Elasticity coefficient

Last updated October 13, 2024

In chemistry, the rate of a chemical reaction is influenced by many different factors, such as temperature, pH, reactant, the 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:

Calculating elasticity coefficients
Algebraic calculation of elasticity coefficients
Summation property of elasticity coefficients
Elasticity with respect to enzyme concentration
Numerical calculation of elasticity coefficients
Interpretation of the log form
Differentiating in log space
Elasticity matrix
See also
References
Further reading

$\varepsilon _{s_{i}}^{v}=\left({\frac {\partial v}{\partial s_{i}}}{\frac {s_{i}}{v}}\right)_{s_{j},\,s_{k},\,\ldots }={\frac {\partial \ln v}{\partial \ln s_{i}}}\approx {\frac {v\%}{s_{i}\%}}$

where $v$ denotes the reaction rate and $s$ denotes the substrate concentration. Be aware that the notation will use lowercase roman letters, such as $s,$ to indicate concentrations.

The partial derivative 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 and was introduced in the early 1970s and possibly earlier by Henrik Kacser and Burns^[1] in Edinburgh and Heinrich and Rapoport^[2] in Berlin.

The elasticity concept has also been described by other authors, most notably Savageau^[3] in Michigan and Clarke^[4] at Edmonton. In the late 1960s Michael Savageau^[3] developed an innovative approach called biochemical systems theory that uses power-law expansions to approximate the nonlinearities in biochemical kinetics. The theory is very similar to metabolic control analysis 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^[4] 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.^[5]

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, 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}^{n_{1}}s_{2}^{n_{2}}

where $v$ is the reaction rate, $k$ the reaction rate constant, $s_{i}$ is the ith chemical species involved in the reaction and $n_{i}$ the ith reaction order, then the elasticity, $\varepsilon _{s_{1}}^{v}$ can be obtained by differentiating the rate law with respect to $s_{1}$ and scaling:

\varepsilon _{s_{1}}^{v}={\frac {\partial v}{\partial s_{1}}}{\frac {s_{1}}{v}}=n_{1}\ k\ s_{1}^{n_{1}-1}s_{2}^{n_{2}}{\frac {s_{1}}{k\ s_{1}^{n_{1}}s_{2}^{n_{2}}}}=n_{1}

That is, the elasticity for a mass-action rate law is equal to the order of reaction of the species.

For example the elasticity of A in the reaction $2A\rightleftharpoons C$ where the rate of reaction is given by: $v=kA^{2}$ , the elasticity can be evaluated using:

$\varepsilon _{a}^{v}={\frac {\partial v}{\partial a}}{\frac {a}{v}}={\frac {2kaa}{ka^{2}}}=2$

Elasticities can also be derived for more complex rate laws such as the Michaelis–Menten rate law. If

v={\frac {V_{\max }s}{K_{m}+s}}

then it can be easily shown than

\varepsilon _{s}^{v}={\frac {K_{m}}{K_{m}+s}}

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.

A. The slope of the reaction rate versus the reactant concentration scaled by both the reactant concentration and reaction rate yields the elasticity. If the log of the reaction rate and the log of the reactant concentration is plotted, the elasticity can be read directly from the slope of the curve. Curves were generated by assuming v = s/(2 + s) ElasticityCurveEnzyme.png — A. The slope of the reaction rate versus the reactant concentration scaled by both the reactant concentration and reaction rate yields the elasticity. If the log of the reaction rate and the log of the reactant concentration is plotted, the elasticity can be read directly from the slope of the curve. Curves were generated by assuming v = s/(2 + s)

For the reversible Michaelis–Menten rate law:

v={\frac {V_{\max }/K_{m_{1}}(s-p/K_{\text{eq}})}{1+s/K_{m_{1}}+p/K_{m_{2}}}}

where $V_{\max }$ is the forward $V_{max}$ , $K_{m_{1}}$ the forward $K_{m}$ , $K_{eq}$ the equilibrium constant and $K_{m_{2}}$ 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 _{s}^{v}={\frac {1}{1-\Gamma /K_{\text{eq}}}}-{\frac {s/K_{m_{1}}}{1+s/K_{m_{1}}+p/K_{m_{2}}}}

\varepsilon _{p}^{v}={\frac {-\Gamma /K_{\text{eq}}}{1-\Gamma /K_{\text{eq}}}}-{\frac {p/K_{m_{2}}}{1+s/K_{m_{1}}+p/K_{m_{2}}}}

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 {V_{\max }(s/K_{s})^{n}}{1+(s/K_{s})^{n}}}

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 _{s}^{v}={\frac {n}{1+(s/K_{s})^{n}}}

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 _{s}^{v}={\frac {1}{1-\Gamma /K_{\text{eq}}}}-{\frac {s/K_{m_{1}}}{1+s/K_{m_{1}}+p/K_{m_{2}}}}

\varepsilon _{p}^{v}={\frac {-\Gamma /K_{\text{eq}}}{1-\Gamma /K_{\text{eq}}}}-{\frac {p/K_{m_{2}}}{1+s/K_{m_{1}}+p/K_{m_{2}}}}

An interesting result can be obtained by evaluating the sum $\varepsilon _{s}^{v}+\varepsilon _{p}^{v}$ . This can be shown to equal:

$\varepsilon _{s}^{v}+\varepsilon _{p}^{v}={\frac {1}{1+s/K_{m_{1}}+p/K_{m_{2}}}}$

Two extremes can be considered. At high saturation ( $s>K_{m_{1}},p>K_{m_{2}}$ ), the right-hand term tends to zero so that:

$\varepsilon _{s}^{v}\approx -\varepsilon _{p}^{v}$

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 ( $s<K_{m_{1}},p<K_{m_{2}}$ ). Under these conditions we obtain the simpler result:

$\varepsilon _{s}^{v}+\varepsilon _{p}^{v}\approx 1$

Expressed in a different way we can state:

$||\varepsilon _{s}^{v}||>||\varepsilon _{p}^{v}||$

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.^[6]

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.^[7]^[8]

The table below summarizes the extreme values for the elasticities given a reversible Michaelis-Menten rate law. Following Westerhoff et al.^[9] the table is split into four cases that include one 'reversible' type, and three 'irreversible' types.

Approximate values of elasticities under various conditions
Equilibrium State	Saturation Levels	Elasticities
Near Equilibrium	All degrees of saturation	$\varepsilon _{s}^{v}\gg 1,\varepsilon _{p}^{v}\ll -1$
Out of Equilibrium	High Substrate, high product	$\varepsilon _{s}^{v}\approx -\varepsilon _{p}^{v}$
Out of Equilibrium	High Substrate, low product	$\varepsilon _{s}^{v}\approx 0,\varepsilon _{p}^{v}\approx 0$
Out of Equilibrium	Low Substrate, high product	$\varepsilon _{s}^{v}\approx 1,\varepsilon _{p}^{v}\approx -1$
Out of Equilibrium	Low Substrate, low product	$\varepsilon _{s}^{v}\approx 1,\varepsilon _{p}^{v}\approx 0$

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 $k_{cat}$ and total enzyme concentration, $E_{t}$ :

$v={\frac {k_{cat}E_{t}s}{K_{m}+s}}$

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:

$\varepsilon _{E_{t}}^{v}={\frac {\partial v}{\partial E_{t}}}{\frac {E_{t}}{v}}=f(s){\frac {E_{t}}{E_{t}f(s)}}=1$

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.^[10]

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:

$\varepsilon _{s}^{v}\simeq {\frac {v_{1}-v_{o}}{\Delta s_{o}}}{\frac {s_{o}}{v_{o}}}={\frac {v_{1}-v_{o}}{v_{o}}}/{\frac {s_{1}-s_{o}}{s_{o}}}$

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:

$\varepsilon _{s}^{v}\simeq {\frac {1}{2}}{\frac {v_{1}-v_{2}}{s_{1}-s_{o}}}\left({\frac {s_{o}}{v_{o}}}\right)$

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:

${\frac {f(x+h)-f(x)}{f(x)}}$

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 $=$

$\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{hf(x)}}={\frac {1}{f(x)}}\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}={\frac {1}{y}}{\frac {dy}{dx}}$

Using calculus, we know that

${\frac {d\ln y}{dx}}={\frac {1}{y}}{\frac {dy}{dx}}$ ,

therefore the rate of proportional change equals:

${\frac {d\ln y}{dx}}$

This quantity serves as a measure of the rate of proportional change of the function $y$ . Just as $dy/dx$ measures the gradient of the curve $y=f(x)$ plotted on a linear scale, $d\ln y/dx$ 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:

${\frac {d\ln y}{d\ln x}}={\frac {x}{y}}{\frac {dy}{dx}}$

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 _{s}^{v}={\frac {\partial \ln v}{\partial \ln s}}

differentiating in log space is an obvious approach. Logarithmic differentiation is particularly convenient in algebra software such as Mathematica or Maple, where logarithmic differentiation rules can be defined.^[11]

A more detailed examination and the rules differentiating in log space can be found at Elasticity of a function.

Elasticity matrix

The unscaled elasticities can be depicted in matrix form, called the unscaled elasticity matrix, ${\mathcal {E}}$ . Given a network with $m$ molecular species and $n$ reactions, the unscaled elasticity matrix is defined as:

{\mathcal {E}}={\begin{bmatrix}{\dfrac {\partial v_{1}}{\partial s_{1}}}&\cdots &{\dfrac {\partial v_{1}}{\partial s_{m}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial v_{n}}{\partial s_{1}}}&\cdots &{\dfrac {\partial v_{n}}{\partial s_{m}}}\end{bmatrix}}.

Likewise, is it also possible to define the matrix of scaled elasticities:

\mathbf {\varepsilon } ={\begin{bmatrix}\varepsilon _{s_{1}}^{v_{1}}&\cdots &\varepsilon _{s_{m}}^{v_{1}}\\\vdots &\ddots &\vdots \\\varepsilon _{s_{1}}^{v_{n}}&\cdots &\varepsilon _{s_{m}}^{v_{n}}\end{bmatrix}}.

Related Research Articles

In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. This state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as dynamic equilibrium.

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

In engineering, deformation may be elastic or plastic. If the deformation is negligible, the object is said to be rigid.

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as $or in Leibniz's notation as$

A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of change of the fluid's velocity vector.

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 towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

In chemistry, the rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters only. For many reactions, the initial rate is given by a power law such as

In biochemistry and pharmacology, the Hill equation refers to two closely related equations that reflect the binding of ligands to macromolecules, as a function of the ligand concentration. A ligand is "a substance that forms a complex with a biomolecule to serve a biological purpose", and a macromolecule is a very large molecule, such as a protein, with a complex structure of components. Protein-ligand binding typically changes the structure of the target protein, thereby changing its function in a cell.

The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.

Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner. Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.

<span class="mw-page-title-main">Metabolic control analysis</span> Mathematical model of biochemical pathways

In biochemistry, metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and species concentrations, depend on network parameters. In particular, it is able to describe how network-dependent properties, called control coefficients, depend on local properties called elasticities or elasticity coefficients.

Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.

The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressure; i.e., at these conditions the adsorbate's partial pressure $is related to its volume V adsorbed onto a solid adsorbent. The adsorbent, as indicated in the figure, is assumed to be an ideal solid surface composed of a series of distinct sites capable of binding the adsorbate. The adsorbate binding is treated as a chemical reaction between the adsorbate gaseous molecule and an empty sorption site S . This reaction yields an adsorbed species with an associated equilibrium constant :$

Anatoly Alexeyevich Karatsuba was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series.

The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology.

Equilibrium chemistry is concerned with systems in chemical equilibrium. The unifying principle is that the free energy of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to the reaction coordinate is zero. This principle, applied to mixtures at equilibrium provides a definition of an equilibrium constant. Applications include acid–base, host–guest, metal–complex, solubility, partition, chromatography and redox equilibria.

The non-radiative dielectric (NRD) waveguide was introduced by Yoneyama in 1981. In Fig. 1 the crosses shown: it consists of a dielectric rectangular slab of height (a) and width (b), which is placed between two metallic parallel plates of a suitable width. The structure is practically the same as the H waveguide, proposed by Tischer in 1953. Due to the dielectric slab, the electromagnetic field is confined in the vicinity of the dielectric region, whereas in the outside region for suitable frequencies, the electromagnetic field decays exponentially. Therefore, if the metallic plates are sufficiently extended, the field is practically negligible at the end of the plates and therefore the situation does not greatly differ from the ideal case in which the plates are infinitely extended. The polarization of the electric field in the required mode is mainly parallel to the conductive walls. As it is known, if the electric field is parallel to the walls, the conduction losses decrease in the metallic walls at the increasing frequency, whereas, if the field is perpendicular to the walls, losses increase at the increasing frequency. Since the NRD waveguide has been devised for its implementation at millimeter waves, the selected polarization minimizes the ohmic losses in the metallic walls.

Control coefficients measure the response of a biochemical pathway to changes in enzyme activity. The response coefficient, as originally defined by Kacser and Burns, is a measure of how external factors such as inhibitors, pharmaceutical drugs, or boundary species affect the steady-state fluxes and species concentrations. The flux response coefficient is defined by:

Enzymes are proteins that act as biological catalysts by accelerating chemical reactions. Enzymes act on small molecules called substrates, which an enzyme converts into products. Almost all metabolic processes in the cell need enzyme catalysis in order to occur at rates fast enough to sustain life. The study of how fast an enzyme can transform a substrate into a product is called enzyme kinetics.

The stoichiometric structure and mass-conservation properties of biochemical pathways gives rise to a series of theorems or relationships between the control coefficients and the control coefficients and elasticities. There are a large number of such relationships depending on the pathway configuration which have been documented and discovered by various authors. The term theorem has been used to describe these relationships because they can be proved in terms of more elementary concepts. The operational proofs in particular are of this nature.

References

↑ Kacser, Henrik; Burns, J. (1973). "The control of flux". Symposia of the Society for Experimental Biology. 27: 65–104. PMID 4148886.
↑ Heinrich, Reinhart; A. Rapoport, Tom (1974). "A Linear Steady-State Treatment of Enzymatic Chains: General Properties, Control and Effector Strength". European Journal of Biochemistry. 42 (1): 89–95. doi: 10.1111/j.1432-1033.1974.tb03318.x . PMID 4830198.
1 2 A. Savageau, Michael (1976). Biochemical Systems Analysis. Addison Wesley Longman Publishing Company.
1 2 L. Clarke, Bruce (1980). "Stability of Complex Reaction Networks". Advances in Chemical Physics. 43: 1–215. doi:10.1002/9780470142622.ch1. ISBN 9780470142622.
↑ Christensen, Carl D.; Hofmeyr, Jan-Hendrik S.; Rohwer, Johann M. (28 November 2018). "Delving deeper: Relating the behaviour of a metabolic system to the properties of its components using symbolic metabolic control analysis". PLOS ONE. 13 (11): e0207983. Bibcode:2018PLoSO..1307983C. doi: 10.1371/journal.pone.0207983 . PMC 6261606 . PMID 30485345.
↑ Sauro, Herbert (2013). Systems biology: an introduction to metabolic control analysis (1st, version 1.01 ed.). Seattle, WA: Ambrosius Publishing. ISBN 978-0982477366.
↑ Ringemann, C.; Ebenhöh, O.; Heinrich, R.; Ginsburg, H. (2006). "Can biochemical properties serve as selective pressure for gene selection during inter-species and endosymbiotic lateral gene transfer?". IEE Proceedings - Systems Biology. 153 (4): 212–222. doi:10.1049/ip-syb:20050082. PMID 16986623.
↑ Heinrich, Reinhart; Klipp, Edda (October 1996). "Control Analysis of Unbranched Enzymatic Chains in States of Maximal Activity". Journal of Theoretical Biology. 182 (3): 243–252. Bibcode:1996JThBi.182..243H. doi:10.1006/jtbi.1996.0161. PMID 8944155.
↑ Westerhoff, Hans V.; Groen, Albert K.; Wanders, Ronald J. A. (1 January 1984). "Modern theories of metabolic control and their applications". Bioscience Reports. 4 (1): 1–22. doi:10.1007/BF01120819. PMID 6365197. S2CID 27791605.
↑ Yip, Evan; Sauro, Herbert (8 October 2021). "Computing Sensitivities in Reaction Networks using Finite Difference Methods". arXiv: 2110.04335 [q-bio.QM].
↑ H. Woods, James; M. Sauro, Herbert (1997). "Elasticities in Metabolic Control Analysis: Algebraic Derivation of Simplified Expressions". Computer Applications in the Biosciences. 13 (2): 23–130. doi: 10.1093/bioinformatics/13.2.123 . PMID 9146958.