## Parameters in Systems of Differential Equations

Kinetic models of cellular systems are usually described by arrays of simultaneous differential equations. There are two features of these arrays that make parameter estimation very difficult. First, the number of parameters in the models tends to be large which means that the process of optimization of parameters can be very slow. More will be discussed about this aspect in Section 6.7. Second, for most realistic models there is no analytical solution to the array of differential equations....

## Rate Equation Deriver

RateEquation rcm,el derives the steady-state rate equation for an enzyme mechanism defined in the rate constant matrix (rcm - see below). The argument el is optional and is a list of user-defined names for the enzyme forms of the reaction mechanism. Rate constant matrix (rcm) this is constructed by drawing a square grid with n x n cells, where n is the total number of enzyme forms (free enzyme plus complexes) in the reaction scheme. On the left, adjacent to the first column, form a list down...

## Upper Limit of Values for Unitary Rate Constants

In the first question in Section 3.2.4 we sought the value of three unitary rate constants. However, there were only two steady-state parameters so it was necessary to assume the value of one of either k1 or k-1 k1 was chosen because there is a rational choice that can be made for the value of such a second-order rate constant. Could we have chosen a much larger value than the 1 x 107mol-1 L s 1 that was used The answer is yes, sort of as there is an upper bound on the value this is called the...

## Response Coefficients and Partitioned Responses

Another class of question that is important to answer in relation to the behavior of a metabolic pathways is How does the model respond to different external effectors For example, experimental work on 2,3-BPG metabolism has shown that its concentration is very sensitive to changes in intracellular pH.(2) One way of characterizing this sensitivity is through the concentration response coefficient. Q What is the value of the response coefficient of 2,3-BPG concentration with respect to...

## Conservation Relations

A defining feature of any metabolic system is its metabolite-conservation relationships. For example, in the human erythrocyte under normal conditions the sums of the concentrations of the adenine nucleotides ( ATP + ADP + AMP ) and the nicotinamide adenine dinucleotides ( NAD + NADH ) are both constant. The stoichiometry matrix, N, which contains the stoichiometric structure of the model of the metabolic system, implicitly contains these conservation relationships in it. A function in the...

## Variances of Parameters

Calculating the uncertainties in parameter estimates is a very important and yet very technical topic. Fortunately, the Mathematica functions Regress and NonlinearRegress automatically yield the standard errors of the estimates. On the other hand, when FindMinimum is used to minimize a merit function, no error estimates are given. So it is important to have a strategy to make these estimates. This can be done via a so-called Monte Carlo simulation. The idea behind the Monte Carlo approach is to...

## Simple Model of the Urea Cycle

The previous Sections of this chapter were concerned mainly with models of enzymes in isolation. Specifically, the concern was how to derive steady-state rate equations for various mechanisms and how to relate the unitary rate constants of these mechanisms to the steady-state parameters, the latter usually being the only parameters that have been measured experimentally. Thus, having described how to model individual enzymes we now turn to the simulation of metabolic pathways. As a simple...

## Lineweaver Burk plot

The most commonly used transformation of Eqn 2.1 entails taking the reciprocal of each side of the equation to yield the Lineweaver-Burk, or double reciprocal, plot A plot of the transformed data pairs (1 A 0i, 1 v0i) i 1, , N, gives a straight line with ordinate and abscissa intercepts at 1 Vmaxand -1 Km, respectively. Q Generate a Lineweaver-Burk plot for the enzyme described in the previous worked example. A First we need to generate a Table of ordered pairs (1 A 0i, 1 v0i) using our...

## Graphical evaluation of Vmax and Km

The challenge faced by an experimenter is the determination of the two key parameters of the Michaelis-Menten equation, Vmax and Km, given a set of data pairs ( A 0i, v0ii i 1, ,NA where N is usually 5 - 10. Actually, a deeper challenge awaits the experimenter, and that is to determine whether the fitting equation is a realistic description of the data. Nevertheless, suppose that we have established, from inspecting the Michaelis-Menten plot, that the data conform, at least roughly, to a...

## Using Matrix Notation in Simulating Metabolic Pathways

For large metabolic networks, it is often simpler to express the set of simultaneous differential equations that describe the reaction system in matrix form. The use of matrix notation is also useful for analysing the existence and stability of steady states and for performing MCA (see Chapter 5). Thus the set of differential equations describing the rate of production and utilization of all metabolites, S,, in a metabolic network of reactions is given by where vj (j 1, ,r) is the rate of...

## Linear least squares

Most rate equations that describe biological processes are nonlinear. However, there are a few cases, such as passive transport of a solute across a membrane, that are describable as first-order processes. In addition, some nonlinear rate equations are readily transformed into linear ones e.g., in Section 2.1 the linear Lineweaver-Burk plot was derived from the nonlinear Michaelis-Menten equation. The Mathematica package, Statistics'LinearRegression , contains several functions that perform...

## Numerical integration in Mathematics NDSolve and stiffness

As discussed previously, NDSolve is the Mathematica function that is used for numerically solving arrays of simultaneous differential equations. The function uses a variety of methods of numerical integration and takes into account an important characteristic known as the 'stiffness' of the array of equations. Stiffness refers to the extent to which the set of equations has members that describe slow processes and members that describe very fast processes. A large range of rate constants means...

## Degree of inhibition

The rate of an enzymic reaction is often affected by substances other than the reactant(s). Inhibitors slow the rate and activators increase it. In dealing with inhibitors it is important to distinguish between the effects that are observed e-perimentally on the various analytical plots mentioned above, and the molecular mechanism (or models) proposed to explain the effects. The purpose behind studies of enzyme inhibition is often to enhance an understanding of the mechanism of the enzyme by...

## Models of Erythrocyte Metabolism

Over the last 25 years many mathematical models of erythrocyte metabolism have been developed 4-7 These have been very successful in identifying the key features of the regulation and control of the metabolism of the cell. Indeed the erythrocyte is the best modelled of all biochemical systems and there are a number of reasons for this. The first is the ease of obtaining them by simple venipuncture. Second, the erythrocyte has relatively simple metabolism as a result of lacking mitochondria and...

## Eisenthal and Cornish Bowden equation direct linear plot

A totally different approach to analyzing enzyme kinetic data involves plotting many lines onto the experimental data it was introduced in 1974 by Eisenthal and Cornish-Bowden.(3) For this procedure a Cartesian a-is-system is drawn with v0 on the ordinate and A 0 on the abscissa. With the data pairs ( A 0i,v0i), i 1, ,N, a straight line is drawn to pass through the points (- A 0i, 0) and (0, v0i). The intersection of this line with others, similarly drawn, occurs at as many different points as...

## When Cell Volume Changes with Time

All the metabolic pathways that we have considered so far have involved the assumption that the reactions take place in free solution and no consideration has been given to partitioning of reactants and enzymes into various compartments. To actually simulate such a system introduces another order of complexity into formulating the model. The principal 'trick' in such models is to express the rate equations in terms of the rate of change of amounts (moles) rather than concentrations. If we do...

## Eadie Hofstee plot

The Eadie-Hofstee equation is derived by multiplying both sides of Eqn 2.1 by (Km + A 0), dividing by A 0, and then rearranging the terms to give V0 -Km TIT + Vmax . 2.3 Hence, a plot of the data pairs consisting of (v0ji A 0ji, v0i) gives a straight line with a slope that has the value -Kmand an ordinate intercept that is the value of Vmax. Q Generate an Eadie-Hofstee plot for the enzyme described in the worked example in Section 2.1.2. A First we define the Eadie-Hofstee equation vo vC>...

## Elasticity Coefficients

Another derived parameter in MCA is the elasticity coefficient. This coefficient is different from the previous two in that elasticity coefficients are properties of individual enzymes in isolation, rather than of the metabolic system as a whole. Essentially, elasticity coefficients measure how the velocity of an enzymic reaction changes in response to changes in substrate, product, inhibitor, and activator concentrations. Hence, these coefficients quantify what enzyme kineticists have been...

## Relationships between Unitary Rate Constants and Steady State Parameters

3.2.1 Progress curve of a Michaelis-Menten reaction Consider the irreversible Michaelis-Menten scheme - it is the simplest enzyme mechanism of all (see also Eqn 2.31 ). Eqn 3.2 expresses the fact that in a conventional steady-state kinetic analysis of such an enzyme only two parameters are estimated, Km and Vmax, and these encapsulate all the information that is required to describe the progress of the reaction after the establishment of a steady state of the EA concentration. In other words,...

## Deriving Expressions for Steady State Parameters

Enzymes are often studied by using a range of concentrations of various substrates, products, and inhibitors in order to determine the reaction mechanism. The choice of particular experiments is made on the basis of the well-trodden path of standard enzyme kinetic practice, as described in many authoritative texts.(4-7) As noted in Section 3.4.2, the function RateEquation provides a means of rapidly deriving the steady-state rate equations for virtually any enzyme mechanism. But for the present...

## Generating the Stoichiometry Matrix

In the above examples the task of specifying the stoichiometric matrix, N, was a simple one. However, for larger and more complex metabolic networks, the generation of these matrices can be tedious and error prone. The functions StoichiometryMatrix and NMatrix automate this process, thus generating the stoichiometic matrix from the list of reactions that constitute the metabolic scheme of interest. defined in the equation list eqn it takes into account the fact that the parameters in the list,...

## Determining Steady State Concentrations

In general, a metabolic pathway can be thought of as a network of biochemical reactions whose function is to take in substrates from the environment and transform them into something that is required by the organism. Thus the organism may take in fuel molecules which are used to produce ATP which is, in turn, used as the main energy 'currency' of the cell. Or alternatively, the organism may take in substrates which it uses as the building blocks for different subcellular components. Thus we can...

## Flux Control Coefficients

As is noted in Chapter 5, flux control coefficients give information on the relative importance of a particular reaction in controlling a particular flux in a metabolic pathway. Often, the easiest way to calculate the flux control coefficient for a model, that is formulated using matrix notation, is with the function FluxControlMatrix from the MetabolicControlAnalysis add-on package. Q Calculate the non-normalized flux control coefficient values for the erythrocyte model. A The matrix is...

## Internal Response Coefficients

Two more coefficients that are useful for describing the regulation of metabolic pathways are the internal response coefficients. These are defined in a manner that is analogous to the response coefficients but they use e-elasticities instead of p-elasticities. The definition of the response that characterizes the effects of a fluctuation in concentration of Sy on the concentration of S, is and for the dependence of the flux J, on the concentration of Sy the definition of the Q Calculate the...

## Response Coefficients

The final type of coefficients of major importance in MCA are the response coefficients. These characterize the effect of an infinitesimally small change in a parameter value in the system on concentrations or fluxes in the system of reactions. Thus, the concentration response coefficient is defined by and the flux response coefficient is defined by respect to the kth parameter of parameter list. Returns a matrix where the element mjk is the flux respect to the kth parameter of parameter list....