Generalized Consumption Model

In this section, we reduce the full model (4.1) to a generalized version of the consumption model by matching their steady state behaviors. In Section 4.2, we modeled the creation of RNA as a single step transcription reaction, with no intermediate RNA transcripts, and discussed a method for incorporating nucleotide consumption in this model. While this suffices for models where the only regulation of gene expression comes from transcription factor proteins, it becomes inadequate when we wish to model the regulation of transcrip- tion via interactions with nascent RNA transcripts, for instance by non-coding RNA like the pT181 transcriptional attenuator [8, 43, 50]. Thus, we wish to have a transcription scheme where the creation of the final RNA transcript occurs in steps, and each intermediate RNA piece is free to interact with other elements of the environment. For an RNA transcript of length n, the detailed model shown in Equation (4.1) has approximately2nchemical re- actions and about the same number of differential equations. For typical RNAs of length 500−2000bases, these models quickly become unwieldy, especially when modeling large systems composed of multiple genes and regulatory pathways. Thus, we derive a way to

ics by such reductions.

We begin by re-indexing the detailed model (4.1) as follows. Divide the length RNA (of length ⁿ) into^K segments of lengthsⁿi, for ⁱ∈ {1, 2, . . . ,K}. The species ^P:D_k,j^:m_k,j−1, with k ∈ {1, 2, . . . ,K}, refers to the species from the kth segment, with the polymerase attached to the jth DNA base pair site of this segment, where j is in {1, . . .n_k}, and an RNA of length ^∑k_i=⁻₁¹n_i + (j−1) attached to the polymerase molecule. Finally, identify the last element of a block with the first element of the next block, i.e., P:D_k,n

k+1:m_k,n

k =

P:D_k+1,1:m_k+1,0. This new indexing scheme is shown on the left in Figure 4.2. We would like to reduce this model to the one shown in Equation (4.76), where the RNA and DNA have been divided into^Kblocks. The concentration of each species in the reduced model (4.76) is the sum of the corresponding species in the full model (4.1). For example, [P:D:R_k] =

∑_nk

h=1[P:D_(k,h):m_(k,h−1)].

P+D

k^′_{P f}

−−* )_k−−′ P r

P:D RNA polymerase binding, P:D+N

k^′_{N f}

−−* )_k−−′ N r

P:D:N nucleotide binding, P:D:N−−−→^kcon(1) P:D consumption reaction, P:D:N−−−→^{kt x(1)} P:D:R₁ lumped RNA production,

...

P:D:R_K−1+N

k^′_{N f}

−−* )_k−−_′

N r

P:D:R_K−1:N nucleotide binding, P:D:R_K−1:N−−−→^kcon(K) P:D:R_K−1 consumption reaction, P:D:R_K−1:N−−−→^{kt x(K)} P:D_t+R_K lumped RNA production, P:D_t−−→^kterm P+D termination.

(4.76)

Let ^DT be the total DNA concentration in in either model. Mass conservation gives

D_T =

∑K i=1

n_i

∑

h=1

€[P:D_(i,h):m_(i,h−1)] + [P:D_(i,h):m_(i,h−1):N]Š

+ [D] + [P:D_t], (4.77)

P+D P:D^1,1:m^1,0 P:D^1,1:m^1,0:N k_{P f}

kP r

+N,k_{N f}

−N,kN r

P:D^1,2:m^1,1 P:D^1,2:m^1,1:N k_{t x}

+N,k_{N f}

−N,k_{N r}

P:D^1,n1:m^1,n1−1 P:D^1,n1:m^1,n1−1:N +N,k_{N f}

−N,k_{N r}

P:D^2,1:m^2,0 P:D^2,1:m^2,0:N +N,k_{N f}

−N,k_{N r}

P:D^2,2:m^2,1 P:D^2,2:m^2,1:N

kt x +N,kN f

−N,k_{N r}

P:D^2,n2:m^2,n2−1 P:D^2,n2:m^2,n2−1:N +N,kN f

−N,k_{N r} k_{t x}

P:D^α,1:m^α,0 P:D^α,1:m^α,0:N +N,k_{N f}

−N,kN r

P:D^α,2:m^α,1 P:D^α,2:m^α,1:N k_{t x}

+N,k_{N f}

−N,k_{N r}

P:D^α,nα:m^α,nα−1 P:D^α,nα:m^α,nα−1:N +N,k_{N f}

−N,k_{N r} P:D^t + m^α,0

kt x

kt er m

P+D P:D P:D:N

P:D:R1 k_{P f}

kP r

+N,k_{N f}

−N,kN r

k_con(1)

kt x(1)

P:D:R1:N +N,k_{N f}

−N,kN r

k_con(2)

P:D:R_α−1 ^+N, P:D:R_α−1:N k_{N f}

−N,kN r

kcon(α)

P:D^t+R_α k_{t x(α)} k_{t er m}

Figure 4.2: The model reduction procedure. Corresponding species are color coded. Each grey ‘block’ represents the part of the model that transcribes the corresponding segment of the RNA, with the output of the grey block as the intermediate RNA species which will be modeled by the reduced model.

D_T =

∑K i=1

[P:D:R_i−1] + [P:D:R_i−1:N]

+ [D] + [P:D_t], (4.78)

withP:D:R₀≜P:DandP:D:R₀:N≜P:D:N, for the reduced model.

Using the mass balance equations (4.21), (4.22) and rapid equilibrium assumption (4.75), we can express[P:D_(k,h):m_(k,h−1)]in the full model as:

[P:D_(k,h):m_(k,h−1)] = 1 n_k



D_T−[D]−[P:D_t]

€ 1+^k_k^{N f}

N r[N]Š −

∑K i=1,i̸=k

n_i[P:D_(i,1):m_(i,0)]



. (4.79)

The rapid equilibrium assumption for the reduced model can be derived in exactly the same way as that for the full model, and using it gives

[P:D:R_k] =



D_T−[D]−[P:D_t]

 1+^k_k^{N f′}′

N r[N]

‹ − ^K−1∑

i=0,i̸=k[P:D:R_i]



. (4.80)

We would like the reduced model to be an approximation of the full model at steady state (noting that the steady state is defined in the sense of the discussion preceding Proposition 3, when the nucleotide concentration is sufficiently large to be assumed to be essentially constant in the time-scale of interest). A set of sufficient conditions for this to be true is presented for the remainder of this section.

We first note that the discussion around equations (4.14) to (4.22) in Section 4.3.2 can be applied to the species within a single block to obtain the result that at steady state, corresponding species within a block are in mass balance (i.e., (4.21) and (4.22) hold within each block). This allows us to define and simplify the correspondence[P:D:R_k] =

∑_nk

h=1[P:D_(k,h):m_(k,h−1)]between the full and reduced models to

[P:D:R_k] =n_k[P:D_(k,h):m_(k,h−1)], Condition 1. (4.81)

The relation (4.81) is a condition we have imposed upon the model to create a corre-

spondence between the species between corresponding blocks of the full and reduced models. This condition, applied to equations (4.79) and (4.80), gives us further correspon- dences between the full and reduced models. In Equation (4.79), the term^DT−[D]−[P:D_t] is the total concentration of species in the grey boxes in Figure 4.2, the_(1+(k ¹

N f/k_{N r})[N])term, which arises out of the analysis of Section 4.4.1, picks out the proportion of[P:D_(k,h):m_(k,h−1)] from the total[P:D_(k,h):m_(k,h−1)]+[P:D_(k,h):m_(k,h−1):N], and so applying it toD_T−[D]−[P:D_t] gives the total concentration of terms of the form[P:D_(k,h):m_(k,h−1)]. Equation (4.80) has a similar interpretation, with the difference that [P:D:R_k] is now the concentration of a block in the reduced model, and thus corresponds to the total concentration of terms of the form[P:D_(k,h):m_(k,h−1)]in a single block of the full model, which by Equation (4.81) is n_k[P:D_(k,h):m_(k,h−1)].

The next step in using the reduced model to approximate the full model is to equate the fluxes out of corresponding blocks in the two models. We can then obtain a set of (not necessarily unique) relations between the parameters of the two models so that the steady state behaviors match. Consider the rate of production of the speciesP:D_(k+1,1):m_(k+1,0), which corresponds to the output of thekth block, orkth intermediate RNA transcript, in the full model,

[P:D_(k+1,1):m_(k+1,0)]production=k_{t x}[P:D_(k,n

k):m_(k,n

k−1):N]

=k_{t x}k_{N f}

k_{N r}[P:D_(k,h):m_(k,h−1)][N], (4.82) where we recall thatP:D_(k,n

k+1):m_(k,n

k)is identified withP:D_(k+1,1):m_(k+1,0). The corresponding expression for the reduced model is

[P:D:R_k+1]production=k^′_{t x(k)}[P:D:R_k:N]

=k^′_{t x(k)} k^′_{N f}

k^′_{N r}[P:D:R_k][N].

(4.83)

Equating the equations (4.82) and (4.83), and comparing terms, we can draw a corre-

k^′_{t x(k)}= k_{t x}

n_k, (4.84)

k^′_{N f} =k_{N f}, (4.85)

k^′_{N r} =k_{N r}, Conditions 2 - 4. (4.86) We now state a result that generalizes the result of Proposition 3 to the model lumped into blocks, and states that the rate of consumption of nucleotides within the kth block isn_ktimes the rate of production of the kth nascent transcript. This, in turn, will allow us to derive a generalization for the consumption reaction for the reduced model (4.76).

Proposition 4. Consider the full model(4.1)under the same assumptions as Proposition 3, now considered with the indexing scheme that divides it into separate transcription blocks, as described at the beginning of Section 4.6. Then, in thekth block, fork∈ {1, 2, . . . ,K}, the rate of nucleotide consumption within the block is proportional to the rate of production of thekth intermediate transcript,P:D_k,n

k+1:m_k,n

k (with theKth transcript beingm_K,0).

Proof. From the proof of Proposition 3, we know that the total rate of nucleotide consump- tion is

d[N_uninc]

dt =−kt x

∑n i=1

[P:D_i:m_i−1:N]. (4.87) This sum can be partitioned into terms using the new indexing scheme into contributions

to the consumption rate by each block as follows:

d[N_uninc]

dt =−k_{t x}

∑K k=1

n_k

∑

h=1

[P:D_(k,h):m_(k,h−1):N]

=−k_{t x}

n1

∑

h=1

[P:D_(1,h):m_(1,h−1):N]

−k_{t x}

n2

∑

h=1

[P:D_(2,h):m_(2,h−1):N] ...

−k_{t x}

n_K

∑

h=1

[P:D_(k,h):m_(k,h−1):N]

=−n₁k_{t x}[P:D_(1,n

1):m_(1,n

1−1):N]

−n₂k_{t x}[P:D₍2,n₂):m_(2,n2−1):N] ...

−n_Kk_{t x}[P:D_(K,n

K):m_(K,n

K−1):N],

(4.88)

where the last line follows from the fact that the species within a block are in flux balance as given by equations (4.21) and (4.22). In each of the terms of the form−nkk_{t x}[P:D_(k,n

k):m_(k,n

k−1):N]

in the above partition, the term^kt x[P:D_(k,n

k):m_(k,n

k−1):N]is the production rate of the^kth transcript (bound to the RNA polymerase and DNA) for all^k ∈ {1, 2, . . . ,K−1}while it is the production rate ofm_K,0 fork=K. On the other hand, the full term is the contribution of thekth block to the total consumption rate of the nucleotides not yet incorporated into the any RNA.

The rate of consumption of nucleotides is the same in the full model (4.1) and the reduced model (4.76) if we set

k^′_con(k)=k^′_{t x(k)}(nk−1), Condition 5. (4.89) Indeed, from Proposition 4, and equations (4.81) and (4.83), the rate of consumption of

k_{t x}n_k[P:D_(k,n

k):m_(k,n

k−1):N] =k_{t x}n_k

n_k[P:D:R_k:N] (4.90)

= k_{t x}

n_k[P:D:R_k:N] +k_{t x}

n_k (n_k−1)[P:D:R_k:N] (4.91)

= [P:D:R_k+1]production+k_con(k)^′ [P:D:R_k:N]. (4.92)