= 1 = v max ( k i + [ i ] − [ i ] k i + [ i ] ) add [ i ] − [ i ] = 0 to numerator = v max ( 1 − [ i ] k i + [ i ] ) simplify k i + [ i ] k i + [ i ] = 1 = v max − v max [ i ] k i + [ i ] multiply out by v max &=\left(+}}\right)&&=1\\&=\left(+-}+}}\right)&&-=0\\&=\left(1-+}}\right)&&+}+}}=1\\&=v_-v_+}}&&v_\end this rearrangement demonstrates that similar to the michaelis–menten equation, the maximal rate of reaction depends on the proportion of the enzyme population interacting with its substrate. fraction of the enzyme population bound by substrate [ s ] [ s ] + k m fraction of the enzyme population bound by inhibitor [ i ] [ i ] + k i the effect of the inhibitor is a result of the percent of the enzyme population interacting with inhibitor. the only problem with this equation in its present form is that it assumes absolute inhibition of the enzyme with inhibitor binding, when in fact there can be a wide range of effects anywhere from 100% inhibition of substrate turn over to no inhibition. to account for this the equation can be easily modified to allow for different degrees of inhibition by including a delta vmax term.: 361 v max − δ v max [ i ] [ i ] + k i -\delta v_ or v max 1 − ( v max 1 − v max 2 ) [ i ] [ i ] + k i -(v_-v_) this term can then define the residual enzymatic activity present when the inhibitor is interacting with individual enzymes in the population. however the inclusion of this term has the added value of allowing for the possibility of activation if the secondary vmax term turns out to be higher than the initial term. to account for the possibly of activation as well the notation can then be rewritten replacing the inhibitor "i" with a modifier term (stimulator or inhibitor) denoted here as "x".: eq 13 v max 1 − ( v max 1 − v max 2 ) [ x ] [ x ] + k x -(v_-v_) while this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity of the michaelis–menten equation, it highlights potential problems with the term used to describe effects relating to the km. the km relating to the affinity of the enzyme for the substrate should in most cases relate to potential changes in the binding site of the enzyme which would directly result from enzyme inhibitor interactions. as such a term similar to the delta vmax term proposed above to modulate vmax should be appropriate in most situations:: eq 14 k m 1 − ( k m 1 − k m 2 ) [ x ] [ x ] + k x -(k_-k_) dissociation constants lineweaver–burk diagrams of different types of reversible enzyme inhibitors. the arrow shows the effect of increasing concentrations of inhibitor. an enzyme inhibitor is characterised by its dissociation constant ki, the concentration at which the inhibitor half occupies the enzyme. in non-competitive inhibition the inhibitor can also bind to the enzyme-substrate complex, and the presence of bound substrate can change the affinity of the inhibitor for the enzyme, resulting in a second dissociation constant ki'. hence ki and ki' are the dissociation constants of the inhibitor for the enzyme and to the enzyme-substrate complex, respectively.: glossary the enzyme-inhibitor constant ki can be measured directly by various methods; one especially accurate method is isothermal titration calorimetry, in which the inhibitor is titrated into a solution of enzyme and the heat released or absorbed is measured. however, the other dissociation constant ki' is difficult to measure directly, since the enzyme-substrate complex is short-lived and undergoing a chemical reaction to form the product. hence, ki' is usually measured indirectly, by observing the enzyme activity under various substrate and inhibitor concentrations, and fitting the data via nonlinear regression to a modified michaelis–menten equation. v = v m a x [ s ] α k m + α ′ [ s ] = ( 1 / α ′ ) v m a x [ s ] ( α / α ′ ) k m + [ s ] +\alpha ^=)v_)k_+}}} where the modifying factors α and α' are defined by the inhibitor concentration and its two dissociation constants α = 1 + [ i ] k i α ′ = 1 + [ i ] k i ′ . =1+^.} thus, in the presence of the inhibitor, the enzyme's effective km and vmax become (α/α')km and (1/α')vmax, respectively. however, the modified michaelis-menten equation assumes that binding of the inhibitor to the enzyme has reached equilibrium, which may be a very slow process for inhibitors with sub-nanomolar dissociation constants. in these cases the inhibition becomes effectively irreversible, hence it is more practical to treat such tight-binding inhibitors as irreversible (see below). the effects of different types of reversible enzyme inhibitors on enzymatic activity can be visualised using graphical representations of the michaelis–menten equation, such as lineweaver–burk, eadie-hofstee or hanes-woolf plots.: 140–144 an illustration is provided by the three lineweaver–burk plots depicted in the lineweaver–burk diagrams figure. in the top diagram the competitive inhibition lines intersect on the y-axis, illustrating that such inhibitors do not affect vmax. in the bottom diagram the non-competitive inhibition lines intersect on the x-axis, showing these inhibitors do not affect km. however, since it can be difficult to estimate ki and ki' accurately from such plots, it is advisable to estimate these constants using more reliable nonlinear regression methods. partially competitive the mechanism of partially competitive inhibition is similar to that of non-competitive, except that the eis complex has catalytic activity, which may be lower or even higher (partially competitive activation) than that of the enzyme–substrate (es) complex. this inhibition typically displays a lower vmax, but an unaffected km value. substrate or product substrate or product inhibition is where either an enzymes substrate or product also act as an inhibitor. this inhibition may follow the competitive, uncompetitive or mixed patterns. in substrate inhibition there is a progressive decrease in activity at high substrate concentrations, potentially from an enzyme having two competing substrate-binding sites. at low substrate, the high-affinity site is occupied and normal kinetics are followed. however, at higher concentrations, the second inhibitory site becomes occupied, inhibiting the enzyme. product inhibition (either the enzyme's own product, or a product to an enzyme downstream in its metabolic pathway) is often a regulatory feature in metabolism and can be a form of negative feedback. slow-tight slow-tight inhibition occurs when the initial enzyme–inhibitor complex ei undergoes conformational isomerism (a change in shape) to a second more tightly held complex, ei*, but the overall inhibition process is reversible. this manifests itself as slowly increasing enzyme inhibition. under these conditions, traditional michaelis–menten kinetics give a false value for ki, which is time–dependent. the true value of ki can be obtained through more complex analysis of the on (kon) and off (koff) rate constants for inhibitor association with kinetics similar to irreversible inhibition.: 168 multi-substrate analogues tgddf/gddf multi-substrate adduct inhibitor. substrate analogue in black, cofactor analogue in blue, non-cleavable linker in red.peptide-based hiv-1 protease inhibitor ritonavir with substrate binding sites located in enzyme labelled as s2, s1, s1', and s2'.nonpeptidic hiv-1 protease inhibitor tipranavir multi-substrate analogue inhibitors are high affinity selective inhibitors that can be prepared for enzymes that catalyse reactions with more than one substrate by capturing the binding energy of each of those substrate into one molecule. for example, in the formyl transfer reactions of purine biosynthesis, a potent multi-substrate adduct inhibitor (mai) to glycinamide ribonucleotide (gar) tfase was prepared synthetically by linking analogues of the gar substrate and the n-10-formyl tetrahydrofolate cofactor together to produce thioglycinamide ribonucleotide dideazafolate (tgddf), or enzymatically from the natural gar substrate to yield gddf. here the subnanomolar dissociation constant (kd) of tgddf was greater than predicted presumably due to entropic advantages gained and/or positive interactions acquired through the atoms linking the components. mais have also been observed to be produced in cells by reactions of pro-drugs such as isoniazid or enzyme inhibitor ligands (for example, ptc124) with cellular cofactors such as nicotinamide adenine dinucleotide (nadh) and adenosine triphosphate (atp) respectively. examples as enzymes have evolved to bind their substrates tightly, and most reversible inhibitors bind in the active site of enzymes, it is unsurprising that some of these inhibitors are strikingly similar in structure to the substrates of their targets. inhibitors of dihydrofolate reductase (dhfr) are prominent examples. other examples of these substrate mimics are the protease inhibitors, a therapeutically effective class of antiretroviral drugs used to treat hiv/aids. the structure of ritonavir, a peptidomimetic (peptide mimic) protease inhibitor containing three peptide bonds, as shown in the "competitive inhibition" figure above. as this drug resembles the peptide that is the substrate of the hiv protease, it competes with the substrate in the enzyme's active site. enzyme inhibitors are often designed to mimic the transition state or intermediate of an enzyme-catalysed reaction. this ensures that the inhibitor exploits the transition state stabilising effect of the enzyme, resulting in a better binding affinity (lower ki) than substrate-based designs. an example of such a transition state inhibitor is the antiviral drug oseltamivir; this drug mimics the planar nature of the ring oxonium ion in the reaction of the viral enzyme neuraminidase. however, not all inhibitors are based on the structures of substrates. for example, the structure of another hiv protease inhibitor tipranavir is not based on a peptide and has no obvious structural similarity to a protein substrate. these non-peptide inhibitors can be more stable than inhibitors containing peptide bonds, because they will not be substrates for peptidases and are less likely to be degraded. in drug design it is important to consider the concentrations of substrates to which the target enzymes are exposed. for example, some protein kinase inhibitors have chemical structures that are similar to atp, one of the substrates of these enzymes. however, drugs that are simple competitive inhibitors will have to compete with the high concentrations of atp in the cell. protein kinases can also be inhibited by competition at the binding sites where the kinases interact with their substrate proteins, and most proteins are present inside cells at concentrations much lower than the concentration of atp. as a consequence, if two protein kinase inhibitors both bind in the active site with similar affinity, but only one has to compete with atp, then the competitive inhibitor at the protein-binding site will inhibit the enzyme more effectively. irreversible inhibitors types dfp reactionreaction of the irreversible inhibitor diisopropylfluorophosphate (dfp) with a serine proteaseirreversible inhibitors bind to the enzyme's binding site then undergo a chemical reaction to form a covalent enzyme-inhibitor complex (ei*). binding site in blue, inhibitor in green. irreversible inhibitors covalently bind to an enzyme, and this type of inhibition can therefore not be readily reversed. irreversible inhibitors often contain reactive functional groups such as nitrogen mustards, aldehydes, haloalkanes, alkenes, michael acceptors, phenyl sulfonates, or fluorophosphonates. these electrophilic groups react with amino acid side chains to form covalent adducts. the residues modified are those with side chains containing nucleophiles such as hydroxyl or sulfhydryl groups; these include the amino acids serine (that reacts with dfp, see the "dfp reaction" diagram), and also cysteine, threonine, or tyrosine. irreversible inhibition is different from irreversible enzyme inactivation. irreversible inhibitors are generally specific for one class of enzyme and do not inactivate all proteins; they do not function by destroying protein structure but by specifically altering the active site of their target. for example, extremes of ph or temperature usually cause denaturation of all protein structure, but this is a non-specific effect. similarly, some non-specific chemical treatments destroy protein structure: for example, heating in concentrated hydrochloric acid will hydrolyse the peptide bonds holding proteins together, releasing free amino acids. irreversible inhibitors display time-dependent inhibition and their potency therefore cannot be characterised by an ic50 value. this is because the amount of active enzyme at a given concentration of irreversible inhibitor will be different depending on how long the inhibitor is pre-incubated with