gibbs free energy.
(3.56) ' 'o or r rG H T 'oS
The standard state transformed enthalpy of reaction, 'or H , is the amount of heat per
mole of reaction progress ( d ) (as the reaction is written) released to or gained from the
environment under standard state conditions, 298.15K, 1 bar pressure and, unless
otherwise stated, pH 7. As discussed in the last Section, we now group all species
together that differ only by the state of protonation, which makes matters much simpler.
'or H is also called the heat of reaction, and it results from the making and breaking of
bonds and changes in molecular interactions.
Note that the heat of reaction ( 'or H ) is not dependent on the concentrations of
reactants or products present, but only on the conversion of reactant to product. Hence,
' or r
'H H (3.57)
35
There are two ways to determine the value of 'or H : 1) by calorimetry, and 2) by
determining the temperature dependence of . 'orG
3.7.1 Determining 'or H by calorimetry.
Since 'or H is a heat, it can be directly measured using a calorimeter during the
course of the reaction. Since we are now mostly interested in reactions that don�’t involve
gases, but reagents in solution, we can measure the heat directly at constant pressure.
Whereas in Chapter 1 (Section 1.14) we described using a bomb calorimeter to measure
the heat of combustion at constant volume, we can now use a different instrument
maintained at atmospheric pressure (Figure 3.11).
Figure 3.11: Schematic of an isothermal calorimeter used for determining the transformed heat of reaction of an enzyme-catalyzed reaction. The vessels are open to the environment and, therefore, maintained at constant pressure. Hence, the heat measured is directly converted to enthalpy. Reactants are mixed along with the enzyme in one vessel and the amount of heat required to maintain the temperature in comparison to a control vessel is measured. The heat exchange with the environment at constant pressure by the reaction is, by
definition, the enthalpy of the reaction. This is measured by determining the energy
required to maintain a constant temperature compared to a reference. The use of
36
calorimetry to determine the heat of reaction of the reaction catalyzed by tryptophan
synthase is described in the next Section.
3.7.2: Example: Enthalpy of the reaction catalyzed by tryptophan synthase
The enzyme tryptophan synthase is an 2 2 tetramer that catalyzes the following
reactions[Kishore, 1998 #1] (Figure 3.12). An active site on the subunit catalyzes the
first reaction, forming indol, and a separate active site in the subunit catalyzes the
second reaction in which indol reacts with serine to form tryptophan. The two active sites
are connected by a hydophobic channel that is wide enough to allow indol to pass
through and which is 25 Å long. This is one example of several enzymes which
physically couple biochemical reactions by joining separate active sites by an intraprotein
tunnel to convey the product of one reaction to a second active site where it serves as a
substrate[Raushel, 2003 #2].
Figure 3.12: The conversion of 1-indol-3-yl)glycerol-3-phosphate to L-tryptophan under in vivo conditions. (from {Kishore, 1998 #18})
37
The heat of reaction for the reaction catalyzed by tryptophan synthase was
measured by placing IGP + serine dissolved in buffered solution into the calorimeter
vessel (see Figure 3.11) and then adding the enzyme tryptophan synthase. The heat
released was quantified and then divided by the number of moles of substrates converted
to the products, indol + G3P, yielding the value of . ' ( ) 27.8 kJ/molr H cal
Each of the two reactions making up the net reaction can also be measured
independently. Interestingly, under standard state conditions reaction I (see Figure 3.13)
is actually favored to run spontaneously in the reverse direction (reaction Irev) than that
which occurs in vivo (Figure 3.12). The equilibrium constant for reaction I is
approximately 104, favoring the formation of IGP from indol. The enzyme works because
the equilibrium constant for reaction II is approximately 1013 ,favoring the formation of
tryptophan, and the reaction with indol at the second site is rapid. Any indol generated at
the first active site (reaction I) is rapidly removed by the reaction at the second active
site. Hence, the steady state concentrations for the reactant and product of reaction I
favors the reaction to form indol, and ' 0rG in the direction of indol being the product.
We can speak of reaction II "pulling" the entire reaction towards the formation of
tryptophan by maintaining a very low concentration of indol.
In order to measure the heat of reaction of reaction I, it is necessary to use as the
substrates indol + G3P where product is IGP (reaction Irev). Adding an excess of G3P to
the calorimetry vessel makes indol the limiting substrate, determining the limit to the
amount of product that can form. This reaction is also catalyzed by tryptophan synthase,
at the active site in the subunit. The enzyme accelerates the rate of the reaction towards
its equilibrium point, which is not altered by the enzyme. In this reaction, heat is released
38
to the environment (i.e., negative heat of reaction), and the amount of heat divided by the
amount of product formed gives the molar heat of reaction, The molar heat of reaction of
this reaction is . The heat of reaction for reaction III was
determined by using serine and IGP as substrates and catalyzing the reaction with the
enzyme. It was found that . Reaction II is catalyzed by
mixing the substrates serine + indol with the
' ( ) 46.9 kJ/molr H cal
' ( ) 27.8 kJ/molr H cal
2 subunit complex. In this case,
. ' ( ) 74.5 kJ/molr H cal
If reaction Irev is written in the reverse direction (reaction I), the thermodynamic
reaction parameters have the opposite sign. If reactions I + II are added together, one
obtains reaction III, the net reaction catalyzed by tryptophan synthase (Figure 3.12).
We therefore expect
' ' '( ) ( ) ( )
(46.9) (74.5) 27.6 / calculated
r I r II r IIIH cal H cal H cal
kJ mol
This calculated value is very close to the experimental value of -27.8 kJ/mol.
'
'
I G3P + indol IGP ( ) 46.9 kJ/mol
I IGP G3P + indol ( ) 46.9 kJ/mol
II indol + serine tr
rev r
rev r
H calor
H cal
'2
'2
yptophan + H O ( ) 74.5 kJ/mol
III IGP + serine tryptophan + G3P + H O ( ) 27.8 kJ/mol
r
r
H cal
H cal
(3.58)
Figure 3.13: The biochemical reactions studied in relation to the reaction catalyzed by tryptophan synthase. This example demonstrates several points. First, it illustrates how reactions can be
added together, along with their thermodynamic reaction parameters, to yield information
39
about the net reaction. We can also do this for more than two reactions, such as all the
steps in gluconeogenesis, for example. In the example of tryptophan synthase, we might
ask whether the fact that the two active sites are linked influences the net outcome. From
a thermodynamic view, the answer is no. The fact that the two reactions are catalyzed by
the same enzyme is irrelevant to the favored equilibrium. If the two reactions were
catalyzed by separate enzymes, the outcome would be exactly the same, in principle.
Reaction II would still scavenge indol, which would be produced by reaction I.
However, the rate of the reaction might be very slow, which is the advantage of having
the enzyme channel the product from one active site to the second active site. In the
context of a cell, such channeling could also avoid having the substrate participating in
other reactions in which it might be utilized.
The equilibrium constant of reaction I favors the formation of IGP, which is the
reverse of the reaction as it occurs in the cell. This illustrates a very important point. In
order to evaluate what will occur within the cell, the equilibrium constant is in itself
insufficient. The equilibrium constant is related to the Gibbs free energy of reaction
under standard state conditions, in this case 1 M of each reaction component. To
evaluate the Gibbs free energy of reaction under physiological conditions, it is necessary
to know the concentrations of the reaction components, [indol], [G3P] and [IGP] and then
to substitute these values into equation (3.59).
' ' [ ][ 3ln[ ]
or r
indol G PG G RTIGP
] (3.59)
Metabolite concentrations are often not easily measured, particularly in the case of
eukaryotic cells with separate compartments such as the mitochondrion, endoplasmic
reticulum, etc. In expression (3.59), the value of , the standard state transformed 'orG
40
Gibbs free energy of reaction, can be calculated from the apparent equilibrium constant
, writing the reaction with IGP as the reactant, on the left. ' 10K 4
ol (3.60)
' ' 4
'
ln ln(10 ) (8.31)(298)( 9.2) 22.8 /
22.8 /
or
or
G RT K RT kJ m
G kJ mol
Under standard state conditions, the driving force for the reaction is positive, driving the
reaction to the left, generating IGP. Let�’s now calculate under hypothetical
conditions that might exist in the cell. If, for example, the concentrations of [G3P] and
[IGP] are equal in the cell and if the indol concentration is 1 µM (10
'rG
-6 M), then the Gibbs
free energy of reaction will be negative and favor the formation of indol and G3P.
' ' 6
'
[ ][ 3 ]ln 22.8 (8.31)(298) ln[10 ] 22.8 34.2[ ]
7.4 /
or r
r
indol G PG G RTIGP
G KJ mol (3.61)
As long as reaction II removes indol from solution, the low concentration of indol will
favor the generation of indol by reaction I. This is made even more effective when the
indol is rapidly removed from the vicinity of the active site where reaction I is catalyzed,
since virtually any indol molecules generated at this site will be directed to the second
active site where it will be consumed to form tryptophan. The end result is not altered by
this arrangement of the two active sites connected by a tunnel, but the rate of the overall
reaction is greatly accelerated.
3.7.3 Determining 'or H by the temperature dependence of 'o
rG
41
Starting with (3.56), , we can see that if both ' 'o or r rG H T 'oS
'or
' and or H S are constant and not changing with temperature, then a plot of
versus temperature should be a straight line with the slope of and an
intercept of
'orG
'or S
'or H . In general, neither ' or o
r'o
rH S are independent of temperature
over a wide range of temperature. However, over a small temperature range, both
' and or
'orH S can be treated as constants. Typically, it is more accurate to determine
the slope of a plot rather than an extrapolated intercept, so equation (3.56) is divided by
T, yielding
' '
'o o
or rr
G H ST T
(3.62)
After measuring at several temperatures within a narrow range, one plots 'orG
'orGT
versus (1/T), and the slope will give the molar enthalpy of reaction, 'or H . Two
equivalent ways to express this are
'
'
''
, ,
'' 2
, ,
( )(1 )
or
( )( )
oo r
rP pH
oo r
rP pH
G THT
G TH TT
(3.63)
An alternate but equivalent expression is to recognize that since
' lnorG RT 'K (3.20), then
42
' '' '
' ''
ln
ln
o oor r
r
o or r
G HR K ST T
or
H SKRT R
(3.64)
The slope of a plot of versus (1/T) should be a straight line with a slope of 'ln K'o
r HR
if the enthalpy of reaction is a constant over the temperature range.
This last equation is called the van't Hoff equation. One can plot the value of the
apparent equilibrium constant as a function of 1/T and the slope, multiplied by �–R, gives
the value of 'or H . Recall that since there is no concentration dependence on the
enthalpy of reaction, this value applies for any concentration of reactants and products
( ' 'or rH H )
What does the van't Hoff equation tell us? As we increase the temperature (1/T
decreases) the equilibrium constant gets larger, favoring the products, if 'or H is positive.
The molar heat content (or enthalpy) of the products is greater and these are favored as
temperature is increased. In other words, if you know the sign of the reaction enthalpy,
you also know whether the equilibrium constant will favor products or reactants more as
the temperature is increased. If the reaction enthaply is zero, i.e., there is no change in the
enthaply as reactants are converted to products, then the equilibrium constant will not
change with temperature.
43
3.8 Transformed entropy of reaction
If both and 'orG
'or H are known, then from equation (3.56), it is evident that
one can simply calculate the value of . In addition, the slope of a plot of
versus temperature (T) will have a slope of , the transformed standard state
entropy of the reaction.
'or S
'orG
'or S
'
''
, ,
( )oo r
rP pH
GST
(3.65)
3.9 Example: Temperature-dependence of the reaction catalyzed by lysine 2,3
aminomutase.
The enzyme lysine 2,3 aminomutase interconverts L-lysine to L- -lysine. This
involves the migration of the -amino group of L-lysine to the -carbon, shown in Figure
3.13.
Figure 3.13: The reaction catalyzed by lysine 2,3 aminomutase (left) and the van�’t Hoff plot of the natural logarithm of the equilibrium constant plotted as a function of the reciprocal of the absolute temperature.
44
-Amino acids are precursors in the biosynthesis of antibiotics and other useful
compounds. The temperature-dependence of the reaction can help define optimal
conditions if the enzyme is to be used in the manufacture of L- -lysine. One can add
either L-lysine or L- -lysine at the start of the reaction and allow the reaction to reach
equilibrium. HPLC is used to quantify the amounts of L-lysine and L- -lysine at
equilibrium. At 303K (30oC) the equilibrium constant is 8.45, favoring L- -lysine.
' [L- -lysine][L-lysine]eqK (3.66)
The equilibrium constant was measured over a temperature range from 277K to 338K,
and the van�’t Hoff plot is shown in Figure 3.14. The straight line indicates that 'or H is
a constant over this temperature range, and the positive slope indicates that the sign of
'or H is negative. The value obtained was ' 5.8 /o
r H kJ mol . At 303K, from the
equilibrium constant one can obtaine the value of . 'orG
' ln (8.31)(303) ln(8.45) 5.4 /or eqG RT K kJ mol
We can now easily obtain the value of the transformed entropy of reaction.
' ' '
'
'
'
5380 ( 5850) (303)
1.55 / deg
0.47 /
o o or r r
or
or
or
G H T S
S
S J mol
T S kJ mol
These data tell us several things.
1) The reaction is �“enthalpy driven�”. The reaction free energy, , favors L- -lysine
because it has a lower molar enthalpy. The
'orG
'or H term is the main contributor to the
45
reaction free energy, The origin of this is that L- -lysine is simply more stable than
L-lysine, possibly due to a stronger C-N chemical bond. It is speculated that this bond
may be stronger in L- -lysine because of the electron withdrawing effect of the carboxyl
group, making the -carbon more electropositive.
'orG
2) The contribution of the entropic term to is negligible, and actually
favors L-lysine over the product. Recall that the entropic contribution will
reflect the change in the number of microscopic states consistent with the macroscopic
properties as L-lysine is converted to L- -lysine. Generally, the change in the number of
microscopic states could be due to differences in the configurational freedom between L-
lysine and L- -lysine or, could be due to differences in which they are solvated by water.
It appears that neither of these have a significant influence on the thermodynamic
distinction beween between L-lysine and L- -lysine.
'orT S 'o
rG
'orT S
3) The negative value of 'or H means that as the temperature is lowered, the equilibrium
constant will shift to favor L-lysine over L- -lysine because L- -lysine has a lower molar
enthalpy. Hence, for manufacturing purposes, the reaction is more efficient at producing
the desired product, L- -lysine, at lower temperatures. For example, yield of product at
21oC will be 12% higher if the reaction is run at 21oC compared to 37oC. At lower
temperatures, the rate of the reaction is too slow to be practical.
3.10 Transformed Gibbs free energy, enthalpy and entropy of formation
For a limited number of biochemical components, the values of the transformed
Gibbs free energy, enthalpy and entropy of formation have been calculated and are
46
tabulated. These are values referenced to the elements at 298.15K and a pressure of 1 bar,
as discussed in Chapter 1, but take into account the distribution of each protonated state
of the component at the particular pH. If available, these values can be used to calculate
the values of ,'orG
'or H and . Let�’s calculate the transformed reaction Gibbs
free energy for the hydrolysis of ATP at pH 7, 298.15K, 1 bar pressure and ionic strength
of 0.25.M. The values of (kJ/mol) are known for each reaction component under
these conditions (see Appendix I).
'or S
'of G
(3.67)
2
' ' ' ' '2
'
ATP + H O ADP + P
( ) ( ) ( ) (H O
=-1424.7 - 1059.49-(-2292.5)-(-155.66)
35.8 /
i
o o o o or f f i f f
or
G G ADP G P G ATP G
G kJ mol
)
Appendix I is a Table of values at several different pHs. 'of G
3.11 Example: Reactions coupled by enzyme mechanism- acetate-CoA ligase:
The example of tryptophan synthase (Section 3.7.2) is an example of two
reactions catalyzed by the same enzyme, but at different active sites. The two reactions
occur sequentially and they are coupled by virtue of the fact that the substrate of the first
reaction is the product of the second reaction. The enzyme channels the substrate from
one site to another, resulting in accelerating the process, but the thermodynamics are not
altered.
Now let�’s examine another enzyme, acetate-CoA ligase, which couples two
reactions by virtue of the enzyme mechanism. The enzyme catalyzes the following
reaction.
47
ATP + acetate +CoA AMP + PP acetylCoAi (3.68)
The first step of the enzyme mechanism is the covalent attachment of AMP to the acetate,
releasing PPi, and then CoA displaces the AMP, forming acetylCoA. In this way, the
hydrolysis of ATP to AMP + PPi is an obligatory part of the enzyme mechanism.
Looking up the values of for each of the reactants and products (Appendix
I), we can determine the standard state Gibbs free energy of reaction.
'of G
'
' ' ' ' ' ' '
'
( 14160)' (8.31)(298)
( ) ( ) ( ) ( ) ( ) (
= 554.83 66.22 1940.66 ( 2292.5) ( 247.83) ( 7.26)
14.16 /
304o
r
o o o o o o or f f f i f f f
or
GRT
G G AMP G acetylCoA G PP G ATP G acetate G CoA
G kJ mol
K e e
)
We expect the equilibrium constant under standard state temperature, pressure, etc
(Appendix I) to be about 300, favoring the product. Experimental values for the
equilibrium constant are in the range of 1 to 10, probably because the measurement
conditions are different than those tabulated (e.g., inclusion of Mg2+). We can now look
at the two reactions that comprise the net reaction.
48
2
'
' '
2
'
'
I acetate + CoA acetylCoA + H O
( ) 66.22 155.66 ( 247.83) ( 7.26)
( ) 33.21 / 1.5 10
II ATP + H O AMP + PP
( ) 554.83 1940.66 ( 2292.5) ( 155.66)
( ) 4
or
or
i
or
or
G I
G I kJ mol K x
G II
G II ' 87.33 / 2 10kJ mol K x
6
These data indicate that the formation of acetylCoA by the removal of water is highly
unfavorable, but that the reaction is made favorable by its being coupled to the hydrolysis
of ATP to form AMP + PPi.
Under physiological conditions, the Gibbs free energy of reaction depends on the
concentrations of the reactants, as in equation (3.69).
' [ ][ ][14.16 ln[ ][ ][ ]
ir
]AMP PP acetylCoAG RTCoA ATP acetate
(3.69)
If each component in the reaction is present a 1 mM concentration, then
'
'
14.16 ln(1) 14.16 /
304
rG RT kJ
K
mol
If the concentrations of the components are all 1 mM, except for the PPi, which is 1µM,
1000-fold less, then
' 3
' 5
14.16 ln[10 ] 14.16 (8.31)(298)( 6.90) 14.16 17.11
31.27 / ' 3 10
r
r
G RT
G kJ mol K x
49
In fact, under physiological conditions, the concentration of pyrophosphate is kept
very low due to the action of a pyrophosphatase, ( 2 2iPP H O Pi ). Hence, the driving
force for the reaction is increased much further to favor product formation. This is known
as an effect of �“mass action�”. Lowering the concentration of one of the products will
make the driving force for the reaction, , more negative, favoring product formation. 'rG
This is also an example of Le Châtelier�’s Principle, which is that any change of
the system away from equilibrium (in this case removal of the pyrophosphate by the
action of pyrophosphatase) is met by a spontaneous adjustment of the system in the
opposite direction which, in this example, is the generation of more pyrophosphate as the
system moves to regain equilibrium. Other examples of Le Châtelier�’s Principle are
observed if you increase the temperature of the system, the system responds by shifting to
favor components with the higher enthalpy content, or if you increase the pressure on the
system, the system responds by shifting its equilibrium in such a way as to reduce the
volume.
Another way to describe the effect of pyrophosphatase on the reaction catalyzed
by acetyl-CoA ligase is to consider that the maximum work available from the hydrolysis
of ATP is increased by lowering the concentration of PPi. The Gibbs free energy of the
biochemical hydrolysis of ATP to AMP + PPi is
' '
'
[ ][ ]ln[ ]
[ ][47.33 ln[ ]
o ir r
ir
AMP PPG G RTATP
AMP PPG RTATP
]
If the concentration of each component is 1 mM, then
50
' 3
'
47.33 ln[10 ] 47.33 17.11
64.44 /
r
r
G RT
G kJ mol
The transformed Gibbs free energy of the reaction is substantially lower (greater driving
force or work potential) than calculated with standard state concentrations (1 M), .
If the PP
'orG
i concentration is reduced to 1 µM, then the value of is reduced further. The
work capacity from the hydrolysis of ATP is substantially greater under these conditions.
'rG
' 6
'
( ) 47.33 ln[10 ] 47.33 34.21
( ) 81.54 /
r
r
G II RT
G II kJ mol
On the other hand, if the ATP concentration were very low, say 10-6 M, while all the
other components were present at 1 mM, the maximum work per mole of ATP
hydrolyzed would be , and the transformed Gibbs free energy
of reaction for the entire reaction
' ( ) 47.33 /rG II kJ mol
(3.69) would be
' 3
'
14.16 ln[10 ] 14.16 17.11
2.95 /
r
r
G RT
G kJ mol
Under these conditions, the reaction (3.68) will spontaneously go to the left, favoring the
reactants over the products.
3.12 Diagrammatic representations of free energy relationships of coupled reactions It is useful to use free energy diagrams to schematically represent the free energy
relationships in biochemical reactions. We will illustrate this with a hypothetical
51
example in which we have a reaction of A B coupled to the formation of ATP from
ADP + Pi, and a second reaction in which we have B C .
'1
'1
'1
'2
Reaction 1: A + ADP + P B + ATP
1a: A B
1b: ADP + P ATP
Reaction 2: B C
Net reaction: A + ADP + P
or
or a
or b
or
r
i G
G
i G
G
i ' '1 2
B + ATP C + ATP
o orG G
(3.70)
Note that in the last line of (3.70) the second part of the reaction has the same as
the reaction because the ATP remains unchanged and is carried along just as a
formality to balance the line of sequential reactions.
'2o
rG
B C
We would ordinarily think of reaction 1 as one in which the conversion
A B drives the formation of ATP. Reactions 1a and 1b are mechanistically coupled,
so separating the two reactions is a conceptual or mathematical convenience. Any
thermodynamics must consider the sum of the two reactions together, i.e., reaction 1. To
make things more interesting, say that , so that under standard state
conditions (1 M concentrations),
'1o
r a r bG G '1o
' '1 1 0o o
r a r bG G , and the spontaneous direction of the
reaction would be from right to left.
We will also specify that reaction 2 is strongly favored under standard state
conditions, 1 M concentrations, ( '2 0o
rG ). Reactions 1 and 2 are coupled by virtue of
sharing reactants and products so that at equilibrium, both equilibrium constants must be
52
satisfied. We can express the thermodynamic parameters diagrammatically as in Figure
3.14 by showing the relative values of 'o values, recalling that ' '1o o
r a B BG 'o , etc.
Note that
(3.71) ' '1 1o o
r r a rG G G '1ob
1K
1
b
Since , etc., we can also write this as '1 lno
rG RT
1 1
1 1 1 1 1
1 1 1
ln ln ln
ln ln ln ln( )
a b
a b a
a b
RT K RT K RT K
K K K K K
K K K
(3.72)
We can see that when add two reactions together, the free energies of the reactions are
simply added together to obtain the free energy of reaction of the coupled reactions,
which is equivalent to multiplying the equilibrium constants. This is why it is often much
easier to visualize free energy relationships in a simple diagram, since they are related by
simple addition and subtraction. Mathematically, one can also use equilibrium
expressions to get to the same point.
If we have two sequential reactions, as in reactions 1 and 2 in (3.70), the treatment
is the same. Let�’s write the an equilibrium expression for the net reaction in (3.70)
'
' ' '1 2
' ' ' ' '1 2 1 2
' ' '1 2
[ ][ ] [ ][ ] [ ][ ] [ ][ ] [ ][ ][ ][ ] [ ][ ][ ] [ ][ ] [ ][ ][ ] [ ]
and since ln ln ln ln
neti i i
net
or net net
o o or net r r
C ATP B ATP C ATP B ATP CK xA ADP P A ADP P B ATP A ADP P B
K K K
G RT K RT K K RT K RT K
G G G
'
x
(3.73)
53
Figure 3.14: Free energy diagram showing the relative values of the standard state free energy values for the products and reactants of the coupled reactions 1, 1a, 1b and 2. See(3.70). The notation of each species indicates the standard state chemical potential, so �“ATP�” designates 'o
ATP , etc. The direction of the arrows indicates the direction in which the reaction is written, products indicated at the arrowhead. Now let us consider two situations.
1. Equilibrium: If we are given the initial amounts of A, ADP, Pi and, if present
initially, ATP, B and C, would could calculate the concentrations of each biochemical
component at equilibrium. The equilibrium constants for each reaction must be consistent
with the final concentrations. The equilibrium condition is
' ' ' ' ' ' 'iA ADP P B ATP C ATP (3.74)
Since reaction 1 mechanistically couples reactions 1a and 1b, at equilibrium it is '1K that
must be satisfied and not '1 or a
'1bK K separately. However, it is important to note that at
equilibrium, 'netK must be satisfied, in addition to '
1 and '2K K . If reaction 2 strongly favors
formation of C, then '2
[ ] 0[ ]CKB
, which would result in a very low concentration of
components A and B at equilibrium. Reaction 2 pulls reaction 1 forward by depleting
54
component B in solution, and every mole of A converted to B is accompanied by the
formation of one mole of ATP. Hence, even though the thermodynamics of reaction 1
may not favor the phosphorylation of ADP to yield ATP, the sequential reaction can
provide the driving force to do so, even though the coupling is not mechanistic but
through shared reaction components.
2. Steady state: In the context of a metabolic network, components A, B and C
can be participants on other reactions, and the concentrations present under conditions of
steady state will be determined by the rates of the various reactions. If the direction of the
direction is from A B C , then ATP will be generated by reaction 1. Furthermore,
since the directions of reaction 1 and reaction 2 are both from left to right, we know that
the concentrations must be consistent with
' ' ' ' ' ' 'iA ADP P B ATP C ATP (3.75)
since the reaction free energy of each reaction must be less than zero:
. ' '1 20 and 0r rG G
We can visualize both the equilibrium and steady state situations as in the free
energy diagram Figure 3.15, in which we are now showing the relative values of the
transformed chemical potentials with the concentrations specified as either equilibrium
conditions or steady state conditions.
55
Figure 15: Free energy diagram showing the transformed Gibbs chemical potentials of the reactants and products of reactions 1 and 2. The notation of each species indicates the chemical potential, so �“ATP�” designates '
ATP , etc. The top panel represents the equilibrium condition where the system has evolved to the minimum free energy and the reaction free energies of each reaction 1 and reaction 2 is zero. The bottom panel represents a steady state situation as one might encounter within a metabolic network in the steady state in which the direction of the reaction sequence is A B C , which means that the reaction free energies for each reaction 1 and reaction 2 must be < 0. 3.13 Metabolic networks Understanding cellular physiology requires an understanding of how the cell
responds to environmental changes, including alterations in metabolites, for example.
Attempts to mathematically model the kinetics of complex systems such as, for example,
the metabolism of E. coli, yeast or an hepatic cell, are at a relatively early stage, and will
doubtless become more accurate tools for predictive purposes in the future. Motivation is
provided by the need to genetically engineer prokaryotes to carry out desired functions,
such as the biosynthesis of useful molecules. Also, understanding the pathology caused
56
by alterations in enzyme function will require in many cases, an understanding of how
the entire network of coupled biochemical reactions responds to such a change.
There are a number of issues that are critical to the thermodynamic analysis of
complicated systems in which many reactions coupled by sharing reaction components.
1. A useful starting place is knowledge of for all the reactions taking place within
the cell. The reactions can be predicted in many cases from a knowledge of which
enzymes are present. The values of can be readily obtained if the transformed
Gibbs free energies of formation, , which are tabulated for many biochemical
components. However, applies to the standard state concentrations of 1 M
concentration of each reaction component and this is not sufficient to know whether a
particular reaction will go to the left or right (as written) under physiological conditions.
'orG
'of G
'orG
2. In order to know the spontaneous direction of a particular reaction within the
cell, it is necessary to know the concentrations of each reaction component. This turns
out to be a very substantial experimental challenge and, in many cases, it is currently
necessary to either guess a range of reasonable values or to use experimental data which
themselves will have a significant range of error. As a first guess, as we did in the last
section, one could assign a concentration value of 1 mM to each reactant. However, as we
also saw in the example above, this can be wrong.
3. Mathematical modeling of metabolic networks involves many unknowns, and
obtaining any solution, or sets of feasible solutions, requires introducing constraints. For
example, for an hepatic cell making glycogen{Beard, 2005 #33}, we could specify the
rate of glucose utilization. Other constraints are the requirement for mass balance and that
the system is in a steady state. The steady state constraint means that each of the
57
metabolites within the cell is made and consumed at equal rates so that their
concentrations remain constant
Thermodynamics introduces additional constraints. It is necessary that the value
of for each reaction is consistent with the direction of the reaction. This provides a
test to judge the accuracy of both the mathematical modeling as well as the metabolite
concentrations used as input into the model {Kummel, 2006 #35}. It is clear that in
steady state, many of the reactions in metabolic systems run close to equilibrium,
meaning that values are below -10 kJ/mol. Other reactions have much larger values
of , meaning that these reactions are operating under conditions far from
equilibrium. Enzymes operating under conditions far from equilibrium are more likely to
be involved in metabolic regulation {Wang, 2004 #7}, though this is not necessarily the
case.
'orG
'rG
'rG
3.13.1 Examples from hepatocyte metabolism
The metabolism of the liver cell (hepatocyte) depends on physiological needs and
is highly regulated. For example, glucose from the blood can be converted to glycogen
(glycogenesis) and stored or, when needed, glycogen can be hydrolyzed to generate
glucose (glycogenolysis). Glucose can also be synthesized from substrates such as
lactate, and the required ATP and NADH can be provided, for example, by the oxidation
of fatty acids and the TCA cycle. Our interests at this point is very limited but students
interested in this have a large and growing literature to examine, e.g. {Beard, 2004 #3;
Chalhoub, 2007 #6; Henry, 2007 #5}. In these different metabolic pathways, in a number
of instances the same enzymes are used to catalyze the next flux in opposite directions.
58
We will consider a couple of examples of reactions that are part of the network of
reactions in the hepatocyte.
1) Glycolysis: The set of reactions by which glucose is converted to pyruvate
have a range of values (0.25M ionic strength, pH 7) ranging from about -24
kJ/mol to +23 kJ/mol for the reactions as shown in Figure 3.16 and Table 3.3. Adding up
all the individual reactions yields the net value of (one must
consider each of the last 5 reactions twice).
'orG
' 80.6 /orG kJ mol
(3.76) 2
'
glucose + 2 P + 2 ADP + 2 NAD 2 pyruvate + 2 ATP + 2 NADH + 2 H O
80.6 /
i
orG kJ mol
One model {Beard, 2004 #3}, used simply to be illustrative, determined the
steady state conditions that would apply to the physiological state when glucose is taken
up by the cells and converted to glycogen for storarge. About 10% of the glucose is
broken down by glycolysis to form pyruvate, which is further oxidized by the TCA cycle
in the mitochondria. This provides the ATP required for glycogen formation and is
required for cell maintenance. Given the assumptions for glucose uptake and glycogen
formation and that there is a steady state condition (each intermediate metabolite is being
made and consumed at equal rates), the model yielded concentrations and values of the
transformed Gibbs free energies of reaction, . These are shown in Table 3.3. Note
that the values of are all negative, as they must be for glycolysis to proceed, and
most are very small. Most of the reactions under physiological conditions, according to
this particular model are operating near equilibrium. This is generally found for
'rG
'rG
59
metabolic systems, by modeling or, where possible, using experimentally determined
concentrations.
Figure 3.16: Schematic of the glycolysis pathway, from glucose to pyuvate, also showing glycogen synthesis and hydrolysis pathways in the hepatic cell, all intersecting at G6P. The values of and calculated values for a steady 'o
rG'
rGstate model of the metabolic reactions are in Table 3.3. Abbreviations: GLC, glucose; G6P, glucose 6-phosphate; G1P, glucose 1-phosphate; F6P, fructose 6-phosphate; F1,6P, fructose 1,6-bisphosphate; T3P1, glyceraldehyde 3-phosphate; T3P2, dihydroxyacetone phosphate; 1,3DPH, 1,3 bisphosphoglycerate; 3PG, 3 phosphoglycerate; 2PG, 2 phosphoglycerate; PEP, phosphoenolpyruvate; PYR, pyruvate.
60
The major exception in Table 3.3 is the reaction catalyzed by phosphofructokinase to
form fructose 1,6-bisphosphate. This enzyme is known to be highly regulated. Although
the sign of determines the direction of the reaction, the absolute rate of conversion
of the substrate to products is not determined by but by the properties of the enzyme
catalyzing the reaction (Michaelis-Menten parameters, V
'rG
'rG
max and KM) and by the amount
of enzyme present.
Table 3.3: Comparison of the standard transformed reaction Gibbs free energy of reaction of the enzyme-catalyzed reactions in glycolysis and the transformed reaction Gibbs free energy under physiological conditions in an hepatic cell synthesizing glycogen, calculated using metabolic modeling {Beard, 2005 #33}. REACTIONS in GLYCOLYSIS 298.15K, pH 7, 0.25 ionic strength
Standard transformed reaction Gibbs free energy
'orG (kJ/mol)
Steady state model Transformed reaction Gibbs free energy
'rG (kJ/mol)
Glucose + ATP Glucose 6-phosphate + ADP
-24.42 -28.63
Glucose 6-phosphate fructose 6-phosphate
3.19 -6.52
fructose 6-phosphate + ATP fructose 1,6-bisphosphate + ADP
-23.25 -19.44
fructose 1,6-bisphosphate dihydroxyacetone phosphate + glyceraldehyde phosphate
23.03
-0.75
dihydroxyacetone phosphate glyceraldehyde phosphate
7.66 -0.29
glyceraldehyde phosphate + NADox +Pi 1,3-bisphosphoglycerate + NADred
1.12 -0.13
1,3-bisphosphoglycerate + ADP 3-phosphogycerate + ATP
-8.22
-0.42
3-phosphogycerate 2-phosphogycerate 5.94 -0.16 2-phosphogycerate phosphoenolpyruvate
+ H2O -3.60 -1.36
phosphoenolpyruvate + ADP pyruvate + -28.85 -0.75
61
ATP
2) Glycogenesis versus Glycogenolysis: Let's consider the reactions involving
glucose 6-phosphate under conditions favoring formation of glycogen (Figure 3.16).
When glycogen is being synthesized, glucose is converted to G6P by glucose kinase,
which is then converted to G1P by phosphoglucomutase. The values of are
calculated to be negative under the steady state reaction conditions, consistent with the
direction of the spontaneous reactions. However, the reaction catalyzed by glucose 6-
phosphatase, hydrolyzing G6P to glucose + P
'rG
i is also negative .
This enzyme must be regulated under the physiological conditions in which glycogen is
synthesized to prevent the hydrolysis of G6P. When the physiological demands are
different and glucose is produced from glycogen, then the hydrolysis of G6P to glucose +
P
' 25.7 /rG kJ mol
1
i is still thermodynamically favored and, in this case, the enzyme is present and active.
Note that the reaction catalyzed by phosphoglucomutase, , must be
favored in opposite directions, depending on whether glycogen is being synthesized from
glucose or being hydrolyzed go form glucose. When glycogen is being hydrolyzed and
the enzyme glucose 6-phosphatase is active (
6G P G P
26 iG P H O GLC P ),the steady state
concentration of G6P is maintained at a low value due to its hydrolysis to glucose + Pi ,
and the value of favors the conversion of G1P to G6P. 'rG
Let�’s quantify this as an exercise. We can start with the values of the standard
state free energies of formation, which are in Appendix I .
62
´
´
´ ´ ´1 6
for G6P: 1318.92 /
for G1P: 1311.89 /
hence, for the reaction G6P G1P
( 1311.89 ( 1318.92)) 7.03 /
of
of
o o or f G P f G P
G kJ mol
G kJ mol
G G G kJ mol
From the model of hepatic cell metabolism in {Beard, 2004 #3} steady state
concentrations are calculated. Under conditions in which glycogen is synthesized
3
' '3
'
[ 6 ] 9.91 and [ 1 ] 0.0372
[ 1 ] (0.0372 10 )ln 7.03 (8.31)(298) ln[ 6 ] (9.91 10 )
5.97 /
or r
r
G P mM G P mM
G P xG G RTG P x
G kJ mol
The direction of the reaction is towards making G1P as the product. The fact the
concentration of G6P is high and the G1P concentration is kept low by the reactions
leading to the formation of glycogen is the key to this working. Under the conditions
where glycogen is hydrolyzed and glucose is formed the concentration of G6P is lower
and G1P is considerably higher since this is the product of glycogen hydrolysis.
3
' '3
'
[ 6 ] 3.3 and [ 1 ] 1.72
[ 1 ] (1.72 10 )ln 7.03 (8.31)(298) ln[ 6 ] (3.3 10 )
5.42 /
or r
r
G P mM G P mM
G P xG G RTG P x
G kJ mol
The higher concentration of G1P changes the transformed reaction Gibbs free energy to a
positive value, meaning that the reaction forms G6P as the product.
63
In metabolic networks, there is an intricate interplay between the rates that
enzymes catalyze each reaction, the resulting steady state concentrations, and the
thermodynamics of the reactions. The values of the standard transformed Gibbs reaction
free energies, , remain fixed but the actual transformed reaction Gibbs free energies
are entirely dependent on the concentrations of the reacting species. In many cases, the
concentrations result in values of are small, near zero, meaning that the{Chalhoub,
2007 #6} driving force is not large, but also resulting in a situation where relatively small
changes in concentrations of the reactants and products will result in reversing the
direction of the reaction. In a number of cases where the values of are large and
negative, these reactions are often at the beginning of a linear sequence of reactions and
the enzymes catalyzing these reactions are subject to metabolic regulation{Henry, 2007
#5} .
'orG
'rG
'rG
3.14 Summary
In this Chapter we have shown how minimizing the Gibbs free energy for a
system of reactants leads naturally to the concept of an equilibrium constant and a Gibbs
free energy of reaction. Utilizing these concepts requires that we select standard states in
such a way to simplify solving problems. For biochemical reactions, there are special
conditions, because the reactions occur in buffered aqueous solution. This means that the
number of bound protons can be increased or decreased as a reaction proceeds, but the
number of free protons does not change. This requires the use of the transformed Gibbs
free energy. The benefit of this, however, is that it allows us to deal with groups of
64
chemical species that are biochemically equivalent as reaction components. This greatly
simplifies dealing with biochemical systems.
Finally, we saw several ways in which biochemical reactions in the cell are
coupled. This can be through an obligatory mechanism of an enzyme, or, more often,
through sharing common reactants. The implication is that to understanding metabolic
systems it is necessary to treat the entire system (metabolomics or systems biology).
Thermodynamics provides constraining information necessary for this purpose.
65
