.ig
:!cat %|tbl|eqn|troff |dpost >%.ps;ghostview %.ps&
:!cat %|tbl|eqn|troff |dpost >%.ps
..
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.EQ
delim $$
.EN
.EQ
define vmodel	% v sup model %
define fmodel	% f sup model %
define vobs	% v sup observed %
define fobs	% f sup observed %
define KA	% K sub m sup A %
define KB	% K sub m sup B %
define KAB	% K sub m sup AB %
define KiA	% K sub i sup A %
define KiB	% K sub i sup B %
define VappA	% V sub app sup A %
define VappB	% V sub app sup B %
define Vapp	% V sub app %
define v0	% v sub o %
define e0	% e sub o %
define Vf	% V sup f %
define Vr	% V sup r %
define KP	% K sub m sup P %
define KQ	% K sub m sup Q %
define KPQ	% K sub m sup PQ %
define k1	% k sub 1 %
define k2	% k sub 2 %
define k3	% k sub 3 %
define k4	% k sub 4 %
define k_1	% k sub -1 %
define k_2	% k sub -2 %
define k_3	% k sub -3 %
define k_4	% k sub -4 %
.EN
.ce
\fBAPPENDIX\fP
.SP 2
.ce
\fBEQUATIONS, FIGURES AND TABLES\fP
.ce
.SP
\fBfor\fP
.SP 3
.ce
\fBSTEADY STATE KINETICS:\fP
.ce
\fBTWO-SUBSTRATE, TWO-PRODUCT REACTIONS\fP
.SP 4
.ce
\fBDIAGRAMS OF MECHANISMS\fP
.SP
.ce
\fBINITIAL RATE EQUATIONS\fP
.SP
.ce
\fBDERIVATION BY KING-ALTMAN METHOD\fP
.SP
.ce
\fBANALYSIS BY PRIMARY AND SECONDARY PLOTS\fP
.SP
.ce
\fBFULL RATE LAW\fP
.SP
.ce
\fBPRODUCT INHIBITION\fP
.SP
.ce
\fBTABLES\fP
.SP 
.SK
.ce
\fBDIAGRAMS OF MECHANISMS\fP
.BP mech_1.eps 7i 9.5i c
.SK
.BP mech_2.eps 7i 9.5i c
.SK
.pl +20 \" to keep table all on one page
.ce
\fBINITIAL RATE EQUATIONS\fP
.P
The initial rate law for the forward direction (starting concentrations
of products, $p$ and $q$, equal to zero) for the random order (eq. 2),
compulsory order (eq. 3) and Theorell-Chance (eq. 4) mechanisms is
.SP
.EQ (7)
v0 ~=~ V over {1 ~+~ KA over a ~+~ KB over b ~+~ KAB over {a b}}
.EN
.SP
The expression of the parameters of eq. (7) in terms of the step
constants differs according to mechanism.
.SP
.TS
box,center;
cfB s s
c c c
c c c.
Relationship of kinetic parameters to mechanism step constants
_
parameter	compulsory order (eq. 3)	Theorell-Chance (eq. 4)
_
.sp
$V$	${k3 k4 e0} over {k3 ~+~ k4}$	$k3 e0$
.sp
$KA$	${k3 k4} over {k1 ( k3 ~+~ k4 )}$	$k3 over k1$
.sp
$KB$	${k4 ( k_2 ~+~ k3 )} over {k2 ( k3 ~+~ k4 )}$	$k3 over k2$
.sp
$KAB$	${k_1 k4 ( k_2 ~+~ k3 )} over {k1 k2 ( k3 ~+~ k4 )}$	${k_1 k3} over {k1 k2}$
.sp
_
.sp
$KiA ~=~ KAB over KB$	$k_1 over k1$	$k_1 over k1$
.sp
$KiB ~=~ KAB over KA$	${k_1 ( k_2 ~+~ k3 )} over {k2 k3}$	$k_1 over k2$
.sp
.TE
.pl \" restore old page length
.SK
.ce
\fBDERIVATION BY KING-ALTMAN METHOD\fP
.SP
.ce
\fBINITIAL RATE LAW\fP
.SP
.ce
\fBCOMPULSORY ORDER MECHANISM IN THE FORWARD DIRECTION\fP
.P
Construct the King-Altman patterns for the compulsory order mechanism
(eq. 3 or equivalently, eq. 6b).  Because for the forward direction the
initial concentrations of products, $p$ and $q$, are zero and steps 3
and 4 are unidirectional, there are only the following seven patterns:
.BP mech_pat_ordered.eps
From inspection of the patterns, construct expressions for the
fraction of total enzyme in each enzyme form:
.SP
.EQ
[E] over e0 "\h'|.7i'" =~ {k_1 k_2 k4 ~+~ k_1 k3 k4 ~+~ k2 k3 k4 b} over D
.EN
.SP
.EQ
[EA] over e0 "\h'|.7i'" =~ {k1 k_2 k4 a ~+~ k1 k3 k4 a} over D
.EN
.SP
.EQ
[EAB] over e0 "\h'|.7i'" =~ {k1 k2 k4 a b} over D
.EN
.SP
.EQ
[EQ] over e0 "\h'|.7i'" =~ {k1 k2 k3 a b} over D
.EN
.SP
.EQ
D "\h'|.7i'" =~ k_1 k4 ( k_2 ~+~ k3 ) ~+~ k1 k4 ( k_2 ~+~ k3 ) a ~+~
k2 k3 k4 b ~+~ k1 k2 ( k3 ~+~ k4 ) a b
.EN
.SP
The reaction rate is equal to the rate of breakdown of the $EAB$ complex
.SP
.EQ
v0 ~=~ k3 [EAB]
.EN
.SP
Substituting for $[EAB]$
.SP
.EQ
v0 ~=~ {k1 k2 k3 k4 a b e0 } over D
.EN
.SP
Dividing numerator and denominator by $k1 k2 ( k3 ~+~ k4 ) a b$, 
to throw the expression for $v0$ into the form of eq. (7)
.SP
.ll +3
.EQ (7a)
v0 ~=~ {{k3 k4 e0} over {( k3 ~+~ k4 )}} over
{left [ 
1 ~+~ 
{k3 k4} over {k1 ( k3 ~+~ k4 ) } 1 over a ~+~
{k4 ( k_2 ~+~ k3 )} over {k2 ( k3 ~+~ k4 )} 1 over b ~+~
{k_1 k4 ( k_2 ~+~ k3 )} over {k1 k2 ( k3 ~+~ k4 )} 1 over {a b }
right ]}
.EN
.ll
.SP
The relationship of the kinetic parameters of eq. (7) 
to the step constants is made clear by comparison of eqs. (7) and (7a).
.SK
.ce
\fBDERIVATION BY KING-ALTMAN METHOD\fP
.SP
.ce
\fBINITIAL RATE LAW\fP
.SP
.ce
\fBTHEORELL-CHANCE MECHANISM IN THE FORWARD DIRECTION\fP
.P
Construct the King-Altman patterns for the Theorell-Chance mechanism
(eq. 4 or equivalently, eq. 6a).  
Again, because the initial concentrations of products, $p$ and $q$, 
are zero and steps 3 and 4 are unidirectional, 
there are only the following four patterns:
.BP mech_pat_tc.eps
The expressions for the fraction of total enzyme in each enzyme form are:
.SP
.EQ
[E] over e0 "\h'|.7i'" =~ {k_1 k3 ~+~ k2 k3 b} over D
.EN
.SP
.EQ
[EA] over e0 "\h'|.7i'" =~ {k1 k3 a} over D
.EN
.SP
.EQ
[EQ] over e0 "\h'|.7i'" =~ {k1 k2 a b} over D
.EN
.SP
.EQ
D "\h'|.7i'" =~ k_1 k3 ~+~ k1 k3 a ~+~ k2 k3 b ~+~ k1 k2 a b
.EN
.SP
The reaction rate is equal to the rate of breakdown of the $EQ$ complex
.SP
.EQ
v0 ~=~ k3 [EQ]
.EN
.SP
Substituting for $[EQ]$
.SP
.EQ
v0 ~=~ {k1 k2 k3 a b e0 } over D
.EN
.SP
Dividing numerator and denominator by $k1 k2 a b$, 
.SP
.EQ (7b)
v0 ~=~ {k3 e0} over
{left [ 
1 ~+~ 
k3 over k1 1 over a ~+~
k3 over k2 1 over b ~+~
{k_1 k3} over {k1 k2} 1 over {a b }
right ]}
.EN
.SP
Compare eqs. (7) and (7b).
.SK
.ce
\fBANALYSIS BY PRIMARY AND SECONDARY PLOTS\fP
.SP
.ce
\fBLACTATE DEHYDROGENASE AS AN EXAMPLE\fP
.SP
.ce
\fBI\fP. FIXED $b = [pyruvate]$, VARIED $a = [NADH]$
.SP
.ul
PRIMARY PLOT:
.P
Taking the reciprocal of eq. (7)
.SP
.EQ
1 over v0 ~=~ 1 over V ~+~ 1 over V KA over a ~+~ 1 over V KB over b ~+~ 1
over V KAB over {ab}
.EN
.SP
Equating $a$ with $[NADH]$ and $b$ with $[pyruvate]$, and
collecting terms in $1 / [NADH]$,
.SP
.EQ (8)
1 over v0 ~=~  1 over [NADH] 1 over V left ( KAB over [pyruvate] ~+~ KA right
) ~+~ 1 over V left ( KB over [pyruvate] ~+~ 1 right )
.EN
.SP
For initial rate data (values of $v0$) measured over ranges of
concentrations of pyruvate and NADH, 
eq. (8) defines a family of lines of $1 / v0$ \fIvs\fP $1 / [NADH]$.
.BP mech_pri_a.eps
Each line represents a different fixed concentration of pyruvate.
The slopes of the lines decrease with increasing pyruvate concentration,
as required by the quantity in parenthesis of the first term
on the right hand side of eq. (8).
.SP
The intercept on the $1 / v0$ axis is given by the second term on
the rhs of eq. (8):
.SP
.EQ (8a)
1 over VappA ~=~ 1 over V left ( KB over [pyruvate] ~+~ 1 right )
.EN
.SP
The intercept, like the slope and for a similar reason, decreases
with increasing pyruvate concentration.
.SP
The lines of constant $[pyruvate]$ meet in quadrant II or III
at the point with coordinates
.SP
.EQ (8b)
left ( 1 over v0 , ~ 1 over [NADH] right ) sub intersection ~=~ 
left ( { 1 over V left ( 1 ~-~ {KA KB} over KAB right ) } , ~ 
- KB over KAB right )
.EN
.SP
(To show this, expand eq. (8), collect terms in $1 / [pyruvate]$,
find the value of $1 / [NADH]$ that makes zero the 
coefficient of $1 / [pyruvate]$, and then find the value of
$1 / v0$ for this value of $1 / [NADH]$.)
.SP
The kinetic parameter $KiA$ is defined by the point of intersection
.SP
.EQ (8c)
KiA ~=~ KAB over KB ~=~ -[NADH] sub intersection
.EN
.SP 2
.ul
SECONDARY PLOT:
.P
Expanding the rhs of eq. (8a)
.SP
.EQ (9)
1 over VappA ~=~ 1 over V KB over [pyruvate] ~+~ 1 over V
.EN
.SP
Thus a plot of the $1 / VappA$ values determined from the primary
plot \fI vs\fP $1 / [pyruvate]$ defines a single line, with
$1 / v0$ intercept equal to $1 / V$, and $1 / [pyruvate]$
intercept equal to $- ( 1 / KB )$.
.BP mech_sec_a.eps
Note that three of the four parameters of eq. (7) are determined
by one pair of primary and secondary plots, 
in this case, $KB$ and $V$ from
the secondary plot and $KAB$ from $KB$ and the value of $KiA$ 
from the primary plot.
.SK
.ce
\fBII\fP. VARIED $b = [pyruvate]$, FIXED $a = [NADH]$
.SP
.ul
PRIMARY PLOT:
.P
As with eq. (8), but collecting terms in $1 / [pyruvate]$,
.SP
.EQ (10)
1 over v0 ~=~  1 over [pyruvate] 1 over V left ( KAB over [NADH] ~+~ KB right
) ~+~ 1 over V left ( KA over [NADH] ~+~ 1 right )
.EN
.SP
Similar to eq. (8),
eq. (10) defines a family of lines of $1 / v0$ \fIvs\fP $1 / [pyruvate]$,
each line for a fixed concentration of NADH.
.BP mech_pri_b.eps
The intercepts on the $1 / v0$ axis, the coordinates of the point of
intersection, and the kinetic parameter $KiB$ are given by
eqs. (10a-c).  
.SP
.EQ (10a)
1 over VappB ~=~ 1 over V left ( KA over [NADH] ~+~ 1 right )
.EN
.SP
.EQ (10b)
left ( 1 over v0 , ~ 1 over [pyruvate] right ) sub intersection ~=~ 
left ( { 1 over V left ( 1 ~-~ {KA KB} over KAB right ) } , ~ 
- KA over KAB right )
.EN
.SP
.EQ (10c)
KiB ~=~ KAB over KA ~=~ -[pyruvate] sub intersection
.EN
.SP
Note that the $1 / v0$ coordinate of the intersection
point is the same for cases I and II, i.e., regardless of which substrate
is fixed.
This requirement of eqs. (8) and (10) must be taken into account when
drawing the lines of the primary plots.
.SP 2
.ul
SECONDARY PLOT:
.P
The values of $1 / VappB$ determined from the primary plot are
described by the equation
.SP
.EQ (11)
1 over VappB ~=~ 1 over V KA over [NADH] ~+~ 1 over V
.EN
.SP
A plot of $1 / VappB$ \fIvs\fP $1 / [NADH]$
gives a line with
$1 / v0$ intercept equal to $1 / V$, and $1 / [NADH]$
intercept equal to $- ( 1 / KA )$.
.BP mech_sec_b.eps
Values of $KA$ and $V$ are determined from
the secondary plot and $KAB$ from $KA$ and the value of $KiB$ 
from the primary plot. 
Note that the graphical procedures of case I and case II
each determine values for $V$ and $KAB$.
This duplication can be used as a check on the analysis or
can be taken into account in construction of the graphs.
.SK
.ce
\fBFULL RATE LAW\fP
.SP
.ce
\fBFOR MECHANISMS OF EQ. (1-4)\fP
.SP 2
.P
A King-Altman analysis for the two-substrate, two-product mechanisms,
without the restriction that the 
starting concentrations of products, $p$ and $q$,
be equal to zero, gives the following initial rate laws.
.SP
For compulsory order:
.SP
.EQ
v0 ~=~ 
left [ { 
{V sup f a b} over KAB  ~-~ {V sup r p q } over KPQ 
} right ]
~~~
"\v'1'\h'-.5'\D'l 1 -2'\v'1'"
left [ { 
1 ~+~ 
KB over KAB a ~+~ KA over KAB b ~+~ 1 over KAB ab ~+~ 
} right ""
.EN
.SP
.EQ
"\h'2.75i'"
KP over KPQ q ~+~ KQ over KPQ p ~+~ 1 over KPQ pq ~+~
.EN
.SP
.EQ
"\h'2.75i'"
{KA KP} over {KAB KPQ} bq ~+~ {KB KQ} over {KAB KPQ} ap ~+~
.EN
.SP
.EQ (12)
"\h'2.75i'"
left "" {
KQ over {KAB KPQ} abp ~+~ KA over {KAB KPQ} bpq
} right ]
.EN
.SP
For random order, omit terms in: bq, ap, abp, bpq.
.SP
For Theorell-Chance, omit terms in: abp, bpq.
.SP
For substituted enzyme, omit terms in: abp, bpq, and the constant term, 1.
.SP
Note the symmetry between reactant and product contributions.
Note also that eq. (7) is obtained from eq. (12) by setting $p$ and $q$ at
zero and multiplying numerator and denominator by $KAB / ab$.
.SK
.ce
\fBPRODUCT INHIBITION\fP
.SP 2
.ce
\fBI\fP. PRODUCT INHIBITION BY PRODUCT $P$, WITH $q$ = 0.
.P
Product inhibition by be described for a two-substrate, two-product
mechanism by setting the initial concentration of one product
at zero in eq. (12).  Here, set $q$ = 0 in eq. (12), in order to describe
inhibition
by the first product, $P$.  
Reciprocal relationships are given below.
.SP
For compulsory order:
.SP
.EQ
1 over v0 ~=~ 1 over V left [ { 1 ~+~ KQ over KPQ p ~+~ KA over a ~+~ 
KB over b left ( 1 ~+~ KQ over KPQ p right )  ~+~ } right ""
.EN
.EQ (13a)
"\h'2i'" left "" { KAB over {ab} left ( 1 ~+~ KQ over KPQ p right ) } right ]
.EN
.SP
For random order:
.SP
.EQ (13b)
1 over v0 ~=~ 1 over V left [ { 1 ~+~ KA over a ~+~ 
KB over b ~+~ 
KAB over {ab} left ( 1 ~+~ KQ over KPQ p right ) }
right ]
.EN
.SP
For Theorell-Chance:
.SP
.EQ (13c)
1 over v0 ~=~ 1 over V left [ { 1 ~+~ KA over a ~+~ 
KB over b left ( 1 ~+~ KQ over KPQ p right )  ~+~ 
KAB over {ab} left ( 1 ~+~ KQ over KPQ p right ) }
right ]
.EN
.P
Inspection of eq. (13a-c) verifies line 4 of Table I, 
for $a$ fixed, $b$ varied, $q$ zero, and $p$ non-zero.
For the random order and Theorell-Chance mechanisms,
a sufficiently high
concentration of $B$ will make negligible any contribution from $P$,
because every term in eq. (13b-c) containing $p$ in the numerator has $b$
in the denominator.
Eq. (13b-c) show that for fixed $a$ and varied $b$, the
presence of product $P$ as inhibitor ($p$ non-zero) has no effect
on the $1 / v0$ intercept and the inhibition is competitive
(entries of "C" in columns 3 and 5 of line 4 of Table I).
In contrast, for the compulsory order mechanism, there is a term
with $p$ in the numerator 
that does not contain $b$ in the denominator
(the second term within the rhs brackets of eq. 13a).
Thus for the compulsory order mechanism, for fixed $a$ and varied $b$,
product $P$ is a non-competitive inhibitor (entry of "NC" in column 4
of line 4 of Table I).
Product inhibition by $P$ with $B$ the varied substrate distinguishes
the compulsory order mechanism from the random order and Theorell-Chance
mechanisms. 
In the absence of product inhibitor the three mechanisms are 
indistinguishable through steady-state kinetic measurements.
.P
Inspection of terms with $p$ in eq. (13a-c) shows that inhibition
by $P$ with $a$ varied shows noncompetitive behavior (terms without
$a$ in the denominator) for the
compulsory order and Theorell-Chance mechanisms and competitive behavior
for the random order mechanism (in agreement with line 3, columns 3-5
of Table I).
.SK
.ce
\fBII\fP. PRODUCT INHIBITION BY PRODUCT $Q$, WITH $p$ = 0.
.P
A similar analysis with $p$ = 0 describes inhibition by $Q$, the
second product.
.SP
For compulsory order:
.SP
.EQ (14a)
1 over v0 ~=~ 1 over V left [ { 1 ~+~ 
KA over a left ( 1 ~+~ KP over KPQ q right ) ~+~ 
KB over b ~+~ 
KAB over {ab} left ( 1 ~+~ KP over KPQ q right ) }
right ]
.EN
.SP
For random order:
.SP
.EQ (14b)
1 over v0 ~=~ 1 over V left [ { 1 ~+~ KA over a ~+~ 
KB over b ~+~ 
KAB over {ab} left ( 1 ~+~ KP over KPQ q right ) }
right ]
.EN
.SP
For Theorell-Chance: as for compulsory order.
.SP
Lines 1 and 2 of Table I can be verified by inspection of eq. (14a-b).
.SK
.SP 5
.ce
\fBTable I. Product Inhibition Patterns\fP
.SP 2
.TS
delim($$), center, box;
c c | c s s
c c | c c c .
		Inhibition Pattern
		_	_	_
		random	compulsory	Theorell-
variable	product	order	order	Chance
substrate	inhibitor	(eq. 2)	(eq. 3)	(eq.4)
_
.sp
A	Q	C	C	C
.sp
B	Q	C	NC	NC
.sp
A	P	C	NC	NC
.sp
B	P	C	NC	C
.sp
.TE
.BP mech_tbl.eps 
