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### Control Analysis

This page summarizes some of the main terms used in biochemical control analysis (BCA). By biochemical control analysis, I mean what is often called metabolic control analysis (MCA) as developed by Kacser and Burns and Heinrich and Rapoport. I have purposefully changed the name to BCA to reflect the fact that MCA doesn't just apply to metabolism but applies to all biochemical or chemical reaction networks, including gene regulatory and protein signaling networks. Although called control analysis, BCA is a distinct theoretical approach to classical control theory although they are both closely linked in a number of ways.

#### Concentration Control Coefficient

A concentration control coefficient measures the relative steady-state change in a species concentration (S)  in response to a relative change in enzyme activity, $e_i$ often through changes in enzyme concentration.  This definition assumes that the enzyme concentration is under the direct control of the experimenter and as such can be classed as a parameter. It is important to note that concentration control coefficients are properties of the intact system and cannot be measured from the isolated reaction step or enzyme. See flux control coefficient for more details.

$$C^S_{e_i} = \frac{dS}{de_i} \frac{e_i}{S} \right) = \frac{d\ln S}{d\ln e_i}$$

#### Connectivity Theorem

The connectivity theorem states the sum of the product of flux control coefficients and elasticity coefficients around a particular molecular intermediate is zero. Likewise the sum of the product of concentration control coefficients and elasticities with respect to the same species is -1 but for different species is 0.

$$\sum_i C^J_i \varepsilon^i_S = 0$$

$$\sum_i C^{S_n}_i \varepsilon^i_{S_m} = 0 \quad n \neq m$$

$$\sum_i C^{S_n}_i \varepsilon^i_{S_m} = -1 \quad n = m$$

#### Control

The term control has a special meaning in control analysis. Control refers to the ability of a system parameter to affect a system variable. For example, changes to the external glucose concentration in a microbial culture will most likely change the culture's growth rate. The concentration of glucose, therefore, has 'control' over the growth rate. Engineering an enzyme in pathway so that its kcat is larger will result in changes to the pathways flux and metabolite concentrations. Changes to the promoter consensus sequence of a particular gene will result in changes to the concentration of the expressed protein and any other variables that depends on that protein. It is possible to quantify control by either measuring or computing control coefficients.

#### Elasticity Coefficient

The elasticity coefficient, $\varepsilon^v_S$, is the degree to which a substrate, product or other effector can change the reaction rate, $v$, of the isolated process. This means that elasticities can be measured by purifying an enzyme and studying its kinetics in vitro. Elasticity coefficients do not therefore have to be measured in the intact system. As a result, elasticities are classed as local properties whereas control coefficients are considered system properties.

$$\varepsilon^v_S = \frac{\partial v}{\partial S} \frac{S}{v} = \frac{\partial \ln v}{\partial \ln S}$$

#### Flux

The flux is the steady state flow of mass through a pathway.

#### Flux Control Coefficient

A flux control coefficient measures the relative steady-state change in pathway flux (J)  in response to a relative change in enzyme activity, $e_i$ often through changes in enzyme concentration.  This definition assumes that the enzyme concentration is under the direct control of the experimenter and as such can be classed as a parameter. Flux Control coefficients are properties of the intact system.

$$C^J_{e_i} = \frac{dJ}{de_i} \frac{e_i}{J} \right) = \frac{d\ln J}{d\ln e_i}$$

A more generalized definition in terms of changes to the local reaction rate $v_i$ of step $i$ is given by:

$$C^J_{v_i} = \left( \frac{dJ}{dp} \frac{p}{J} \right) \bigg/ \left( \frac{dv_i}{dp}\frac{p}{v_i} \right) = \frac{d\ln J}{d\ln v_i}$$

so that the definition is now independent of the particular parameter, $p$, used to perturb the reaction step.

#### Parameter

A parameter is a measurable characteristic of a system that can in principle be controlled by the observer. Parameters are also often called independent variables. By definition, a parameter cannot be changed by the system itself, if it can then it is called a variable. Examples of parameters include external concentrations such as glucose fed to a culture or externally added drug compounds, internal parameters such as kinetic constants, and depending on the system under study, enzyme concentrations.

#### Regulation

Regulation may be defined as the capacity to achieve control. Such control may involve homeostasis or the ability to move from one state to another in a particular manner.

#### Summation Theorem

The summation theorem for flux control coefficients states that the sum of all the flux control coefficients in a pathway should equal 1. Similarly, the concentration control coefficients will sum to 0.

$$\sum_i C^J_{v_i} = 1$$

$$\sum_i C^S_{v_i} = 0$$

#### Rate Limiting Step

Much of the old literature from the 1960s onwards and even some of the current literature including textbooks refer to a concept called the rate-limiting step. This concept has been also called the pathway's bottleneck, committed step or pacemaker. This concept states that in a biochemical pathway, there is a single step that completely controls the pathway's steady-state flux. Unfortunately, this idea is too simplistic and both theory and experimental evidence suggests that control is shared in a pathway and no single reaction step has full control of the steady-state flux. The concept of the rate-limiting step should therefore be abandoned in favor of a more quantitative definition, such as the flux control coefficient.

#### Response Coefficient

A response coefficient measures the relative steady state change in pathway flux (J) or species concentration  (S) in response to a relative change in an external species, eg a drug or supplying substrate.

$$R^J_x = \frac{dJ}{dX} \frac{X}{J} = \frac{\ln J}{\ln X}$$

$$R^J_x = \frac{dS}{dX} \frac{S}{J} = \frac{\ln S}{\ln X}$$

#### Variables

A variable also called a dependent variable or state variable is a measurable characteristic of a system that can only be changed by an observer through changes to a suitable parameter. Variables are by definition determined by the system. Examples of possible variables include the pathway flux and species concentrations, such as metabolites or proteins.