4.3 Mathematical Preliminaries
4.3.2 Relationship Between Nucleotide Consumption Rate and RNA Pro-
We state a few results used in carrying out the model reduction in Section 4.6. Ideally, we would like to determine the relationship between the rate of production of RNA and the rate of consumption of nucleotides in the full model (4.1). The approach we will take involves first partitioning the corresponding mass action equations (4.2) into subsets of equations as follows:
dξ
dt =F(ξ,[N]), ξ∈R2n+3≥0 , (4.7a) d[N]
dt =G(ξ,[N]), (4.7b)
d[mn]
dt =H(ξ), (4.7c)
whereξis a vector comprising the concentrations of all the species except the completed RNA transcript mn and the free nucleotides, N. F, G and H are functions defined using mass action kinetics, which, with their respective arguments, give the rates of change of the vectorξand the scalars[N]and[mn]. This decomposition allows us to consider the rate of production of RNA, a species that does not participate anywhere else in the network, separately from the rate of consumption of the nucleotides, which affect dynamics of many reactions in the network. In particular, we note that in equations (4.7a)–(4.7c), the functions F, G andH do not have[mn]as an argument, while bothF andG depend on[N].
We will show that when the concentration of nucleotides, [N] as an argument of F in Equation (4.7a) is held constant, the trajectories of ξ reach an asymptotically stable equilibrium, ξe. At this equilibrium, which can be thought of as an operating point for the local dynamics of [N] in Equation (4.7b) and of [mn] in Equation (4.7c), the rate of consumption of nucleotides is proportional to the rate of production of RNA.
The assumption to hold the concentration of some species constant in order to de- termine the properties of a network requires some justification. To this end, we note that it has been used in the theory of chemical reactions, for instance by Feinberg ([18], Remark 4.3.1), who notes that when a species is in great excess, then over some ‘reason-
appreciably, while the remaining species can display non-constant dynamics. One do- main where nucleotide concentration is in excess for most of the duration of interest is in cell-free extracts, which were the primary motivation for this study. Another domain of relevance for the constancy of nucleotides is in cells, where nucleotide concentrations are regulated, and one might wish to calculate the consumption rate to obtain a measure for the loading of the cell’s metabolic machinery.
We now state a proposition which establishes the relationship between the rate of production of RNA and that of the consumption of nucleotides at this steady state, and furthermore provides steady state relationships among species concentrations, which will turn out to be useful for the model reduction procedure in Section 4.6.
Proposition 3. Consider the full model given by equations(4.1)and(4.2), and its decom- position into subsystemsF,GandHgiven by equations (4.7a)–(4.7c). When the nucleotide concentration is held constant, [N] = [N]const, in the subsystem F, the trajectories ofξ reach an asymptotically stable equilibrium,ξe, in the sense of Remarks 12 and 13. Further- more, substitutingξeand[N]constinto the subsystemsG andH gives the relationship
d[N]uninc
dt =−nd[mn]
dt , (4.8)
wherenis the length of the RNA,mn, in nucleotides, and[N]unincis the total concentration of nucleotides not incorporated into RNA, i.e.,[N]uninc≜[N] +∑n
k=1[P:Dk:mk−1:N]. Proof. We first prove the stability of the equilibriumξeof the subsystem
dξ
dt =F(ξ,[N]const). (4.9)
Using the technique from Feinberg ( [18], Section 4.3), we can write out the dynamical Equation (4.9) as a chemical reaction network with certain rate constants modified by the
constant scalar[N]const
P+D
kP f
−−* )k−−
P r
P:D1:m0 P:D1:m0
kN f[N]const
−−−−−−−* )−−−−−−k
N r
P:D1:m0:N P:D1:m0:N−−→kt x P:D2:m1
...
P:Dn:mn−1
kN f[N]const
−−−−−−−* )−−−−−−k
N r
P:Dn:mn−1:N P:Dn:mn−1:N−−→kt x P:Dt
P:Dt−−→kterm P+D.
(4.10)
According to Theorem 2, if we can show that the network given by (4.10) has deficiency zero and is weakly reversible, we would have shown that it possesses an asymptotically
stable equilibrium. The set of complexes in the network is{P+D, P:D1:m0, P:D1:m0:N, . . . , P:Dn:mn−1, P:Dn:mn−1:N, P:Dt}. Thus, there are c ≜ 2n+2 complexes in the network. Note that there is a cyclic path
through the set of complexes, given byP+D→P:D1:m0→P:D1:m0:N→ · · · →P:Dn:mn−1→ P:Dn:mn−1:N→P:Dt→P+D. Thus, the network is weakly reversible, and has only one link- age class (l=1). Finally, we compute the rank of the stoichiometric matrix as follows. The network can be written in matrix form as
dξ
dt =Mν(ξ,[N]const),
dξ dt = d
dt
[P] [D] [P:D1:m0] [P:D1:m0:N]
[P:D2:m1] [P:D2:m1:N]
...
[P:Dn:mn−1] [P:Dn:mn−1:N]
[P:Dt]
, ν(ξ,[N]const) =
kP f[P][D] kP r[P:D1:m0] kN f[N]const[P:D1:m0]
kN r[P:D1:m0:N]
kt x[P:D1:m0:N] kN f[N]const[P:D2:m1]
kN r[P:D2:m1:N] kt x[P:D2:m1:N]
...
kN f[N]const[P:Dn:mn−1] kN r[P:Dn:mn−1:N]
kt x[P:Dn:mn−1:N]
kterm[P:Dt]
, (4.11)
and
M=
c1 c2 c3 c4 c5 c6 c7 c8 . . . c3n c3n+1 c3n+2 c3n+3
r1 −1 1 0 0 0 0 0 0 0 0 0 1
r2 −1 1 0 0 0 0 0 0 0 0 0 1
r3 1 −1 −1 1 0 0 0 0 0 0 0 0
r4 0 0 1 −1 −1 0 0 0 0 0 0 0
r5 0 0 0 0 1 −1 1 0 0 0 0 0
r6 0 0 0 0 0 1 −1 −1 0 0 0 0
... ...
r2n+1 0 0 0 0 0 0 0 0 −1 1 0 0
r2n+2 0 0 0 0 0 0 0 0 1 −1 −1 0
r2n+3 0 0 0 0 0 0 0 0 0 0 1 −1
,
where we denote the rows and columns of the matrix M using ri andci, i=1, . . . 2n+2, respectively. We determine the rank of M as follows. Remove r1 since it is a dupli- cate of r2, and therefore does not affect the rank of the matrix. Also remove columns {c2,c4,c7, . . . ,c3i+1, . . . ,c3n+1}, which are all scalar multiples of the columns preceding them.
We are then left with the2n+2×2n+2matrix
M˜ =
˜
c1 ˜c3 ˜c5 ˜c6 ˜c8 . . . ˜c3n ˜c3n+2 ˜c3n+3
˜r2 −1 0 0 0 0 0 0 1
˜r3 1 −1 0 0 0 0 0 0
˜r4 0 1 −1 0 0 0 0 0
˜r5 0 0 1 −1 0 0 0 0
˜r6 0 0 0 1 −1 0 0 0
... ...
˜r2n+1 0 0 0 0 0 −1 0 0
˜r2n+2 0 0 0 0 0 1 −1 0
˜r2n+3 0 0 0 0 0 0 1 −1
,
which has the same rank asM. The sub-matrixM˜1obtained by removing˜c3n+3and˜r2n+3is lower triangular with nonzero diagonal entries, and thus has a (full) rank of2n+1, giving rank(M)≥2n+1. We also know thatrank(M) =rank(M˜)≤2n+2. Finally, note that˜r2n+3 can be written as a linear combination of the remaining rows in M˜ as
˜
r2n+3=−
2n∑+2 i=2
˜ ri.
Thusq≜rank(M) =2n+1. I.e., this network has zero deficiency δ=c−l−q=2n+2− 1−(2n+1) = 0, and is weakly reversible and using Theorem 2, we conclude that there exists a positive equilibrium of the subsystem (4.7a), asymptotically stable relative to its stoichiometric compatibility class.
d[mn]
dt =kt x[P:Dn:mn−1:N], (4.12)
d[Nuninc]
dt =
d
[N] +
∑n i=1
[P:Di:mi−1:N]
dt
= d[N]
dt +
∑n i=1
d[P:Di:mi−1:N] dt
=
∑n k=1
kN r[P:Dk:mk−1:N]−kN f[P:Dk:mk−1][N] +
∑n
k=1
kN f[P:Dk:mk−1][N]−kN r[P:Dk:mk−1:N]−kt x[P:Dk:mk−1:N]
= −kt x
∑n k=1
[P:Dk:mk−1:N]. (4.13)
For the model (4.7a) to be at steady state, the net flux into and out of every species must be zero, and individual fluxes are constant in time. Consider a set of three consecutive reactions from the subsystem (4.10) at an arbitrary[N]const
P:Di−1:mi−2:N−−→kt x P:Di:mi−1, (4.14) P:Di:mi−1
kN f[N]const
−−−−−−−* )−−−−−−k
N r
P:Di:mi−1:N, (4.15) P:Di:mi−1:N−−→kt x P:Di+1:mi. (4.16) Since the instantaneous flux into and out ofP:Di:mi−1 is zero, the flux in due to (4.14) and the flux out due to the reversible reactions (4.15) must balance, we have
kt x[P:Di−1:mi−2:N] =kN f[P:Di:mi−1][N]const−kN r[P:Di:mi−1:N]. (4.17) Similarly, considering the speciesP:Di:mi−1:Nin (4.15) and (4.16), we have
kN f[P:Di:mi−1][N]const−kN r[P:Di:mi−1:N] =kt x[P:Di:mi−1:N]. (4.18)
Thus,
[P:Di−1:mi−2:N] = [P:Di:mi−1:N] (4.19) [P:Di−1:mi−2] = [P:Di:mi−1], (4.20)
and by induction, we have that for alli,j in{1, 2, . . .n},
[P:Di:mi−1] = [P:Dj:mj−1], (4.21) [P:Di:mi−1:N] = [P:Dj:mj−1:N]. (4.22)
Thus, Equation (4.13) can be reduced to d[Nuninc]
dt = −kt x
∑n
k=1
[P:Dk:mk−1:N] (4.23)
= −n·kt x[P:Dn:mn−1:N] (4.24)
= −nd[mn]
dt , (4.25)
which completes the proof.