Design principles of autocatalytic cycles constrain enzyme kinetics and force low substrate saturation at flux branch points

Abstract

A set of chemical reactions that require a metabolite to synthesize more of that metabolite is an autocatalytic cycle. Here, we show that most of the reactions in the core of central carbon metabolism are part of compact autocatalytic cycles. Such metabolic designs must meet specific conditions to support stable fluxes, hence avoiding depletion of intermediate metabolites. As such, they are subjected to constraints that may seem counter-intuitive: the enzymes of branch reactions out of the cycle must be overexpressed and the affinity of these enzymes to their substrates must be relatively weak. We use recent quantitative proteomics and fluxomics measurements to show that the above conditions hold for functioning cycles in central carbon metabolism of E. coli. This work demonstrates that the topology of a metabolic network can shape kinetic parameters of enzymes and lead to seemingly wasteful enzyme usage.

Keywords: E. coli; computational biology; dynamic analysis; enzyme kinetics; metabolism; systems biology.

Figure 1.. A basic autocatalytic cycle requires an internal metabolite to be present in order to assimilate the external metabolite into the cycle, increasing the amount of the internal metabolite by some amount, δ.

DOI: http://dx.doi.org/10.7554/eLife.20667.003

Figure 2.. Three representative autocatalytic cycles in central carbon metabolism: (I) The Calvin-Benson-Bassham cycle (yellow); (II) The glyoxylate cycle (magenta); (III) A cycle using the phosphotransferase system (PTS) to assimilate glucose (cyan).

Assimilation reactions are indicated in green. Arrow width in panels represent the relative carbon flux. DOI: http://dx.doi.org/10.7554/eLife.20667.004

Figure 2—figure supplement 1.. An autocatalytic cycle assimilating ribose-5-phosphate using the pentose phosphate pathway.

This cycle contains a direct input reaction (rpi, dashed line) allowing the cycle to operate with broader sets of kinetic parameters than cycles missing this feature. A knockout strain where rpi is eliminated, does not grow under ribose despite having the theoretical ability to do so. DOI: http://dx.doi.org/10.7554/eLife.20667.005

Figure 2—figure supplement 2.. An autocatalytic cycle assimilating dhap while consuming gap using the fba reaction in the gluconeogenic direction.

This cycle contains a direct input reaction (tpi, dashed line) allowing the cycle to operate with broader sets of kinetic parameters than cycles missing this feature. According to fluxomics data this cycle does not operate in vivo as a more energy efficient alternative in growth under glycerol is to use the tpi reaction and proceed in the glycolitic direction in the lower part of glycolysis. A knockout strain where tpi reaction is eliminated, does not grow under glycerol despite having the theoretical ability to do so. DOI: http://dx.doi.org/10.7554/eLife.20667.006

Figure 3.. Analysis of a simple autocatalytic cycle.

(A) A simple autocatalytic cycle induces two fluxes, $f_a$ and $f_b$ as a function of the concentration of $X$. These fluxes follow simple Michaelis-Menten kinetics. A steady state occurs when $f_a=f_b$, implying that $\dot{X}=0$. The cycle always has a steady state (i.e. $\dot{X}=0$) at $X=0$. The slope of each reaction at $X=0$ is ${\displaystyle V_{max}/K_M}$. A steady state is stable if at the steady state concentration ${\displaystyle \frac{d\dot{X}}{dX}<0}$. (B) Each set of kinetic parameters, $V_{\max ,a},V_{\max ,b},K_{M,a},K_{M,b}$ determines two dynamical properties of the system: If ${\displaystyle V_{max,b}>V_{max,a}}$, then a stable steady state concentration must exist, as for high concentrations of $X$ the branching reaction will reduce the concentration of $X$ (cyan domain, cases (I) and (II)). If ${\displaystyle \frac{V_{max,b}}{K_{M,b}}<\frac{V_{max,a}}{K_{M,a}}}$, implying that ${\displaystyle \frac{V_{max,b}}{V_{max,a}}<\frac{K_{M,b}}{K_{M,a}}}$, then zero is a non-stable steady state concentration as if $X$ is slightly higher than zero, the autocatalytic reaction will carry higher flux, further increasing the concentration of $X$ (magenta domain, cases (I) and (IV)). At the intersection of these two domains a non-zero, stable steady state concentration exists, case (I). DOI: http://dx.doi.org/10.7554/eLife.20667.007

Figure 4.. Analysis of an autocatalytic cycle with input flux.

(A) The effect of a fixed input flux, $f_i$, on the possible steady states of a simple autocatalytic cycle. A steady state occurs when $f_a+f_i=f_b$. If ${\displaystyle V_{max,b}>V_{max,a}+f_i}$ then there is always a single stable steady state (I). If ${\displaystyle V_{max,b}<V_{max,a}+f_i}$ then there can either be no steady states (II), or two steady states where the smaller one is stable (III). DOI: http://dx.doi.org/10.7554/eLife.20667.008

Figure 5.. Generalization of analysis to multiple-reaction autocatalytic cycles with a single assimilating reaction.

(A) A two reaction system. (B) A generic $n$-reaction system. The system is at steady state when the total consumption of intermediate metabolites by the branch reactions is equal to the flux through the autocatalytic reaction, because the autocatalysis is in a 1:2 ratio. A sufficient condition for the stability of a steady state in these systems is that the derivative of at least one branch reaction with respect to the substrate concentration is larger than the derivative of the equivalent autocatalytic reaction at the steady state concentration. Given the connection between derivatives of fluxes and saturation levels of reactions (see methods), this condition implies that at a stable steady state, the saturation level of at least one branch reactions is smaller than the saturation level of the corresponding autocatalytic reaction. DOI: http://dx.doi.org/10.7554/eLife.20667.009

Figure 6.. Major branch points and relative enzyme saturation in operating autocatalytic cycles.

Solid arrow width represents carbon flux per unit time. Shaded arrow width represents the maximal carbon flux capacity per unit time, given the expression level of the catalyzing enzyme. In all cases there is enough excess capacity in the branching reactions to prevent the cycle from overflowing. A $4\%$ flux from pep to biomass was neglected in growth under glucose and fructose. Only in one out of the nine branch points observed (the branch point at fbp in growth under fructose), the outgoing reaction is significantly more saturated than the autocatalytic reaction. (*) A branch point at which the branching reaction is more saturated than the autocatalytic reaction, which may result from neglecting fructose transport directly as f6p when deriving fluxes (see text). DOI: http://dx.doi.org/10.7554/eLife.20667.010