4.1. Response Coefficients

In any system we choose to study there will be qunatities which we can measure and other quantities which we can change.

In a metabolite pathway the kinds of qunatiies we can measure include the pathwy flux and the pathwya metabolite concentations. If the pathwya understudy is sigaling network or a gen resulatory network we migth instead meaure the levels of transcription factor or the levels of certian phosporylatde proteins.

We will call these measurable quantities variables or the dependent variables.

In constrast there are also independent variables which we will call parameters which expermentalist can directly control.

Examples of parameters include any environmental condition, this includes externally applied inhibitors or drug molecules, and parmaeters internal to a cell such as the levels of enzyme expression.

One of the core ideas in MCA is meauring how much a variable such as a flux, metabolite or protein (if its a signaling network) is influenced by a parmaeter such as an inhibitor.

4.1.1. Measuring Influence

When we talk about influence we are primarly going to consider how a given parameter influences the steady-state. Therefore, in all subsequent discussions we will be assuming our pathway is at steady-state.

Imagine a simple metabolite pathway such as the one below is at ateady-state:

\[X_o \stackrel{e_1}{\longrightarrow} S_1 \stackrel{e_2}{\longrightarrow} S_2 \stackrel{e_3}{\longrightarrow} S_3 \stackrel{e_4}{\longrightarrow} X_1\]

We will assume that the boundary species \(X_o\) and \(X_1\) are fixed n order to sustain a steady-state. At steady-state there will be a pathway flux, \(J\) and steady-state level of three metabolites, \(S_1, S_2\) and \(S_3\). We will also asusme ther is an inhibitor called \(I\) that inhibites the second enzymes \(E_2\). To guage how much influence the inhibior has on the pathwya, we can pick a dependent variable to measure, for example \(S_3\).

With the pathway at steady-state, let us make a small change to the inhibitor by an amount \(\delta I\). As a rsult the system is no longer in steady-state and will start to change and move to a new steady-state with a new flux and new concentsations for the metabolites.

Let’s say the concentation of \(S_3\) has changed by \(\delta s_3\). We could estimate the influence the inhiitor has on \(S_3\) by taking the ratio:

\[\frac{\delta s_3}{\delta I}\]

This ratio however has units. For convenince we can eliminate the units by multiplying and dividing by \(I\) and \(s_3\) to give us a unit-less measure of influence:

\[\frac{\delta s_3}{\delta I} \frac{I}{s_3}\]

This can be reexpressed in the following form:

\[\frac{\delta s_3}{s_3} / \frac{\delta I}{I}\]

This can now be seen as a ratio of relative changes. In general, the relationship between changes in enzyme and say a flux is non-linear. This means that the ratio will depend on the size of the perturbation in \(\delta I\). To remidie this, we can reduce the size of perturbation such that in the limit, we obtain a ratio of differentials independent of the size of the perturbation. We call this ratio the response coefficient, \(R^{s_3}_{I}\):

\[R^{s_3}_{I} = \frac{d s_3}{d I} \frac{I}{J}\]

This form can still be interpreted, at least aproximately, as a ratio of percentage changes.

Definition

The Flux Response Coefficient:

\[R^J_I = \frac{dJ}{dI} \frac{I}{J} \approx \frac{J\%}{I\%}\]

Definition

The Concentration Response Coefficient:

\[R^s_I = \frac{ds}{dI} \frac{I}{s} \approx \frac{J\%}{I\%}\]

Example

When the concentration of a drug is increased from 2 mM to 2.5 mM, the cocentration of a given metabolite changes its steady-state level from 15 mM to 18 mM. Estimate the response cofficient.

Since the percentage change in the metabolite is (18-15)/18 and the percntage change in drug is (2.5-2.0)/2, then the response coefficient is estimated to be:

\[R = ((18-15)/18)/(2.5-2.0)/2 = 0.66667\]

This means that a 1% change in the drug concentration will lead to a 0.66667% change in the metabolite.

If a given pathway has \(m\) metabolites then it means there will be \(m\) response coefficients with respect to a given external factor.