Elasticity Coefficient
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In
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
, the rate of a
chemical reaction A chemical reaction is a process that leads to the chemistry, chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an Gibbs free energy, ...
is influenced by many different factors, such as
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
, pH, reactant, the
concentration In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
of products, and other effectors. The degree to which these factors change the reaction rate is described by the elasticity coefficient. This coefficient is defined as follows: \varepsilon_^v = \left(\frac \frac\right)_ = \frac \approx \frac where v denotes the reaction rate and s denotes the
substrate Substrate may refer to: Physical layers *Substrate (biology), the natural environment in which an organism lives, or the surface or medium on which an organism grows or is attached ** Substrate (aquatic environment), the earthy material that exi ...
concentration. Be aware that the notation will use lowercase roman letters, such as s, to indicate concentrations. The
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
in the definition indicates that the elasticity is measured with respect to changes in a factor S while keeping all other factors constant. The most common factors include substrates, products, enzyme, and effectors. The scaling of the coefficient ensures that it is dimensionless and independent of the units used to measure the reaction rate and magnitude of the factor. The elasticity coefficient is an integral part of
metabolic control analysis In biochemistry, metabolic control analysis (MCA) is a mathematical framework for describing Metabolic pathway, metabolic, Cell signaling#Signaling pathways, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and Chemi ...
and was introduced in the early 1970s and possibly earlier by
Henrik Kacser Henrik Kacser FRSE (22 September 1918 – 13 March 1995) was a Austro-Hungarian-born biochemist and geneticist who worked in Britain in the 20th century. Kacser's achievements have been recognised by his election to the Royal Society of Edi ...
and Burns in Edinburgh and Heinrich and Rapoport in Berlin. The elasticity concept has also been described by other authors, most notably Savageau in Michigan and Clarke at Edmonton. In the late 1960s Michael Savageau developed an innovative approach called
biochemical systems theory Biochemical systems theory is a mathematical modelling framework for biochemical systems, based on ordinary differential equations (ODE), in which biochemical processes are represented using power-law expansions in the variables of the system. ...
that uses power-law expansions to approximate the nonlinearities in biochemical kinetics. The theory is very similar to
metabolic control analysis In biochemistry, metabolic control analysis (MCA) is a mathematical framework for describing Metabolic pathway, metabolic, Cell signaling#Signaling pathways, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and Chemi ...
and has been very successfully and extensively used to study the properties of different feedback and other regulatory structures in cellular networks. The power-law expansions used in the analysis invoke coefficients called kinetic orders, which are equivalent to the elasticity coefficients. Bruce Clarke in the early 1970s, developed a sophisticated theory on analyzing the dynamic stability in chemical networks. As part of his analysis, Clarke also introduced the notion of kinetic orders and a power-law approximation that was somewhat similar to Savageau's power-law expansions. Clarke's approach relied heavily on certain structural characteristics of networks, called extreme currents (also called elementary modes in biochemical systems). Clarke's kinetic orders are also equivalent to elasticities. Elasticities can also be usefully interpreted as the means by which signals propagate up or down a given pathway. The fact that different groups independently introduced the same concept implies that elasticities, or their equivalent, kinetic orders, are most likely a fundamental concept in the analysis of complex biochemical or chemical systems.


Calculating elasticity coefficients

Elasticity coefficients can be calculated either algebraically or by numerical means.


Algebraic calculation of elasticity coefficients

Given the definition of the elasticity coefficient in terms of a
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
, it is possible, for example, to determine the elasticity of an arbitrary rate law by differentiating the rate law by the independent variable and scaling. For example, the elasticity coefficient for a mass-action rate law such as: : v = k\ s_1^ s_2^ where v is the
reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per u ...
, k the
reaction rate constant In chemical kinetics, a reaction rate constant or reaction rate coefficient () is a proportionality constant which quantifies the rate and direction of a chemical reaction by relating it with the concentration of reactants. For a reaction between ...
, s_i is the ith
chemical species Chemical species are a specific form of chemical substance or chemically identical molecular entities that have the same molecular energy level at a specified timescale. These entities are classified through bonding types and relative abundance of ...
involved in the reaction and n_i the ith reaction order, then the elasticity, \varepsilon^v_ can be obtained by differentiating the rate law with respect to s_1 and scaling: : \varepsilon^v_ = \frac \frac = n_1\ k\ s_1^ s_2^ \frac = n_1 That is, the elasticity for a mass-action rate law is equal to the
order of reaction In chemistry, the rate equation (also known as the rate law or empirical differential rate equation) is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and ...
of the species. For example the elasticity of A in the reaction 2 A \rightleftharpoons C where the rate of reaction is given by: v = k A^2 , the elasticity can be evaluated using: \varepsilon^v_a = \frac \frac = \frac = 2 Elasticities can also be derived for more complex rate laws such as the Michaelis–Menten rate law. If : v = \frac then it can be easily shown than : \varepsilon^v_s = \frac This equation illustrates the idea that elasticities need not be constants (as with mass-action laws) but can be a function of the reactant concentration. In this case, the elasticity approaches unity at low reactant concentration (s) and zero at high reactant concentration. For the reversible Michaelis–Menten rate law: : v = \frac where V_\max is the forward V_, K_ the forward K_m , K_ the
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
and K_ the reverse K_m , two elasticity coefficients can be calculated, one with respect to substrate, S, and another with respect to product, P. Thus: : \varepsilon^v_s = \frac - \frac : \varepsilon^v_p = \frac - \frac where \Gamma is the mass-action ratio, that is \Gamma = p/s . Note that when p = 0, the equations reduce to the case for the irreversible Michaelis–Menten law. As a final example, consider the Hill equation: : v = \frac where n is the Hill coefficient and K_s is the half-saturation coefficient (cf. Michaelis–Menten rate law), then the elasticity coefficient is given by: : \varepsilon^v_ = \frac Note that at low concentrations of ''S'' the elasticity approaches ''n''. At high concentrations of ''S'' the elasticity approaches zero. This means the elasticity is bounded between zero and the Hill coefficient.


Summation property of elasticity coefficients

The elasticities for a reversible uni-uni enzyme catalyzed reaction was previously given by: : \varepsilon^v_s = \frac - \frac : \varepsilon^v_p = \frac - \frac An interesting result can be obtained by evaluating the sum \varepsilon^v_s + \varepsilon^v_p . This can be shown to equal: \varepsilon^v_s + \varepsilon^v_p = \frac Two extremes can be considered. At high saturation ( s > K_, p > K_ ), the right-hand term tends to zero so that: \varepsilon^v_s \approx -\varepsilon^v_p That is the absolute magnitudes of the substrate and product elasticities tends to equal each other. However, it is unlikely that a given enzyme will have both substrate and product concentrations much greater than their respective Kms. A more plausible scenario is when the enzyme is working under sub-saturating conditions ( s < K_, p < K_ ). Under these conditions we obtain the simpler result: \varepsilon^v_s + \varepsilon^v_p \approx 1 Expressed in a different way we can state: , , \varepsilon^v_s, , > , , \varepsilon^v_p, , That is, the absolute value for the substrate elasticity will be greater than the absolute value for the product elasticity. This means that a substrate will have a great influence over the forward reaction rate than the corresponding product. This result has important implications for the distribution of flux control in a pathway with sub-saturated reaction steps. In general, a perturbation near the start of a pathway will have more influence over the steady state flux than steps downstream. This is because a perturbation that travels downstream is determined by all the substrate elasticities, whereas a perturbation downstream that has to travel upstream if determined by the product elasticities. Since we have seen that the substrate elasticities tends to be larger than the product elasticities, it means that perturbations traveling downstream will be less attenuated than perturbations traveling upstream. The net effect is that flux control tends to be more concentrated at upstream steps compared to downstream steps. The table below summarizes the extreme values for the elasticities given a reversible Michaelis-Menten rate law. Following Westerhoff et al. the table is split into four cases that include one 'reversible' type, and three 'irreversible' types.


Elasticity with respect to enzyme concentration

The elasticity for an enzyme catalyzed reaction with respect to the enzyme concentration has special significance. The Michaelis model of enzyme action means that the reaction rate for an enzyme catalyzed reaction is a linear function of enzyme concentration. For example, the irreversible Michaelis rate law is given below there the maximal velocity, V_m is explicitly given by the product of the k_ and total enzyme concentration, E_t : v = \frac In general we can expresion this relationship as the product of the enzyme concentration and a saturation function, f(s) : v = E_t f (s) This form is applicable to many enzyme mechanisms. The elasticity coefficient can be derived as follows: \varepsilon^v_ = \frac \frac = f(s) \frac = 1 It is this result that gives rise to the control coefficient summation theorems.


Numerical calculation of elasticity coefficients

Elasticities coefficient can also be computed numerically, something that is often done in simulation software. For example, a small change (say 5%) can be made to the chosen reactant concentration, and the change in the reaction rate recorded. To illustrate this, assume that the reference reaction rate is v_o, and the reference reactant concentration, s_o . If we increase the reactant concentration by \Delta s_o and record the new reaction rate as v_1 , then the elasticity can be estimated by using Newton's difference quotient: \varepsilon_s^v \simeq \frac \frac=\frac / \frac A much better estimate for the elasticity can be obtained by doing two separate perturbations in s_o . One perturbation to increase s_o and another to decrease s_o. In each case, the new reaction rate is recorded; this is called the two-point estimation method. For example, if v_1 is the reaction rate when we increase s_o, and v_2 is the reaction rate when we decrease s_o, then we can use the following two-point formula to estimate the elasticity: \varepsilon_s^v \simeq \frac \frac\left(\frac\right)


Interpretation of the log form

Consider a variable y to be some function f(x), that is y=f(x). If x increases from x to (x+h) then the change in the value of y will be given by f(x+h)-f(x). The proportional change, however, is given by: \frac The rate of proportional change at the point x is given by the above expression divided by the step change in the x value, namely h: Rate of proportional change = \lim _ \frac=\frac \lim _ \frac=\frac \frac Using calculus, we know that \frac = \frac \frac, therefore the rate of proportional change equals: \frac This quantity serves as a measure of the rate of proportional change of the function y. Just as d y / d x measures the gradient of the curve y=f(x) plotted on a linear scale, d \ln y / d x measures the slope of the curve when plotted on a semi-logarithmic scale, that is the rate of proportional change. For example, a value of 0.05 means that the curve increases at 5 \% per unit x. The same argument can be applied to the case when we plot a function on both x and y logarithmic scales. In such a case, the following result is true: \frac=\frac \frac


Differentiating in log space

An approach that is amenable to algebraic calculation by computer algebra methods is to differentiate in log space. Since the elasticity can be defined logarithmically, that is: : \varepsilon^v_s = \frac differentiating in log space is an obvious approach. Logarithmic differentiation is particularly convenient in algebra software such as Mathematica or Maple, where
logarithmic differentiation In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function , (\ln f)' = \frac \quad \implies \quad f' = f \cdot (\ln f)' ...
rules can be defined. A more detailed examination and the rules differentiating in log space can be found at
Elasticity of a function In mathematics, the elasticity or point elasticity of a positive differentiable function ''f'' of a positive variable (positive input, positive output) at point ''a'' is defined as :Ef(a) = \fracf'(a) :=\lim_\frac\frac=\lim_\frac\frac=\lim_\frac\ap ...
.


Elasticity matrix

The unscaled elasticities can be depicted in matrix form, called the unscaled elasticity matrix, \mathcal. Given a network with m molecular species and n reactions, the unscaled elasticity matrix is defined as: : \mathcal = \begin \dfrac & \cdots & \dfrac \\ \vdots & \ddots & \vdots \\ \dfrac & \cdots & \dfrac \end. Likewise, is it also possible to define the matrix of scaled elasticities: : \mathbf = \begin \varepsilon^_ & \cdots & \varepsilon^_ \\ \vdots & \ddots & \vdots \\ \varepsilon^_ & \cdots & \varepsilon^_ \end.


See also

*
Control coefficient (biochemistry) In biochemistry, control coefficients are used to describe how much influence a given reaction step has on the flux or concentration of the species at steady state. This can be accomplished experimentally by changing the expression level of a giv ...
*
Metabolic control analysis In biochemistry, metabolic control analysis (MCA) is a mathematical framework for describing Metabolic pathway, metabolic, Cell signaling#Signaling pathways, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and Chemi ...


References


Further reading

* * *{{cite book, last1=Heinrich, first1=Reinhart, last2=Schuster, first2=Stefan, year=1996, title=The Regulation of Cellular Systems, publisher=Chapman and Hall Chemical kinetics Biochemistry methods Metabolism Mathematical and theoretical biology Systems biology