Supporting Information
for
Stochastic mapping of the Michaelis-Menten mechanism
Éva Dóka, Gábor Lente*
Department of Inorganic and Analytical Chemistry, University of Debrecen, Debrecen,
Hungary. Fax: 36-52-518-660; Tel: 36-52-512-900 Ext.: 22373; E-mail:
lenteg@delfin.unideb.hu
Derivation of equations
Equations (1)-(3):
These are literature equations.
Equation (4):
The number of all possible states can be derived by counting all possible (e,s) pairs that
are physically meaningful. (e,s) is physically meaningful if e, s, es, and p are nonnegative integers.
e0
s0
es = e0 – e  0  e  e0
p = s0  s  e0 + e  0  s  e  s0  e0
The number of possibly states is counted in the following table:
value of s
possible values of e
number of cases
0
0, 1, ....., e0
e0 + 1
1
0, 1, ....., e0
e0 + 1
.......j
.......0, 1, ....., e0
.......e0 + 1
s0  e0
0, 1, ....., e0
e0 + 1
s0  e0 + 1
1, 2, ....., e0
e0
s0  e0 + 2
2, 3, ....., e0
e0  1
.......s0  e0 + j
.......j, j+1, ....., e0
.......e0  j + 1
s0
e0
1
Therefore, the overall number of possible cases is:
e0
m  ( s0  e0  1)(e0  1)   i  ( s0  e0  1)(e0  1) 
i 1
 ( s0 
e0
 1)  (e0  1)
2
This is identical to equation (3).
S2
e0 (e0  1)
2
Equation (5):
The master equation can simply be derived from the scheme of the reaction with the
transition probabilities. The following scheme shows the possible transitions.
S3
Equation (6):
A simple substitution into the formula can be used to check the validity of this
enumerating function. For example, for s0 = 6 and e0 = 3 (m = 22)
( s 0  s  e0  e  1)(e0  1)  e


f (e, s)  
(e  s  1)(e  s)
e
( s0  s  e0  e  1)(e0  1) 
2
e
3
2
1
0
3
2
s
6
5
4
3
5
4
f(e,s)
1
2
3
4
5
6
e
1
0
3
2
1
0
s
3
2
4
3
2
1
f(e,s)
7
8
9
10
11
12
e
3
2
1
0
3
s
3
2
1
0
2
f(e,s)
13
14
15
16
17
e
2
1
3
2
3
if
es
if
es
s
1
0
1
0
0
f(e,s)
18
19
20
21
22
The logic behind the enumerating function can be seen better if it is given in terms of es
(= e0 – e) and p (= s0  s  e0 + e).
p(e0  1)  es  1


( s 0  e0  p  1)( s 0  e0  p)
f ( e, s )  
p
(
e

1
)

es

1

0

2
p
0
1
2
....
s0e0
s0e0+1
....
s01
s0
if
p  s 0  e0
if
p  s 0  e0
es
0
1
2
....
e0  1
e0
(e,s)
f(e,s)
(e,s)
f(e,s)
(e,s)
f(e,s)
(e0,s0)
1
(e0,s01)
(e0+1)+1
(e0,s02)
2(e0+1)+1
....
(e0,e0)
(s0e0)(e0+1)+1
(e0,e01)
(s0e0+1)(e0+1)+1
....
(e0,1)
(s0e0/2+1)(e0+1)2
(e0,0)
(s0e0/2+1)(e0+1)
(e01,s01)
2
(e01,s02)
(e0+1)+2
(e01,s03)
2(e0+1)+2
....
(e01,e01)
(s0e0)(e0+1)+2
(e01,e02)
(s0e0+1)(e0+1)+2
....
(e01,0)
(s0e0/2+1)(e0+1)1
-
(e02,s02)
3
(e02,s03)
(e0+1)+3
(e02,s04)
2(e0+1)+3
....
(e02,e02)
(s0e0)(e0+1)+3
(e02,e03)
(s0e0+1)(e0+1)+3
....
-
....
(1,s0e0+1)
e0
(1,s0e0)
(e0+1)+e0
(1,s0e01)
2(e0+1)+e0
....
(1,1)
(s0e0)(e0+1)+e0
(1,0)
(s0e0+1)(e0+1)+e0
....
-
(0,s0e0)
e0+1
(0,s0e01)
(e0+1)+e0+1
(0,s0e02)
2(e0+1)+e0+1
....
(0,0)
(s0e0)(e0+1)+e0+1
-
-
....
(e,s)
f(e,s)
(e,s)
f(e,s)
(e,s)
f(e,s)
(e,s)
f(e,s)
S4
....
....
....
....
....
-
....
-
Equation (7)-(10):
These are written using the statistical definition of expectation and standard deviation.
Equation (11):
This is the assumption made in the approximation. The analogy of this with the
deterministic steady-state assumption is given by the fact that S values satisfy the
following equation:
 k

k
0    1 es  (k 1  k 2 )(e0  e) S e0 e ,s0  s e e0  1 (e  1)( s  1) S e0 e1,s0  s e e0 
N AV
 N AV

 k 1 (e0  e  1) S e0 e1,s0  s e e0  k 2 (e0  e  1) S e0 e 1,s0  s e e0
Equation (12):
This equation is derived from equation (5), into which equation (11) is substituted:
dRs0  s  ee0 (t )
dt
 k

S e0 e ,s e  e0    1 es  (k 1  k 2 )(e0  e) Rs0  s  e e0 (t ) S e0 e ,s0  s  e e0 
 N AV

k1
(e  1)( s  1) Rs0  s  e e0 (t ) S e0 e 1,s0  s  e e0  k 1 (e0  e  1) Rs0  s  e e0 (t ) S e0 e 1,s0  s  e e0 
N AV
 k 2 (e0  e  1) Rs0  s  ee0 1 (t ) S e0 e 1,s0  s  e e0 1
Summing all the equations from e = 0 to e =  with p = s  e + e0 held constant (the
definition of  is given in equation (13) in the main text) gives:
dR p (t )
dt

α
 Si ,p  
i 0


k1
R p (t ) ((e0  i )( s 0  p  i ) S i , p )  (k 1  k 2 ) R p (t ) (iS i , p ) 
N AV
i 0
i 0
α
α 1
k1
R p (t ) ((e0  i  1)( s 0  p  i  1) S i 1, p )  k 1 R p (t ) ((i  1) S i 1, p ) 
N AV
i 1
i 0
α 1
 k 2 R p 1 (t ) ((i  1) S i 1, p 1 )
i 0
This equation contains most of the terms with both positive and negative signs. It must
also be remembered that equation (13) ensures that (e0  )(s0  p  ) = 0.
dR p (t )
dt
α

α
i 0
i 0
i 1
 Si , p  k 2 R p (t ) (iS i , p )  k 2 R p1 (t ) (iS i , p1 )
S5
Considering the definition given in equation 13 and the fact that

S
i 0
i,p
 1 , this is
identical to equation (12).
Notes on solving equation (12):
This equation is easily solved in an iterative fashion. The general solution is:
p
R p (t )   Ap ,i e
 k 2 ES i t
i 0
The coefficients Ap,i can be calculated iteratively:
A0 ,0  1
A p 1,i ES p 1
for i  p A p ,i 
ES p  ES i
for
i  p
p 1
A p ,i   A p ,i
i 0
Equation (13):
This equation displays the definition of the expectation for the number of ES molecules at
constant p.
Equation (14):
This equation can be directly obtained from statistical thermodynamics using the concept
of partition functions. The following considerations are necessary for this:
The energy term is directly related to the Michaelis constant:
e i  e iG / kT  (e G / kT )  ( VN A K M )i
i
The multiplicities are obtained with a combinatorical line of thought.
There are e0 enzyme and s0  p substrate molecules that can build enzyme-substrate
adducts. If i adduct molecules are formed, there are  e 0  different possible ways to select
 i 
the enzyme molecules,  s 0  p  different possible ways to select the substrate molecules
i


and i! different permutations of enzyme-substrate pair formation. The multiplicity is then:
 e  s  p   e0  ( s 0  p)!
M   0  0
i!   
 i  i   i  ( s 0  p  i)!
The Si,p value is obtained as the energy multiplied by the multiplicity of a given state and
divided by the entire partition function.
S6
Equation (15):
The first part of this equation is simply the definition of a statistical expectation.
For the second part of the equation, the definition of the confluent hypergeometric
function (1F1) is used:
F (a ,b , z )  1 
1 1
a(a  1) z 2 a(a  1)( a  2) z 3
a
z

 ....
b
b(b  1) 2! b(b  1)(b  2) 3!
If a is a positive integer and z is a positive real number, the infinite series is reduced to a
finite sum for 1F1(a,b,z):
1 F1(  a ,b , z )  1 
a

i 0
a
a(a  1) z 2 a(a  1)( a  2) z 3
a(a  1)( a  2)...2  1
za
z

 .... 

b
b(b  1) 2! b(b  1)(b  2) 3!
b(b  1)(b  2)...(b  a  2)(b  a  1) a!
a!
(b  1)! z i
(a  i)! (b  1  i)! i!
Here, only the case  = e0 will be dealt with. The other possibility, i.e.  = s0  p is
handled by an entirely similar sequence of thought. Transforming the original equation in
a step-by step manner:
( s 0  p)!
(VN A K M ) i
i 1
0  p  i )!
e0
 e0  ( s 0  p)!
(VN A K M ) i
i 

(
s

p

i
)!


i 0
0
 i i  (s
e0
e0
e0

i
(VN A K M ) i
i 1 (e0  i )! i! ( s 0  p  i )!

e0
1
i
( s 0  p)! e0 ! 
(VN A K M )
i  0 (e0  i )! i! ( s 0  p  i )!
( s 0  p)! e0 ! 
e0
i
(VN A K M ) i
i 1 (e 0  i )! i! ( s 0  p  i )!

e0
1
( K MVN A ) e0 
(VN A K M ) i
i  0 (e 0  i )! i! ( s 0  p  i )!
( K MVN A ) e0 
e0

i
( K MVN A ) e0 i
i 1
0  i )! i! ( s 0  p  i )!
e0
1
( K MVN A ) e0 i

i  0 (e 0  i )! i! ( s 0  p  i )!
 (e
At this point, a new auxiliary variable j = e0  i is introduced.
S7
e0 1
e0
i
( K MVN A ) e0 i

i 1 (e 0  i )! i! ( s 0  p  i )!
e0
1
( K MVN A ) e0 i

i  0 (e 0  i )! i! ( s 0  p  i )!

e0 ! 
j 0
e0
e0 ! 
j 0
e0  j
( K MVN A ) j
j!(e0  j )! ( s 0  p  e0  j )!
1
( K MVN A ) j
j!(e0  j )! ( s 0  p  e0  j )!
(e0  1)!
( K MVN A ) j
( s 0  p  e0 )! e0 
j!
j  0 (e 0  j  1)! ( s 0  p  e 0  j )!

e0 1

e0 !
( K MVN A ) j
( s 0  p  e0 )! 
j!
j  0 (e 0  j )! ( s 0  p  e 0  j )!
e0
(e0  1)!
( s 0  p  e0 )! ( K MVN A ) j
j!
j  0 (e 0  j  1)! ( s 0  p  e 0  j )!

e0 1

e0 
 e0
e0 !
( s 0  p  e0 )! ( K MVN A ) j

j!
j  0 (e 0  j )! ( s 0  p  e 0  j )!
e0

F (e0  1, s 0  p  e0  1, K M N AV )
1 F1 ( e 0 , s 0  p  e 0  1, K M N AV )
1 1
The last line is the same as the last part of equation (15) if  = e0 is taken into account.
Equation (16):
By definition, the standard deviation is given as
   e  ( s 0  p)!
 e  ( s 0  p)!
i2 0 
(VN A K M ) i   i 0 
(VN A K M ) i

i
i
  ( s 0  p  i)!
 ( s 0  p  i)!
i 1
i 1 
σp 
  

(
s

p
)!
 e0 
 e  ( s 0  p)!
0
(VN A K M ) i
(VN A K M ) i
   0 
i 

i
 ( s 0  p  i )!
i 0 
 i 0   ( s 0  p  i)!

Successive transformation of equation (16) gives:
S8






2
ES
p 
 (e0  ES p )( s 0  p  ES p ) 
( s 0  p )!
(VN A K M ) i

p

i
)!
i 1
0
VN A K M  e0  s 0  p   e0 ( s 0  p)  ES

(
 e0  s 0  p )!
i
(VN A K M )
i 

 ( s 0  p  i )!
i 0 
( s 0  p )!
(VN A K M ) i
i 1
0  p  i )!
VN A K M  e0  s 0  p   e0 ( s 0  p)  ES

 e0  ( s 0  p )!
i
(VN A K M )
i 

 ( s 0  p  i )!
i 0 
2

2

p
 i i  (s

e0
p

( s 0  p )!
 e  ( s 0  p)!
(VN A K M ) i  e0 ( s 0  p ) 0 
(VN A K M ) i
i
(
s

p

i
)!
 ( s 0  p  i )!


i 0
0
 ES

 e0  ( s 0  p )!
i
(VN A K M )
i 

 ( s 0  p  i )!
i 0 
e
 VN A K M  e0  s0  p i i0 

i 1


2
p
e0


 K

ES p  M  e0  s 0  p   e0 s 0  ES
 VN A

 i i  (s


KM
VN A


p
 iK
i 0
M
 e  ( s 0  p )!
VN A  ie 0  is 0  ip  e0 ( s 0  p)  0 
(VN A K M ) i
i
(
s

p

i
)!
  0
 ES

(
s

p
)!
 e0 
i
0
(VN A K M )
i 

 ( s 0  p  i )!
i 0 
2
p
The term containing KM is handled separately for

 iK
i 0

 e  ( s0  p)!
 e  ( s0  p)!
VN A  0 
(VN A K M ) i   i 0 
(VN A K M ) i 1 
i
i
  ( s0  p  i )!
 ( s0  p  i )!
i 1 
M
 1
( s0  p)!
e  j  e0 
( s0  p)!
 e0 
j
(
j

1
)
(
VN
K
)

( j  1) 0
(VN A K M )  j 
 j  1
 j ( s 0  p  j )


A M
(
s

p

j

1
)!
j

1
(
s

p

j
)!

 0
 
j 0
j 0
0
 1
 1

 e0  ( s0  p)!
 e  ( s0  p)!
j

j
)(
s

p

j
)
(
VN
K
)

(e0  j )( s0  p  j ) 0 
(VN A K M )  j



0
0
A M
j
(
s

p

j
)!
 j  ( s0  p  j )!


j 0
0
 (e
j 0
The last part of this equation is true because the definition of  (equation (13)) ensures
that (e0  )(e0  p  )=0. Returning to the original expression handled and substituting
this new formula:
S9
2
p


 iK
i 0
M
 e  ( s 0  p)!
VN A  ie 0  is 0  ip  e0 ( s 0  p)  0 
(VN A K M ) i
 i  ( s 0  p  i )!
 ES

 e0  ( s 0  p)!
i
(VN A K M )
i 

 ( s 0  p  i )!
i 0 


 (e
i 0
0
2
p

 e  ( s 0  p)!
 i )( s 0  p  i )  ie 0  is 0  ip  e0 ( s 0  p)  0 
(VN A K M ) i
 i  ( s 0  p  i )!
 ES

 e0  ( s 0  p)!
i
(VN A K M )
i 

 ( s 0  p  i )!
i 0 
 e0  ( s 0  p)!
(VN A K M ) i
i 
  ( s 0  p  i )!
i 0
 ES

 e0  ( s 0  p)!
i
(
VN
K
)
i 

A
M
 ( s 0  p  i )!
i 0 
2
p


i
2
2
p
   e  ( s 0  p)!
 e  ( s 0  p)!
i  0
(VN A K M ) i   i 0 
(VN A K M ) i

i
i
  ( s 0  p  i )!
 ( s 0  p  i )!
i 0
i 0 

  

 e0  ( s 0  p)!
 e  ( s 0  p)!
(VN A K M ) i
(VN A K M ) i
   0 
i 

i
 ( s 0  p  i )!
i 0 
 i 0   ( s 0  p  i )!

2






2
This proves equation (16).
Equation (17)-20:
These equations follow from the definitions of expectation and standard deviation.
Equation (21):
This equation gives an estimate of the initial time period after which the steady-state
assumption can be used justifiably. This time is obviously related to the time needed to
establish the equilibrium between species E, S and ES. A good estimate of the time can
be given by deterministic kinetics using a sequence of thought similar to relaxation
kinetics:
d [ES]
 k1 [E][S]  k 1 [ES]
dt
The concentrations are given by introducing a new variable x, the distance from the
equilibrium value:
[E]  [E]   x
[S]  [S]   x
[ES]  [ES]   x
S10

The differential equation is then:

dx
 k1 ([E]   x)([S]   x)  k 1 ([ES]   x)  k1 x 2  (k1 [E]   k1 [S]   k 1 ) x
dt
Introducing the new constant k:
k   k1[E]   k1[S]  k 1
The transformed differential equation is then:

dx
 k1 x 2  k  x
dt
The standard solution method for separable ordinary differential equations gives:
x
[ES]  k  e  k  t
k1 [ES]  (1  e  k  t )  k 
Finding the value of t when x falls to a minute  fraction of its initial value:
δ[ES]  
[ES]  k  e  ktδ
k1[ES]  (1  e ktδ )  k 
Rearranging this equation gives:
e  k tδ 
k  δ  k1δ[ES] 
k  k1 [ES] 
 
k   k1δ[ES] 
k  / δ  k1 [ES] 
Expressing t gives:
tδ 
k  k1 [ES] 
1
ln 
k  k  / δ  k1 [ES] 
This is the same as equation (17) with substituting  = 0.05.
Equation (23):
The equation can be derived directly from the definition of expectation for random
variables with continuous distribution:

τ   tk 2 ES 0 e
0
 k 2 ES
0
t

k 2 ES
0
(k 2 ES 0 )
2

1
k 2 ES
0
S11
Figure S1. Comparison of expectations and standard deviations obtained for the enzyme
substrate adduct (ES) and the product (P) obtained with the full solution and the
stochastic steady-state approximation in the Michaelis-Menten mechanism. Solid lines:
full solution; markers: steady-state approximation. k1/NAV = 1 s1, k1 = 1 s1, k2 = 1 s1,
e0 = 10, s0 = 50
S12
Figure S2a Comparison of the dead times (t0.05) of the steady-state approximation in the
Michaelis-Menten mechanism obtained from the full solution and the estimate given in
equation 1. Varied variable: k1/NAV. Other parameters: k1 = 100 s1, k2 = 1 s1, e0 = 10, s0
= 50.
Figure S2b Comparison of the dead times (t0.05) of the steady-state approximation in the
Michaelis-Menten mechanism obtained from the full solution and the estimate given in
equation 1. Varied variable: k1. Other parameters: k1/NAV = 100 s1, k2 = 1 s1, e0 = 10, s0
= 50.
S13
Figure S2c Comparison of the dead times (t0.05) of the steady-state approximation in the
Michaelis-Menten mechanism obtained from the full solution and the estimate given in
equation 1. Varied variable: k2. Other parameters: k1/NAV = 100 s1, k1 = 100 s1, k2 = 1
s1, e0 = 10, s0 = 50.
S14