# Branches, Cycles, Enzyme-Enzyme Interactions and Cascades

### Introduction

Although very often used as examples, linear pathways (as in figure 5) are not the only structures to be found in metabolic maps (see figure 4). The latter are populated with branches and cycles. Also, the restriction that the rate of reactions should be linearly proportional to the enzyme concentration is not needed to apply MCA. A very brief description of how MCA deals with such cases is presented here.

### Branched Pathways

In a previous section it was mentioned that one can calculate the values of all control coefficients of a pathway given the knowledge of the values of all enzyme elasticities using the summation theorem and the connectivity relations. When a pathway contains one or more branches (see figure 6), the number of equations is smaller than the number of steps (enzymes), and so one needs more equations for a complete solution. Those extra equations relate the enzyme elasticities and the control coefficients at the branch points and are known as the branch-point theorems. Including them here would make these web pages too heavy and so the interested reader is pointed to the following papers: Kacser (1983), Fell & Sauro (1985), Westerhoff & Kell (1987) and Small & Fell (1989).

### Moiety-Conserved Cycles

Reich & Sel’kov (1981) introduced the term moiety-conserved cycle to name metabolic cycles in which there is conservation of a certain chemical group (moiety) after a full turn of the cycle. These cycles (see figure 7) are quite common in metabolism, an example of which is the NADH-NAD+ cycle. One important characteristic of these cycles is that the the entry and exit of other moieties into and out of the cycle is done via bimolecular reactions and there cannot be any branches from the cycle (see figure 7). It is easy to mistake other metabolic structures for moiety-conserved cycles when they are not. A notable example is that of the TCA cycle, in which there is no conservation of moieties. In fact the TCA cycle is a branched pathway (that we can draw as a circle…)

Because there is mass-conservation in moiety-conserved cycles, the concentration of one of the metabolites of the cycle is a linear combination of the concentrations of the other metabolites of the cycle, and therefore the former is not an independent variable. Furthermore, the sum of the concentrations of the metabolites of the cycle is constant, this sum is thus an external parameter of the patway (the total [NADH]+[NAD+] is an example). Because of this, one can define response coefficients for the total concentration of the moiety. The existence of one or more of these cycles in a pathway also implies that some connectivity relations and the summation theorem for concentration-control coefficients will be different from those of a linear pathway . Hofmeyr et al. (1986) gives a detailed study of MCA and moiety-conserved cycles and is best consulted for further details. Reder (1988) introduced a general methodology to calculate control coefficients from elasticities that takes into account moiety-conservation (see also the next section).

### Substrate Cycles

Substrate cycles, also known as futile cycles, are metabolic structures in which there are two independent enzymes operating in opposite directions interconverting two metabolites. In some substrate cycles a full turn of the cycle is accompanied by the breakdown of ATP (and this is the origin of the term futile, meaning that the cycle is just “wasting” energy). Figure 8 depicts a substrate cycle.

Inspection of figure 8 reveals that in a substrate cycle there is no conservation of any moiety, rather the cycle is formed by two sequential branches. The application of metabolic control analysis to substrate cycles is based on that of metabolic branches and is well covered in Fell & Sauro (1985).

### Enzyme-Enzyme Interactions

MCA as developed by Kacser & Burns ( 1973) and Heinrich & Rapoport (1974)could not deal with metabolic systems in which the concentration of one enzyme affected reactions other than the one it catalysed. Furthermore, although the text in Kacser & Burns (1973) is general enough to handle systems in which the rate of reaction is not linearly proportional to the enzyme concentration, the concept of control coefficients became equated with the changes in enzyme concentration rather than rate of reaction (see definition of control coefficient in Kacser & Burns 1973 and above). This would result that the summation theorems would be invalid whenever there are enzyme-enzyme interactions.

There are many cases of enzyme-enzyme interactions in metabolism. A well known case is that of the mitochondrial electron-transfer chain in free-energy transducing membranes, or more generally, group transfer chains. In these structures, each enzyme directly transfers a metabolite to the next enzyme (be it electrons or molecules). Metabolic channelling (see srere 1987, Ovádi 1991, Mendes et al. 1995) can also be seen as a form of group transfer. Click here to read something more about metabolic channelling.

Several papers have addressed enzyme-enzyme interactions in the context of MCA and there are a few methods that can be used to describe such systems using MCA. See Westerhoff & Kell (1988), Kell & Westerhoff (1990), Kacser et al. (1990), Sauro & Kacser (1990), Kholodenko & Westerhoff (1993), and van Dam et al. (1993).

### Interconvertible Enzyme Cascades

Interconvertible enzyme cascades are sequences of enzymes catalysing the formation of covalent bonds between other enzymes and chemical moieties. This is a special case of enzyme-enzyme interactions in which enzymes are substrates of other enzymes, forming cycles of active-inactive forms (more and less active forms, to be more precise). A good example of interconvertible enzyme cascades is that of the interconversion of glycogen phosphorylase a and glycogen phosphorylase b catalysed by phosphorylase-b kinase and phosphorylase-a phosphatase (see for example Chock & Stadman 1980). Figure 9 depicts an interconvertible enzyme cascade.

The application of MCA to enzyme cascades was first done on a mechanistic level by Cárdenas & Cornish-Bowden (1989), later followed by Small & Fell ( 1990) and Schuster et al. ( 1993) with mechanism-independent studies. The latter work is also applicable to other modular systems such as protein synthesis.

Last updated: May 31, 2016 at 10:57 am