Linearised dynamics. To get some insight into the general form of the projected equations we rst consider a simpli ed problem, starting from a linearised description for the full network. The linearised reaction equations including copy number noise are ∂tδx = A δx + η (3.2.1) where A is as de xxx just before (3.1.13) and the covariance matrix ϵBBT of the noise η, which normally is δx-dependent, is evaluated at steady state (δx = 0). The corresponding adjoint Fokker-Xxxxxx operator is = L Σij ∂ ϵ Aijδxj ∂δx + 2 i Σ ij (BB T ∂2 i j )ij ∂δx ∂δx (3.2.2) In Section 3.1.4 we showed that in general, the most appropriate choice of subnetwork observables {ai} consists of the subnetwork concentrations and all their products. Now that we are considering linearised dynamics, we will only want to project onto the concentrations themselves, omitting the products. The linearised projected equations can then be written in the general form ∂ ∂tδxi(t) = N s Σ j=1 δxj(t)Ωji + t ∫ dtj N s Σ j=1 δxj(tj)Mji(t − tj) + ri(t) (3.2.3) and our aim will be to nd explicit expressions for the rate matrix entries Ωji and the memory functions Mji(t − tj). Note that, as it should be for a description of the subnetwork dynamics, the sums over j above run only over subnetwork concentrations. We assume here that these concentrations make up the rst entries of the vector δx, i.e. δxj with j = 1 . . . N s where N s is the number of subnetwork species. We will denote the subnetwork part of δx by δxs, and the remaining bulk part by δxb, so that δxT = (δxsT, δxbT). Here T denotes the transpose of a column vector. To nd the rate matrix and memory functions from the general expressions (2.2.9) and (2.2.10), or equivalently (3.1.11), we need to be able to nd the action of the operators L, P and Q on the observables ai = δxi (i = 1, . . . , N s) and evaluate products of the form (a, b). Starting with the latter, we choose for the (approximate) steady-state distribution a Gaussian over δx with mean zero and Poissonian covariance matrix Σ. The elements of this matrix then give the products (δxi, δxj) = Σij. More speci cally, if we partition the covariance matrix depending on whether the relevant molecular species are in the subnetwork or bulk, as we did for the vector δx, it can be written in the form ,Σs,s 0 0 Σ Σ = b,b (3.2.4) The Poissonian form for Σ forces zeros on the o -diagonal blocks as we have written. It also implies that Σs,s and Σb,b are diagonal, although we will not need this property in the followin...
Linearised dynamics. + In linearised dynamics we only consider terms in the mass-action kinetics up to linear order in δx. The dimensionless scaled reaction equations for a reversible Xxxxxxxxx- Xxxxxx reaction are then, from (5.2.12) and (5.2.13), ∂ ∂tδxu = −kc—,ue (ye/yu )δxe − kue,c ye(δxu + δxe) + . . . ∂t e c,ue u e ep,c p p e ∂ δx = −γ[(k— + k— )δx + k+ y (δx + δx ) + k+ y (δx + δx )] (5.3.1) c,ep e ue,c u ∂t p c,ep e ∂ δx = −k— (y /y )δx − k y (δx + δx ) + . . . + p e ep,c e p e Let us partition the matrix form of the adjoint Fokker-Xxxxxx matrix operator (5.2.2) so that the bulk species are split into fast and slow blocks. If ej and e represent the collection of subnetwork and bulk enzymes respectively and s and b represent the other molecular species in the subnetwork and bulk, we partition as , Ls,s Ls,e′ Ls,b Ls,e , L = L L b,e′ = , LS,S LS,B B,S B,B L L L L = e′,s e′,e′ e′,b e′,e b,s b,b b,e m w1 f1 w2 w3 f2 (5.3.2) L L L L Le,s Le,e′ Le,b Le,e w4 w5 f3 where w are slow terms and f are fast terms; the top left block denoted m contains a mixture of fast and slow terms. In writing the last equality above we have grouped s and ej together; the resulting speci c 3 × 3 block structure of slow and fast terms is one that we will nd again in the case of the full nonlinear dymamics. Note that because subnetwork enzymes only have interactions with subnetwork species (s and ej), Lb,e′ and Le,e′ are zero. Similarly, because bulk proteins or enzymes do not interact with subnetwork enzymes, Le′,b and Le′,e vanish. This means that subnetwork enzymes do not feature at all in the calculation of the memory function (5.2.6), which makes intuitive sense. The vanishing of Lb,e′ and Le,e′ is important also as these blocks contain rates for the time evolution of (subnetwork) enzymes, which by our construction scale with γ: if these blocks were nonzero, it would change the character of w2 and w4 from slow to fast. To analyse the memory function (5.2.5) that results from (5.3.2), we note that LB,B has both slow and fast sub-blocks. As a result the memory function should have both slow contributions that decay on O(1) timescales, and fast contributions that decay for time di erences of O(1/γ). As the memory function appears as a weight in an integral over the past (5.2.3), the fast contributions only matter for γ → ∞ if their amplitude is proportional to γ so that the integral over all time di erences remains nite. Accounting also for subleading terms in the a...
