It has been shown theoretically that a simple signalling motif comprising a kinase, phosphatase and their substrate can lead to an ultrasensitive input-response relation when the enzymes are fully saturated by their substrate [92]. This mechanism is termed zero-order sensitivity and can be achieved by having kinetic parameters that favour complex formation among enzymes and the substrates, and by having a large ratio of the total concentration of substrate to that of enzymes [92]. I found that when conditions allow, zero-order sensitivity readily evolves in silico. Of the 30 simulations, which were started with a high ratio of output protein to signalling protein concentrations, 11 have resulted in the emergence of ultrasensitivity and 8 of these successful simulations resulted in kinetic parameters where either or both kinases and phosphatases were saturated (Figure 3.2A, blue points). These results confirm that the in silico simulation framework can recover a known biochemical
mechanism - enzyme saturation by substrate - for achieving ultrasensitivity.
Pro 10, Out 10 Pro 1, Out 1000
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Bifunctional Bipath Cascade
Figure 3.4: Average level of enzymes sequestrated by other proteins and average level of enzymes saturated by substrate (output protein) in all evolved ultrasensitive networks. The orange coloured dots represents kinases and the blue dots represents phosphatases. The numbers on the dots denote the network number.
While enzyme saturation mediated zero-order sensitivity is well understood theoretically, this biochemical mechanism might not be relevant for many biological systems where the ratio of substrate to enzyme concentrations is found to be low [97, 222]. To explore whether ultrasensitivity can still emerge under such conditions, I ran evolutionary simulations with equal starting concentrations for the substrate
and signalling proteins. Although concentration of signalling proteins could freely evolve in these simulations, enzyme saturation was expected to be difficult to evolve, which could lead to evolution of alternative mechanisms for ultrasensitivity. Indeed, the emerging ultrasensitive networks from these simulations (10 out of 30 simula- tions) did not display the kinetic parameters required for enzyme saturation (Figure 3.2A, red points). Together with three ultrasensitive networks that started with high concentrations of the substrate, but did not evolve enzyme saturation, these ultrasensitive networks clearly utilize mechanisms other than enzyme saturation.
Analysing the structure of these networks (Figure 3.3), I did not find any distinct structural features. However, I found that in many evolved networks with parameters in the non-saturating regime, there is a high prevalence of enzyme se- questration (Figure 3.2B) and also allosteric regulation of enzyme activity (Figure 3.2C) by other signalling proteins. In theory, allosteric regulation of enzyme ac- tivity by upstream proteins that are activated by signals could implement a form of ultrasensitivity [87, 88, 223] that could relax the need for enzyme saturation. I found that for at least some networks, the ratio of allosteric forms of the enzymes barely changes across the input range (Figure 3.2D), showing that allosteric reg- ulation is not the main or sole process enabling ultrasensitivity. This suggests a more general role for enzyme sequestration, which prompted us to analyse all of the evolved networks with regard to the prevalence of the different enzyme complexes. In particular, I calculated the average proportions ofES complexes, formed by en- zyme binding to substrate, andET complexes, formed by enzyme binding to other proteins (Figure 3.4). Note that these proportions can be seen as the average level of enzyme saturation by the substrate and sequestration by other proteins in the signalling network. This analysis revealed that most of the ultrasensitive networks evolved parameters that resulted in enzymes being bound in complexes (i.e. they lie close to the line given by [ET] = 1−[ES]). Moreover, contrasting the results of evolutionary simulations where enzyme saturation was made difficult to evolve vs.
Figure 3.5: Analysis of evolved adaptive networks. (A) Structure and dynamics of the evolved adaptive network 1. The upper panel shows a cartoon of the net- work. The oval shapes represent ligand (top) and the output protein (bottom) (e.g. substrate with a phosphorylation site, S), while all other signalling proteins (e.g. receptor/adaptor proteins, kinases, or phosphatases) are shaped as rectangle. Black line represents binding reaction between two sites. Red arrows represent phosphory- lation reactions between a kinase site (red) and a phosphorylation site (purple). Blue arrows represent dephosphorylation reactions between a phosphatase site (blue) and a phosphorylation site. The green coloured rectangle indicates a protein domain, whose conformational switching is allosterically regulated (also indicated by a self- pointing green line with arrows at both ends) [212]. The lower panel shows the dynamics of input signal and output response. The stacked colours represents the compositions of enzyme complexes: blue for proportion of enzyme-substrate com- plexes, green for free form enzymes that are accessible by the substrate, and red for complexes where enzymes are not accessible by the substrate (i.e. titrated enzymes).
(B) Structure and dynamics of the evolved adaptive network 2. Panels are as in (A).
not, showed that the former scenario resulted in enzymes that were mostly titrated by other signalling proteins (see Figure 3.2A and Figure 3.4). These results suggest that when enzyme saturation is not readily achievable, evolution of ultrasensitivity was made possible mostly through enzyme sequestration. I analysed this proposition further with a simpler model (Figure 3.6).
Figure 3.6: Designed signalling cycle motif and parameter space for adaptation and ultrasensitivity. (A)Cartoon showing the designed signalling cycle motif with a se- questering protein. The sequestrating proteinT binds both the kinaseK and phos- phataseP, which catalyse the phosphorylation and dephosphorylation of substrate
S and Sp respectively. (B) The values of key parameters for achieving ultrasensi-
tive (> 0.8) and adaptive response (> 0.3), when assuming an enzyme-saturated regime ([Stotal] = 1, [Ptotal] = 0.1). The upper and lower two panels are distribu-
tion of parameters that generate ultrasensitive and adaptive responses respectively. Panels on the left show the distribution of Michaelis-Menten constants, for kinase:
KM,K = k2k1+k3 (x-axis) and phosphatase KM,P = k5k4+k6 (y-axis). Panels on the
right show the distribution of affinities of sequestrating protein T with kinase and phosphatase: KD,K = k8k7 and KD,P = k10k9 . Note that all four panels are plotted
on the same logarithmic range. Each black dot represents a parameter set and the colours shows density of parameters. (C) Values of key parameters for achieving ultrasensitive (> 0.8) and adaptive response (> 0.3), when assuming an enzyme- non-saturated regime ([Stotal] = 0.1 and [Ptot] = 0.1).
3.3.2 Selection for adaptive dynamics leads to networks employing