Negative feedback successfully decouples two genes

We optimised the key controller parameters (ωρ, ωF, kF, hF andbF) using the cost

function as described in Section 4.2.

To demonstrate the ability of the controller to increase o-ribosome production in response to demand, we consider the induction point when approximately equal amounts of protein are produced (ωA =ωB = 100 mRNAs per min) (Figure 4.2).

We consider the constitutive expression of one circuit gene and simulate the response of the system to the stepped induction of a second gene (Figure 4.2b, c, d). When circuit demand is low (depicted in Figure 4.2a, left), before the induction of the second gene, competition between the circuit and the controller is low. This results in high expression of the controller protein and therefore high repression of the o- rRNA, meaning that few ribosomes are co-opted from the host. Upon induction of the second gene (depicted in Figure 4.2a, right), the demand for o-ribosomes increases (Figure 4.2b). The repressor mRNAs will remain largely unaffected, but their translation falls due to increased competition (Figure 4.2c). The decrease in repressor production results in relief of the inhibition of the o-16S rRNA gene and so increased o-rRNA production and increased co-option of host ribosomes (Figure 4.2b). This results in the maintenance of circuit protein production as other circuit genes are induced (Figure 4.2d), although it should be noted that a small steady state error of∼2% remains. Note that whilst the free o-ribosome is not maintained in the presence of the controller there is only a 5% fall (Figure 4.2b) but in the absence of the controller this fall is 50%.

We simulate the increasing induction of one protein pA and assess the fall in a

constitutive protein pB. The negative feedback controller successfully slows the

fall in pB while removing the resource-mediated saturation effect on pA (Figure

4.3). Note that the mRNA levels in both open and closed loop confirmations are equivalent. To determine the effect of the controller, we simulate the performance of the same circuit in the presence of the o-ribosome system but in the absence of the controller. As we found that protein levels were a large determinant of host response in Chapter 3 (specifically Figure 3.3), we vary the value ofωρ so that both the open

loop (i.e. without the feedback mechanism) and closed loop (i.e. with the feedback mechanism) circuits produce the same protein outputs when the induction of the two genes is matched (i.e. when ωA=ωB = 100 mRNAs per minute) (Figure 4.3).

Comparison of the protein levels with their respective mRNAs shows the controller successfully restores the input–output mapping. In the open loop confirmationpB

Key mRNA Protein o-rRNA hRibosomeso Ribosome flux Inhibition Reaction o-rRNA gene

Genes

a

b

c

d

Figure 4.2: Operation of the F-controller. Simulations of a two-gene circuit

were carried out as described in the main text. pA is induced at t = 0. The

changing distribution of controller and circuit components is shown in (b), (c) and (d). (a) Structure and function of the F-controller. Left, Low demand circuit. When competition is low pF expression is high and so o-rRNA production is low

and hence co-option of host ribosomes is low. Right, High demand circuit. As

circuit demand increases (aspAis induced), the o-ribosome pool redistributes across

circuit and controller genes (width of purple ribosome flux lines) due to competition between mRNAs. This reduces translation the constitutively expressed pF. This

reduces the repression of o-rRNA production, allowing more co-option of ribosomes to the orthogonal pool. This maintains ribosome flux formBtranslation despite the

increase in mA. (b) Changing distribution of the controller components. ρ, o-16S

rRNA; P, free o-ribosome; pF, controller protein. Normalised by their maximum

value. (c)Changing distribution of the translation complexes over time in response topA induction. cY, translation complex of gene Y, Σ P, sum of all o-ribosomes.

Normalised by maximum ΣP. (d) Protein output over time normalised by sum of the final circuit protein concentration.

Figure 4.3: The simple F-controller decouples co-expressed genes. Simula- tion of the action of the negative feedback controller. ωA is varied between 1 and

104 mRNAs per minute. wB is held constant at 100 mRNAs per minute through-

out. The simulation time span is increased until it reaches steady state. Controller

parameters: ωρ = 350 rRNAs per min; ωF = 103 mRNAs per min; Hill function

parameterskF = 104 and hF = 4. Open loopωρ= 1.6 rRNAs per min. Insets,left,

mRNA concentration in the open loop; right, mRNA concentration in the closed

loop.

falls 50% over the first two orders of magnitude of ωA induction, whilst in the

presence of the controller the equivalent fall is negligible at less than 2%. pB is

maintained within 10% of its initial value over three orders of magnitude of ωA

induction. Over the whole range of ωA, pB only falls 40% (compared to 99% in

the open loop). In the presence of the controller pA increases linearly and is not

subjected to the saturation effects of the open loop controller.