A linear pathway

Example 11.9 A linear pathway Consider a linear metabolic pathway composed of five consecutive reversible reactions (Boogerd et al., 2005) where each reaction is catalyzed by an enzyme E 

Metabolite X4 inhibits the rate of enzyme 1. The metabolites X andX5 are maintained constant at all times. The kinetics model for this linear pathway yields 

The first term on the right represents the inhibition effect, while the second term is the net reaction excluding the inhibition. KltX is the equilibrium dissociation constant (in mM), and indicates the ratio (El - ] )/(Eh ), when the binding relaxes to equilibrium. Here, A stands forX0, A, . orX4, and (/ ., - X]) is the concentration of the enzyme-substrate X1 complex, and A, A, is the product of the concentrations of the free enzyme E, and the free substrate Xx. The reaction velocities. /rir and. /rlh are the maximal rates of forward and backward rates of catalysis (in mM/min), respectively. As seen from Eq. (11.65), the rate is a nonlinear function of the concentrations of metabolites. 

The equilibrium constant depends only on the properties of the reactants and temperature. The enzyme shortens the time necessary for the reaction to reach equilibrium and does not affect the equilibrium constant. If the actual ratio of product and substrate is 

The extent of displacement is directly related to the rate. /r, which vanishes at equilibrium where Q is equal to A eq. The displacement is directly related to the Gibbs free energy difference or the chemical potential difference 

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Figure 5.6. The production of NPs using matrix pathways was predicted by Jones and Firn because of the opportunity to produce and retain chemical diversity efficiently. In this diagrammatic scheme, three enzymes (ei, e2 and es) have access to one substrate. The upper panel shows that if each of the enzymes has a strict substrate specificity, a linear pathway producing three new chemicals would be expected, ffowever, if the three enzymes have a broad substrate specificity then the order of conversion can vary and a matrix pathway will result. Now three enzymes will produce 11 novel substances. Furthermore, such matrix pathways are more robust to the loss of any one enzyme activity (see Figure 5.4).

Beginning with Eduard Buchner s discovery (c. 1900) that an extract of broken yeast cells could convert glucose to ethanol and C02, a major thrust of biochemical research was to deduce the steps by which this transformation occurred and to purify and characterize the enzymes that catalyzed each step. By the middle of the twentieth century, all ten enzymes of the glycolytic pathway had been purified and characterized. In the next 50 years much was learned about the regulation of these enzymes by intracellular and extracellular signals, through the kinds of allosteric and covalent mechanisms we have described in this chapter. The conventional wisdom was that 1860 1917 in a linear pathway such as... 

Given that the pathway can consume citrate (see sample 3), if citrate is to act as a catalyst it must be regenerated. If the set of reactions first consumes then regenerates citrate, it must be a circular rather than a linear pathway. 

A concept traditionally held in highest esteem, especially by chemists, is that of a rate-controlling step. The idea is that the overall rate is determined by the slowest step in the mechanism, the "bottleneck." For a linear pathway in which one step is much slower than all others, this may allow the set of simultaneous rate equations for all participants to be reduced to one single rate equation of formation of the product or products. 

In effect, like a linear pathway, the loop is reduced to a pseudo-single step... 

In other words, to obtain the rate equation for a catalytic cycle with a macs, the cycle can be "snipped" at the latter to give a linear pathway. The rate equation of that imaginary pathway approximates that of the cycle (granted the validity of the assumption that practically all of the total catalyst material is present as the macs). This simple rule allows the rate equation of any catalytic reaction with a macs to be written down as quickly and easily as those for linear simple pathways. It will be used extensively in the next section (see also Figure 8.5 in that section). 

None the less, a set of rules can be given for catalyst systems with a macs. As shown in the previous section, the rate equation for a catalytic reaction with a macs is the same as that for an imaginary (linear) simple pathway that "starts" and "ends" with the macs. Think of the catalytic cycle as being "cut" at the macs to give a linear pathway with the macs at both end (see Figure 8.5). That imaginary equivalent pathway has the same rate equation as the actual catalytic cycle. With this principle, the rules for reaction orders deduced in Section 7.3.1 can be reformulated for catalytic cycles with a macs (as in the earlier section, the rules are for the forward rate if the reaction is reversible). 

In practice, catalytic networks often involve more than a single cycle. In a very common type of network, a linear pathway is attached to the cycle. This is so, for instance, in reactions with ligand-deficient catalysts. Here, the coordinatively saturated "catalyst" has no activity but, rather, must first lose one of its ligands to provide access for a reactant. Other examples of cycles with attached pathways include systems with inhibition, activation, decay, and poisoning. Also, the network may consist of two or more cycles with a common member or pathway. This situation is typical for reactions yielding different isomeric products. The Christiansen formula is extended to cover such cases. 

A D az-Quintana LeibI Bottin and S0tif (1998) Electron transfer in photosystem I reaction centers follows a linear pathway in which iron-sulfur cluster Fb is the intermediate electron donor to soluble ferredoxin. Biochemistry 37 3429-3439... 

Purine synthesis is a branched pathway pyrimidine synthesis is a linear pathway with interconversion of products UTP and CTP. 

In the reductive pathway, the Krebs cycle enzymes are assumed to operate as far as a-oxoglutarate, thus forming a linear pathway. A second linear pathway, from oxaloacetate to malate to fumarate to succinate, is suggested to account for the formation of succinic acid [46]. In support of this new pathway are the observations that (/) yeast contains cytoplasmic malate dehydrogenases capable of converting oxaloacetate to malate, (//) several fumarate reductases (FAD-dependent) have been found in the yeast cytoplasm which have high affinity for fumarate and are unable to oxidize succinate [52] and (Hi) succinate is a significant product of fermentation, i.e. an end product . 

A linear pathway of 2 linET is shown below (it can be constructed by skipping the layer 4c of SEN). The 5 cycET are needed to find the stimulus element in layer 2,3 and the response element in the same layer. 

The same is true for vl 11H30A, where (Fp-con)/ET = 2.7. That means, Fp is produced by a process which only needs about 3 ET (presumably a linear local shortcut pathway). I therefore take 4ET right of con for the minimal pathway, remaining 2 cycET left of the median. This makes a linear pathway of 2 linET and a cyclical pathway of 2 + 2 = 4 cycET,... 

In these quasi-Iinear fluxes the rate of the step before the branch is not equal to the rate of the actual chemical reaction, since it can be considered to be composed of two separate fluxes designated Ek and Ex(, ) for /, and 7b. respectively. (This is one reason for describing such systems as fluxes rather than pathways a pathway would imply a series of whole reactions.) Feedback control is an inherent part of these branch fluxes for example, an increased activity of E2 increases 7, by lowering the concentration of B and thus deflecting flux from /i, to 7 in this way an increased E2 is effectively increasing the part-reaction Ei(,) by decreasing the concentration of B. However, it must be stressed that this is not the same mechanism as feedback in a linear pathway, since B has no direct effect on reaction in this particular branched system. The possible metabolic importance of this indirect feedback produced by branching is discussed in Section V. 

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