Derivations of correlation functions from the Reaction Diffusion Master
Equation and the Spatial Chemical Langevin Equation.
Derivation of (3)
O
å Õ u(
To obtain (3) apply
so )
no
to both sides of (2). Evaluating the left hand side
o=1
u
of (2) then leads to:
O
O
å Õ u(
so )
no
o=1
u
O
¶P( u,t )
=
¶t
¶å Õ un( soo ) P( u,t )
u
o=1
¶t
(S.1)
Õ u(
d
=
so )
no
o=1
dt
which matches the left hand side of (3). Now consider the right hand side of (2):
å å å Õ ( u(
R
N
O
r =1 n=1 u
so ) ( n)
no
r
a
N
r ,n
so ) ( n)
no
r
r ,n
o=1
å å å å Õ (u
S
( u - V ( ) ) P( u - V ( ) ,t ) - u(
N
O
( so ) ( s)
no
s=1 m=1 n=1 u
(
( s)
( s,m,n)
dm,n um - ws,m
o=1
a
( u) P( u,t ))
) P( u - W ,t ) - u
( s)
( so ) ( s) ( s)
m,n
no
dm,num P( u,t )
)
. (S.2)
We consider the reaction term only for an illustrative purpose. The diffusion
term follows in the same way. First note that the Master Equation in (2) will
prevent populations from becoming negative for physically relevant reaction
propensities. Thus, we can extend the summation over u to all negative
populations as well as positive populations without affecting the behaviour of
the RDME. This then allows us to apply the transformation u® u+ V ( r ,n) ,
"r Î{1,..., R} and "n Î{1,..., N} to the first term of the first line without
changing the evaluation of the sum in the first line. This leads to:
å å å Õ ((u(
R
N
O
r =1 n=1 u
o=1
so )
no
)
)
+ vs(or ,n,no) ar( n) ( u) - un( soo )ar( n) ( u) P( u,t ) ,
(S.3)
which leads to the first line of (3) when the sum over u is executed. The diffusion
term follows in the same manner.
Derivation of (5) and (7)
We use Ito's lemma as stated in [14] from the main text’s bibliography. Consider
some number n of SDEs with d Wiener processes driving them. For 1 £ i £ n , put
each SDE in the following form:
d
dYi ( t ) = bi ( Y ( t )) dt + ås ij ( Y ( t )) dWj ( t )
(S.4)
j =1
and define the matrix c( Y ( t )) = s ( Y ( t ))s ( Y ( t )) . Then, for any twice
T
continuously differentiable function f :» n ® » , Ito’s lemma states:
2
å n ¶ f ( Y ( t ))
å
1 n n ¶ f ( Y ( t ))
df ( Y ( t )) = åå
bi ( Y ( t )) + å å
cij ( Y ( t ))ådt + M , (S.5)
¶xi
2 i =1 j -1 ¶xi ¶xj
å i=1
å
where M is a martingale whose specific form is unimportant to us, since we will
take expectations of (S.5) anyway. To obtain (5), use Ito’s Lemma with the SDEs
O
from (4) with f = Õ un( soo ) . This gives:
o=1
S N N
å O O ( so' ) å R N ( n)
( r ,n)
( s) ( s) ( s,m,n) å
u
a
U
v
+
dm,n
um wso ,no å
(
)
å
å
å
å
å
å
å
n
r
s
,n
å
o' å
o o
å r =1 n=1
å
s=1 m=1 n=1
å o=1 o'=1
O
o'åo
( so ) å
då uno =
O O O
R N
S N N
å+ 1
( so' ) å
( n)
( r ,n) ( r ,n)
( s) ( s) ( s,m,n) ( s,m,n)
o=1
u
a
U
v
v
+
um wso ,no wsp ,np
(
)
å
å
å
å
å
å å dm,n
å
n
r
s
,n
s
,n
o' å
o o
p p
å 2
å r =1 n=1
o=1 p=1 o'=1
s=1 m=1 n=1
å
o'åo
å
o'å p
(
)
(
)
å
å
å
ådt + M
åå
å
åå
å
å
, (S.6)
where the first and second lines contain terms corresponding to the first and
second order terms from (S.5), respectively. Taking expectations of (S.6), and
rearranging terms gives rise to (5). To derive (7), either use the chain rule, or
interpret the deterministic approach as one with the Wiener coefficients set to
zero. In the latter case, equation (7) follows from (S.6) by simply disregarding
the second line, taking expectations, and rearranging.