## Pages

### Enzyme Kinetics

#### 81-fold

81-fold refers to the fold change in substrate required to change the reaction rate from 10\% of the rate to 80\% of the rate for a simple Michaelis hyperbolic response.

#### Active Site

The active site is that part of an enzyme or other protein where a reactant or effector can bind and take part in a catalytic transformation. The active site will usually only bind specific molecules and is therefore highly selective.

The Adair equation was the first attempt to present a physically reasonable model that described cooperativity. It was first applied to hemoglobin which is a four subunit protein, where the model assumed that as each oxygen molecule binds to a hemoglobin molecule, the binding constant on the same oligomer for subsequent oxygen bind events increases. The model thus describes a gradual change in binding affinities as the hemoglobin molecule becomes saturated with oxygen.

#### Allostery

The word allosteric is derived from the Greek {\em allos} and {\em stereos{, means other-solid. In enzymology an allosteric effector is a ligand that can bind to a site other than the active site.

#### Arrhenius Equation

The Arrhenius equation describes the dependence of a rate constant on temperature, that is:

$$k = A \exp \left(\frac{-E_a}{R\ T}\right)$$

where $k$ is the rate constant and $T$ the temperature. There are three constants in the equation, $A$, $E_a$ and $R$. $A$ is called the pre-exponential factor and $E_a$ the activation energy. $E_a$ can be interpreted as the minimum kinetic energy that reactants must have during a collision in order to form products. $R$ is the gas constant. The equation shows that the higher the temperature the higher the rate constant.

#### Association Constant

The association constant ($K_a$) is the equilibrium constant for the association of a complex from its component parts. For example, the association constant for the reaction, $H + A \rightleftharpoons HA$ is given by:

$$K_a = \frac{HA}{H \cdot A}$$

#### Catalyst

A catalyst is a substance that can speed up a chemical reaction towards equilibrium without itself been consumed.

#### Catalytic Constant

[latexpage]

The catalytic constant is the rate constant that describes the first-order transformation of enzyme-substrate complex into product and free enzyme.

#### Combinatorial Equation

Consider an oligomer with $n$ binding sites. The number of ways, $C$, to place $i$ ligands is given by the combinatorial equation:

$$C(n, i) = \frac{n!}{(n-i)!\ i!} = \binom{n}{i}$$

#### Competitive Inhibitor

A competitive inhibitor is a molecule that is structurally similar to the natural substrate of an enzyme such that it is capable of binding to the active site. The small molecule is also assumed to be catalytically inactive, that is, it is not transformed into any product.  The small molecule can can therefore compete with the substrate and thereby inhibit the reaction rate with respect to the substrate. Due to the competitive nature of the inhibition, the natural substrate can out compete a competitive inhibitor at sufficiently high concentrations. A competitive inhibitor has no effect on the enzyme's $V_m$ but will increase the substrate $K_m$.

#### Compulsory Order Mechanism

In multiractant enzymne catalyzed reactions, if the reactants must bind to the enzyme active site in a specific order,  the reaction type is called a compulsory order reaction or compulsory order mechanism.

#### Cooperativity

In enzymology, cooperativity is where the binding of one ligand affects the binding of subsequent ligands. The effect can either enhance subsequent binding (called positive cooperativity) or reduce binding (called negative cooperativity).

#### Detailed Balance

If a sequence of chemical reactions forms a closed cycle, for example, $A \rightarrow B$, $B \rightarrow C$ and $C \rightarrow A$, then the product of the equilibrium constants around the cycle will equal one, that is the net free energy change around the cycle is zero. This is termed detailed balance. There is a more general definition of detailed balance in statistical mechanics which will not be given here.

#### Disequilibrium Ratio

The disequilibrium ratio ($\rho$) is the ratio of the mass-action ratio to the equiblirum constant, that is:

$$\rho = \frac{\Gamma}{K_{eq}}$$

At equilibrium $\rho = K_{eq}$.

#### Dissociation Constant

The dissociation constant ($K_d$) is the equilibrium constant for the dissociation of a complex into its component parts. For example, the dissociation constant for the reaction, $HA \rightleftharpoons H+ A$ is given by:

$$K_d = \frac{H \cdot A}{HA}$$

#### Elasticity Coefficient

The elasticity ($\varepsilon$) describes the sensitivity of a reaction rate with respect to a given reactant, product or effector. It it defined by the relation:

$$\varepsilon^v_x = \frac{\partial v}{\partial x} \frac{v}{x} = \frac{\ln v}{\ln x} \approx \frac{v \%}{x \%}$$

where $x$ is a reactant, product or other effector that can influence the reaction rate. The derivative is defined such that all other possible factors other than $x$ are fixed. In general substrate elasticities are positive, product elasticities negative, inhibitor elasticities negative and activator elasticities positive.

#### Elementary Reaction

A chemical reaction that does not involve any reaction intermediates is called an elementary reaction.

#### Energy

Classically, energy is defined as the capacity to do work which doesn't really tell us what it is just what we can do with it if we had some. The actual nature of energy is still somewhat mysterious but energy is known to exist in many forms and most importantly of all it appears that when energy is transferred in a process, it is neither created or destroyed. However, during the transfer some energy may be converted to heat. The problem with heat is that it is difficult with 100\% efficiency to convert heat to other more useful forms of energy. Energy in the form of heat is therefore considered a poor form of energy.

#### Enzyme

An enzyme is a protein molecule that behaves as a catalyst.

#### Equilibrium Constant

The equilibrium constant describes the ratio of the product to the reactants for a chemical reaction at equilibrium. That is, for a simple reaction such as $A \rightarrow B$, the equilibrium constant is given by:

$$K_{eq} = \frac{B}{A}$$

In general for a reaction of the form:

$$n_1 A + n_2 B + \ldots \rightarrow m_1 P + m_2 Q + \ldots$$

the equilibrium constant is given by:

$$K_{eq} + \frac{A^{n_1} B^{n_2} \ldots}{P^{m_1} Q^{m_2} \ldots}$$

#### Exclusive MWC Model

In the exclusive model, the ligand can only bind to the relaxed form in the MWC model.

#### Extensive Property

An extensive property is one where the property is proportional to the size of the system. For example, mass, energy and and volume are extensive properties. Mass is extensive because the larger the system the more mass there is.

#### First Law of Thermodynamics

The first law of thermodynamics states that the total energy is a closed system is fixed. In other words it is not possible to create or destroy energy, merely interconvert one form of energy into another.

#### First-Order Reaction

A first-order reaction is one where the reaction rate is a linear function of the concentration of a given reactant. For example, in the rate equation: $v = k A$, the reaction rate is first-order with respect to species $A$.

#### Fractional  Saturation

The fractional saturation refers to the degree of saturation of an oligmeric structure to ligand, that is:

$$Y = \frac{\mbox{Number of bound sites}}{\mbox{Total number of binding sites}}$$

#### Generalized Inhibition Model

The generalized inhibition model, also known as the Botts-Morales scheme merges all the commonly known inhibition patterns, such as competitive, uncompetitive, mixed and others. The Botts-Morales scheme is sufficiently general to also accommodate activation models.

#### Haldane Relationship

When considering reversible enzyme catalyzed reactions, the Haldane relationship describes the relation between the equilibrium constant of the reaction and the forward and reverse kinetic constants, $K_m$ and $V_m$.For a single substrate, single product reversible reaction, the Haldane relationship is given by:

$$K_{eq} = \frac{V_m^f K_m^P}{V_m^r K_m^S}$$

where $V_m^f$ is the forward maximal rate, $V_m^r$ is the reverse maximal rate, $K_m^S$ the Michaelis constant for substrate and $K_m^P$ the Michaelis constant for product.

#### Half-Life

The half-life of a first-order chemical reaction is the time it takes for half the reactant to be consumed. If a first-order reaction has a rate constant, $k$, then the half-life $t_{1/2}$ is given by:

$$t_{1/2} = \frac{\ln 2}{k}$$

#### Heterotropic Regulation

If a ligand affects the binding of another molecule to a protein, then the ligand is called a heterotropic effector. There are many examples of heterotropic effectors. For example, 2,3-diphosphoglycerate is a heterotropic effector of hemoglobin. When 2,3-diphosphoglycerate binds to hemoglobin it lowers the binding affinity of hemoglobin to oxygen.

#### Hill Coefficient

The Hill coefficient, denoted by $h$, is a positive number defined as the midpoint of the Hill plot, $\log [Y/(1-Y)]$ versus $\log x$ where $x$ is the ligand and $Y$ the fractional saturation by ligand. The midpoint is also the point where $\log[Y/(1-Y)] = 0$. When $h=1$, the behavior is hyperbolic, shows positive cooperativity when $h>1$ and negative cooperativity when $h<1$.

#### Hill Equation

The Hill equation is an empirical relationship between the fractional saturation of a protein by ligand and the ligand concentration. The equation can also be derived by assuming that for a protein with $n$ ligand binding sites, the $n$ ligands bind simultaneously to the protein. Physically this is an unrealistic scenario but by assuming a rapid-equilibrium model, this model will lead to the Hill equation:

$$Y = \frac{L^h}{K_d + L^h}$$

where $h$ is the Hill coefficient, $L$ the ligand concentration, $K_d$ the dissociation constant for ligand binding, and $Y$ the fractional saturation of the binding sites with respect to ligand. If the protein is an enzyme, the the rate of reaction is often assumed to be proportional to the fractional saturation by ligand.

#### Homotropic Regulation

If the binding of one ligand affects the binding of similar ligands to a oligomeric protein, then the ligand is called a homotropic effector. An example of a homotropic effector is oxygen binding to hemoglobin. When one molecule of oxygen binds, the binding affinity on the remaining oxygen binding sites is altered.

#### Intensive Property

An intensive property is one where the property is independent of the size of the system. Examples of intensive properties includes density, temperature, and pressure.

#### Km

See Michaelis constant

#### Lin-Log Approximation

[latexpage]

The lin-log equation is a more sophisticated power law approximation. The approximation is based on a logarithmic Taylor expansion around a convenient operating point. One of the chief advantages of the lin-log equation is that it approximates a saturation response at high substrate concentration. The lin-log equation is given by:

$$v= v_o \left[ \frac{e}{e_o} \right] \left( 1 + \sum_i \varepsilon^v_{S_i} \ln \left( \frac{S_i}{S_i^o} \right) \right)$$

#### Macroscopic Rate Constant

The macroscopic or apparent rate constant is the observable rate constant (or equilibrium constant) between two states. For example, consider a dimer with two independent binding sites. For a given dimer molecule, a ligand can bind to either the first or second binding site. However, because the two forms are experimentally indistinguishable all we measure is the the ratio between the sum of the two states and the fully empty state. The equilibrium constant is this case is therefore called the macroscopic or apparent equilibrium constant. The actual constants, though unobservable, are called the microscopic constants. If the sites are independent, the two microscopic constants will be identical.

#### Mass-Action

The law of mass-action states that the reaction rate of an elementary chemical reaction is proportional to the product of the concentrations of the participating reactants, each raised to the power of their respective stoichiometric amounts. For example, the forward reaction rate for the reaction: $2A + B \rightarrow C$ is given by:

$$v = k A^2 B$$

where $k$ is called the reaction rate constant.

#### Mass-Action Ratio

The mass-action ratio ($\Gamma$) of a chemical reaction is the ratio of the concentrations of product to the reactants. At equilibrium, the mass-action ratio and equilibrium constant are equal.

#### Maximal Velocity

The maximal velocity, often denoted, $V_m$, is the reaction rate that an enzyme catalyzed reaction achieves at high reactant concentrations. It is the maximal rate that an enzyme catalyzed reaction can reach.

#### Michaelis Constant

For a simple irreversible Michaelis-Menten reaction, the Michaelis constant ($K_m$) is the concentration of substrate that yields half the maximal rate.

#### Michaelis-Menten Equation

The Michaelis-Menten equation is an equation that describes the rate of an enzyme catalyzed reaction as a function of the substrate concentration by assuming the rapid-equilibrium assumption. The Michaelis-Menten equation is often confused with the Briggs-Haldane equation which assumes steady-state. In both cases however the resulting equation is looks almost identical except for the interpretation of the kinetic constant, $K_d$ or $K_m$. In the Michael-Menten equation, this constant is the dissociation constant ($K_d$) of substrate binding, while in the Briggs-Haldane equation it is the half-maximal saturation constant, ($K_m$).The equation below represents the Briggs-Haldane equation.

$$v = \frac{V_m S}{K_m + S}$$

#### Microscopic Rate Constant

See macroscopic rate constant.

#### Monod-Wyman-Changeux Model

The Monod-Wyman-Changeux (MWC) model is frequently use to model positive cooperativity and allosteric control. The model assumes that a given oligomer can only exist in two states. This assertion stems from the notion that most if not all oligomers have at least one degree of symmetry suggesting that state transition occur throughout the oligomer when a ligand binds. This is in contrast to the Adair model where transitions in state occur one monomer at a time. The MWC can not only describe positive cooperativity but also can accommodate allosteric effectors where a ligand can displace the equilibrium between the two states.

#### Monomer

In protein biochemistry, a monomer is a single folded polypeptide chain. An example of a monomer is myoglobin which is made up of a single polypeptide chain.

#### Noncompetitive Inhibition

If an inhibitor can bind to both free enzyme and enzyme-substrate complex then the inhibition is terms noncompetitive inhibition. In general this type of inhibition is called mixed inhibition but if the binding of inhibitor to has no effect on the binding of substrate then the inhibition is termed pure noncompetitive inhibition otherwise it is called mixed noncompetitive inhibition. In pure noncompetitive inhibition the effect is to change the $V_m$ while the $K_m$ is unaffected. Pure noncompetitive inhibition is however quite rate.

#### Nonexclusive MWC Model

In the nonexclusive MWC model, the ligand can bind to both the tense and relaxed forms. The nonexclusive is also known as the generalized MWC model.

#### Oligomer

[latexpage]

In protein biochemistry, an oligomer is a protein made of two or more monomers. An example of an oligomer is phosphofructokinase from E. coli which is made up of four identical monomers.

#### Ping-Pong Reaction

In multiractant enzyme catalyzed reactions, if the first reactant must bind and unbind to the active site before the second reactant can bind at the active site, then the reaction type is called a ping-pong mechanism, also known as a double-displacement reaction.

#### Power Laws

There are a number of approximate rate laws used in building biochemical models. A common type is the power law:

$$v_i = \alpha_i \PI S_j^{\varepsilon^v_S}$$

where $\varepsilon^v_S$ is the elasticity or kinetic order of the reaction. Like many approximations, the power law is centered around a particular operating point where the approximation is the most accurate. The reaction rate at zero substrate is zero, however at high substrate concentration the reaction rate diverges significantly. One issue with a power law approximation is that the response doesn't saturate at high substrate concentrations.

#### Random Order Reaction

In multiractant enzyme catalyzed reactions, if the reactants can bind to the enzyme active site in any order,  the reaction type is called a random order reaction or random order mechanism.

#### Rapid-Equilibrium Approximation

To simplify the quantitative description of enzyme action, a number of assumptions are invoked. In one such simplification, the assumption of rapid-equilibrium is used. The mechanism of enzyme action proposes the reversible binding of free enzyme to substrate to form the enzyme-substrate complex. The complex then undergoes a transformation with the release of free enzyme and product. In the rapid-equilibrium assumption, it is assumed that the binding and unbinding of substrate to free enzyme is very rapid compared to the release of product, that is, it is assumed that the binding of substrate to enzyme is at thermodynamic equilibrium. See steady state approximation for an alternative.

#### Rate of Change

The rate of change, $dX/dt$, is defined as the rate of change in concentration or amount of a designated chemical species, $X$.

#### Rate Constant

The rate constant is the proportionality factor that appears in mass-action rate laws. For a simple first-order reaction, the rate constants has units of per unit time ($t^{-1}$). The rate constant is a function of temperature as described by the Arrhenius equation.

#### Reaction Rate

The reaction rate, $v_i$ is the rate of change of a given species, divided by the stoichiometric coefficient. For example, in the reaction $2 A \rightarrow B$, the reaction rate with respect to species $A$ is given by: $$v = \frac{1}{-2} \frac{dA}{dt}$$

#### Reversible Briggs-Haldane Equation

Most enzyme texts describe the irreversible Briggs-Haldane equation. For modeling however it is more usefu to consider the reversible form:

$$v = \frac{V_m/K_s (S - P/K_{eq})}{1 + S/K_s + P/K_p}$$

where $K_{eq}$ is the equilibrium constant for the reaction, $K_s$ and $K_p$ are the half-maximal constants for the substrate and product respectively, and $V_m$ is the forward maximal rate.

#### Second Law of Thermodynamics

The second law of thermodynamics comes in many forms but essentially expresses the observation that over time both matter and energy tend to disperse. For example, the Clausius version of the law states that, "No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature".

#### Stoichiometric Amount

The stoichiometric amount is the number of molecules of a particular reactant or product taking part in a reaction.

#### Stoichiometric Coefficient

The stoichiometric coefficient, $c_i$, for a molecular species $A_i$ is the difference between the stoichiometric amount of the species on the product side and the stoichiometric amount on the reactant side.

#### Second-Order Reaction

A second-order reaction is one where the reaction rate is a function of the square of the concentration of a given species. For example, in the rate equation: $v = k A^2$, the reaction rate is second-order with respect to species $A$.

At steady-state the concentrations if all species do not change even though at the same time the net reaction rates are non-zero. In contrast, at thermodynamic equilibrium, the net reaction rates are zero.

To simplify the quantitative description of enzyme action, a number of assumptions are invoked. In one such simplification, the assumption of steady-state assumption is used. The mechanism of enzyme action proposes the reversible binding of free enzyme to substrate to form the enzyme-substrate complex. The complex then undergoes a transformation with the release of free enzyme and product. In the steady-state assumption, it is assumed that the concentration of enzyme-substrate complex very rapidly reaches steady-state before any significant amount of substrate are consumed or product produced.  See rapid-equilibrium approximation for an alternative.

#### Thermodynamic Equilibrium

A chemical reaction is in thermodynamic equilibrium when the forward and reverse rates of the reaction are equal, that is the net reaction rate is zero.

#### Vm

See Maximal Velocity

#### Uncompetitive Inhibition

Uncompetitive inhibition occurs when an inhibitor only binds when substrate is also bound to the enzyme. This results in both the $V_m$ and $K_m$ changing, however the ratio of $V_m/K_m$ remains the same. In reality, uncompetitive inhibition is rare.

#### Zero-Order Reaction

A zero-order reaction is one where the reaction rate is independent of the concentration of a reactant.