Constitutive relationships

To complete the specification of the model, we require expressions for the allocation coefficientsα0,

. . . ,αG. One option is to treat these as forcing functions that drive the model. These functions can be

observed directly, due to recent advances in ribosome profiling [73, 95] and enzyme re-profiling [90]. Alternatively, the allocation coefficients can be treated as control inputs, to be calculated on the basis of a suitable, evolutionarily relevant optimality criterion [156]. Another option to ‘close’ the equations is to posit outright the dynamics for the reserve densitiesxi, for instance setting ˙xi=νi(fi−xi), whereνiis a

positive constant [88] andfias in eqn (2.5). This approach, which defines the allocation implicitly, while

having the advantage of simple dynamics, would seem to require cellular-level stoichiometric parameters to be in fortuitous agreement with the kinetic parameters of the molecules of the regulatory system [150]. Here, we treat the allocation coefficients as a function of the internal state variables and/or environmental parameters [117]. In particular, we assume thatm0, . . . ,mG,x1, . . . ,xn are mapped toα0, . . . ,αG by a

suitable R2(n+1) 7→Rn+2 function. Recalling that αi is the fraction of ribosome time devoted to the

production of machinery of typei, we propose the following:

αi= e

ri

e

r0+er1+···+ern+erG

, (2.9)

where the_erirepresent, roughly speaking, the concentrations of translationally active mRNA for the corre-

sponding types of machinery (corrected for relevant molecular properties, such as affinity for the ribosome and mobility within the cytosol, as mRNA species of various lengths and tertiary structures will differ with respect to this properties, in particular, might affect the arrival rate of the various mRNA species at the ribosome, which potentially translates into a skewing of the relative amounts of ‘ribosome time’ devoted to each of them, which we tacitly assume can be done via suitable weighting coefficients; synthesis rates inE. coliare predominantly under translational, rather than transcriptional control [95]). For the sake of simplicity,_er0is assumed to be constant, corresponding to constitutive expression. We scale the other_eri

by this constant:

For j=1, . . . ,n, rj is assumed to be a decreasing function ofxj (we shall take this as a generic sig-

moid for the sake of convenience); as the reserve density increases, less of the machinery that feeds it is synthesised. The central mechanism in the present theory resides in a feedback loop connecting re- serve densities and allocation of building blocks to machinery; the control logic here is related to that of I-control in control engineering, cf. [74]. The building blocks are fed from core metabolism into the synthesis routes; the allotment is achieved effectively by an allocation of ribosome time (cf. the Scott-Hwa-model [134, 135, 136]). Therj can be thought of as corresponding to levels of mRNA for

the various types of molecular machinery, although issues such as differences in stability of the mRNA molecule, affinity for ribosomes may distort a direct 1-to-1 correspondence (which can be compensated to some extent by assuming that appropriate correction factors have been assimilated into the scaling).

We assume further thatrGis an increasing function ofm0. For the sake of simplicity, we represent it

as a piecewise affine function:

rG[m0] =      0 ifm0_≤1₋ε rG,max/2+K(m0−1) if 1−ε<m0≤1+ε rG,max ifm0>1+ε, (2.11) whereK is the slope, andε=rG,max/(2K). The mid-point of this function is set atm0=1 (we here

exercise our freedom to choose a natural unit for the scaling factorm_b which we identify as the physio- logical optimum for type-zero machinery;m0=1 follows from this choice). Equation (2.11) expresses

the hypothesis that the safeguarding of core catalytic machinery takes precedence over growth [10]. This relationship is suggested by, and consistent with, Herbert’s [68] classic observations on the relationship between RNA content and growth rate (the componentm0corresponding to rRNA). The slope of the

relationship observed by Herbert [68] is inversely proportional toK, that is, the larger the value ofK, the smaller the variation of RNA content with growth rate. Figure 2.2 illustrates the relationship between rel- ative growth rate and RNA content for the micro-organismAerobacter aerogenesgrown in a continuous culture with glycerol as a limiting factor, as observed by Herbert [68]. This relationship appears to be linear and thus can be represented as follows:

e

µ=Ke_×RNA+b,

whereµ_e is relative unscaled growth rate, Ke andb are, correspondingly, slope and offset parameters.

0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 Rela tiv e gro wth ra te (h 1)

RNA content (g RNA/g cell dry weight)

Figure 2.2: Relationship between relative growth rate and RNA content for the micro-organismAerobacter aero-

genesgrown in a continuous culture with glycerol as a limiting factor, together with the optimal fit of eqnµe= e

K×RNA+bwith parametersKe=7.58 h g DW g RNA−1,b=−0.45 h−1. Original data taken from [68].

haveµ_e=ψWrGφe0, giving rG= e K ψWφe0 ×RNA+ b ψWφe0 .

According to Section 4.5.1, we have:

RNA=M0/β0,

whence together with the scaling forM0(cf. Section 2.2.1) we obtain: rG= e KWmˆ ψWφe0β0 m0+ b ψWφe0 .

Given the values for stoichiometric coefficients provided (as detailed in Section 4.5) and using the esti- mate for the slope parameterKeobtained from fitting the Herbert data to the linear model (fit shown in Fig. 2.2), we can calculate the slopeKfrom eqn (2.11) by means of the following equation:

K= KWe mˆ ψWφe0β0