Theory on the accurate estimation of Michaelis-Menten enzyme kinetic parameters from steady state and progress curve datasets

Theory on the accurate estimation of Michaelis-Menten enzyme kinetic parameters from steady state and progress curve datasets | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Theory on the accurate estimation of Michaelis-Menten enzyme kinetic parameters from steady state and progress curve datasets View ORCID Profile Rajamanickam Murugan doi: https://doi.org/10.1101/2025.04.02.646753 Rajamanickam Murugan 1 Department of Biotechnology, Indian Institute of Technology Madras Chennai , India Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Rajamanickam Murugan For correspondence: rmurugan{at}gmail.com Abstract Full Text Info/History Metrics Data/Code Preview PDF ABSTRACT We show that neither pure steady state nor pure progress curve analysis yield reliable estimate of enzyme K M values since these methods are valid only at specific timescales and also use different type of datasets. All the currently proposed validity conditions of these methods assume a priori knowledge on K M . Hence, there is no way to check whether the obtained K M from a given dataset is a reliable estimate or not. Here we propose an integrated approach in which the same time course dataset will be analysed both in the progress curve as well as steady state perspectives at different reaction timescales across replications and substrate concentrations. Our theory shows that there exists an optimum reaction time at which the error in the estimation of K M using various progress curve and steady state methods show the least possible value so that the coefficient of variation of the median K M values obtained across various methods attains a minimum. Using detailed stochastic simulations, we confirm that the K M value obtained with minimum coefficient of variation across various methods is actually the reliable estimate that is close to the original K M value. We further show that using multiple nonlinear regression methods, the type of inhibition viz. competitive, uncompetitive and mixed can be accurately classified apart from obtaining the accurate inhibitor constants and IC 50 values from Dixon type average velocity steady state datasets. Introduction Enzymes catalyse several crucial biochemical reactions especially at physiologically relevant timescales and environmental conditions [ 1 - 3 ]. The mechanistic view of single substrate enzyme catalysis can be well described by the Michaelis-Menten scheme (MMS) [ 4 , 5 ]. In this scheme ( Fig. 1A ), the enzyme first binds the substrate to form an enzyme-substrate complex which reduces the activation energy barrier of substrate to product conversion. Subsequently, the substrate will be converted into the respective product and released. The MMS scheme is characterized by three different rate constants viz. diffusion-controlled bimolecular forward k 1 (1/mol/lit/second), unimolecular reverse k -1 (1/second) and product formation rate constants k 2 (1/second). These rate constants along with the initial enzyme level (e 0 ) are generally sumarized by two important enzyme kinetic parameters viz. K M = ( k -1 + k 2 ) / k 1 (mol/lit) and v max = k 2 e 0 (mol/lit/s). Here k 2 is also termed as k cat and the ratio k E = k cat / K M is defined as the catalytic efficiency of the enzyme. When k 2 ≫ k - 1 , then k E reaches the diffusion-controlled rate limit k 1 . Download figure Open in new tab FIGURE 1. Dynamics of product formation (p, µM) in Michaelis-Menten scheme of single substrate enzyme catalyzed reactions. Here t c ≅ 1⁄ k 1 ( K M + s 0 ), t m ≅ (( K M + s 0 )⁄ k 2 e 0 ) are the approximate pre-steady state and reaction-end timescales (t, seconds) and numerical integration settings from Eqs. 1 are k 1 = 0.1 µM -1 s -1 , k -1 = 0.01 s -1 and k 2 = 0.2 s -1 , with K M = 2.1 µM. A . s 0 = 100 µM and e 0 was iterated in (0.1, 1, 10, 100) µM. B . e 0 = 100 µM and s 0 was iterated in (1, 10, 100-1000, with increment of 100) µM along the arrow. Scheme B describes the fully competitive inhibition and Scheme C describes various partial inhibition schemes viz. competitive (k z = 0), uncompetitive (k i = 0) and mixed types. In fully competitive inhibition both substrate and inhibitor will be converted into their respective products by the same enzyme. Whereas, in case of partial competitive inhibition the enzyme-inhibitor will be a dead-end complex. Exact closed form analytical solution to the nonlinear set of ODEs corresponding to MMS scheme is not known though the integral solution can be expanded in terms of ordinary [ 6 ] and singular perturbation series [ 7 ] or over slow manifolds [ 8 , 9 ]. Singular perturbation expansions always yield inner and outer solutions corresponding to the pre-steady state and post-steady state regimes which should be combined via proper matching at the temporal boundary layer [ 7 , 10 - 15 ] to obtain the complete solution. Whereas, ordinary perturbation series can give the complete solution that is valid for both pre- and post-steady state regimes [ 6 ]. Obtaining ( K M , k cat , v max ) from the steady state and progress curve experimental datasets is essential to characterize an enzyme. Since complete closed form analytical solution to MMS is unknown, one needs to rely on the widely used standard quasi steady state approximation (sQSSA) of the reaction velocity v = d p /d t as v = v max s / ( K M + s ) that is valid only when k 2 ≪ (k 1 s 0 ) [ 16 ]. Here p represents the product concentration. When e 0 ≪ s 0 , then one can use the stationary reactant assumption to further simplify as v ≅ v 0 = v max s 0 / (K M + s 0 ) which connects K M , v max and initial substrate concentration s 0 . Nonlinear least square fitting of the data on (v, s 0 ) over original hyperbolic function or linearization techniques such as Lineweaver-Burk on (1/v, 1/s 0 ) and Eadie-Hofstee on (v, v/s 0 ) [ 17 ] can be used to obtain the estimates of kinetic parameters [ 18 , 19 ]. Several statistical analysis procedures were also proposed [ 20 , 21 ] to compute the error on these parameter estimates. Enzymes of pathogenic organisms are important targets for drug molecules to control various diseases. Accurate estimation of various MMS parameters especially K M along with K I of a competitive inhibitor is essential to compute the IC50 values of drug molecules against a given target enzyme [ 22 - 25 ]. In steady state experiments, a fixed amount of enzyme will be incubated with a series of different substrate concentrations (s 0 ) for fixed amount of time (t r ) at constant temperature and pH conditions from which the average product formation velocity will be calculated by the ratio = total product formed during incubation time / total incubation time (t r ). Generally, t r will be fixed approximately at 1 % depletion of s 0 however without any theoretical basis. Under sQSSA conditions with e 0 ≪ s 0 , it will be approximately assumed that the system evolves with constant reaction velocity v 0 = v max s 0 / (K M + s 0 ) after a short pre-steady state timescale t c ≪ t r so that the total product formed over time t r will be approximately equal to (p = v 0 t r ) and therefore one sets the average reaction velocity . This equation is being used widely across enormous volumes of literature to estimate K M and v max from the dataset on obtained from steady state experiments. To estimate the inhibition constants K I , this average velocity experiments will be conducted at different inhibitor concentrations i 0 , and the data on ( , s 0 , i 0 ) will be used to classify the type of inhibition (i.e. competitive, uncompetitive or mixed type) and estimate v max , K M and K I using Dixon plots [ 26 ] or sequential or simultaneous nonlinear least square fitting procedures [ 27 , 28 ]. Using these parameters IC 50 value of the inhibitory drug will be computed using Cheng-Prusoff equation [ 23 ]. Here is the average velocity = product formed in a given total reaction time / total reaction time. Detailed studies proposed the essential and sufficient conditions for the validity of the approximation viz. (a) k 2 ≪ (k 1 s 0 ) [ 16 ] (b) e 0 ≪ s 0 [ 29 , 30 ] (c) (e 0 / (s 0 + K M )) ≪ 1 [ 29 , 30 ]. However, except (b), conditions (a) and (c) strictly require a priori knowledge on k 1 , k 2 and K M . Otherwise, one needs to set very high values of s 0 along with condition (b) so that all the conditions (a-c) are satisfied. Although the inequality (b) is possible under in vitro laboratory conditions, there are several situations such as single molecule enzyme kinetics and other in vivo conditions where one cannot manipulate the ratio of substrate to enzyme concentrations much. Setting high s 0 along with infinitesimal e 0 ≪ s 0 would eventually drives towards infinitesimal values below the background noise level which results in noisy experimental data on ( , s 0 ) that leads to enormous amount of error in the estimation of K M . Several recent studies [ 32 - 35 ] reported this issue and also put forth alternate methods such as total QSSA (tQSSA) to obtain accurate K M values from such experimental datasets. The source datasets for the estimation of K M will be either time dependent progress curve type [ 36 ] or steady state average velocity type. Here progress curve data on (p, s, t) contains information on both pre-as well as post steady state dynamics that in turn depends on the timescale of experimental data collection. Whereas, steady state experimental setups generally capture the details of only the post-steady state regime. Progress curve methods can yield accurate estimates of K M only when the data collection done at those timescale regimes with maximum change in the curvature of the kinetic trajectory [ 37 ]. Further, the role of total reaction time t r is not considered anywhere in the sQSSA calculations i.e., though the final results strongly depend on t r . In this context, integrated form of sQSSA (isQSSA) is more superior than the general sQSSA since it also considers t r . Since the conditions of validity of various methods like sQSSA, isQSSA and progress curve analysis are different from each other, K M values of the same enzyme obtained from each of these methods under identical conditions using the same source dataset will not be consistent. Further, absence of a priori knowledge on K M and other kinetic parameters will eventually introduces uncertainty in setting up the appropriate experimental conditions to minimize the error in the estimation of these parameters using sQSSA methods with stationary reactant assumption or progress curve methods. Apart from the sequential and simultaneous nonlinear fitting, analysis of Dixon type inhibition datasets on ( , s 0 , i 0 ) warrants multiple nonlinear regression methods (MNLR) to obtain reliable estimates of IC 50 of drug molecules. In this article, we develop an integrated theoretical and computational approach which can produce consistent estimate of v max , K M , K I and IC50 upon considering all these uncertainty aspects. Theoretical Methods Single substrate Michaelis-Menten kinetics The nonlinear coupled differential rate equations corresponding to MM Scheme A shown in Fig. 1 can be written as follows [ 16 ]. Here k 1 (1/mol/lit/second), k -1 (1/second) and k 2 (1/second) are the rate constants corresponding to bimolecular enzyme-substrate complex formation, dissociation and product formation respectively. Further, ( e, s, x, p ) (mol/lit) are the concentrations of enzyme, substrate, enzyme-substrate complex, product respectively and v (mol/lit/second) is the reaction velocity. The initial conditions at t → 0 are ( e, s, x, v, p ) = ( e 0 , s 0 , 0,0,0) and the reaction trajectory ends at ( e, s, x, v, p ) = ( e 0 , 0,0,0, s 0 ) as t → ∞. The dynamical variables ( e, s, x, p ) obey the mass conservation laws viz. s = ( s 0 − x − p ), e = ( e 0 − x ) where one can also replace x = v ⁄ k 2 . Here the required enzyme kinetic parameters are defined as K M = ( k −1 + k 2 )⁄ k 1 and v max = k 2 e 0 . From the definition of K M , one can conclude that accurate estimation of K M requires detailed data points representing both pre- and post-steady state dynamics. To simplify the system of Eqs. 1 , we first introduce the following set of scaling transformations [ 16 ]. In Eqs. 2 , the variables (e, s, x, p) are normalized so that 0 ≤ ( E, S, X, P ) ≤ 1 with the mass conservations laws S = 1 − εX + P, E + X = 1. We define V = εX so that V + P + S = 1 Where . Eqs. 1 can be rescaled in multiple ways at different variable spaces by the following set of dimensionless nonlinear ODEs [ 16 ]. Here Eq. 3.1 describes the MMS dynamics in (P, τ) space, Eqs. 3.2 in (V, P, τ), Eqs. 3.3 (V, S, τ), Eq. 3.4 in (V, S), Eq. 3.5 in (V, P) and Eq. 3.6 in (P, S) spaces. Various dimensionless parameters in Eqs. 3 are defined as follows. When η → 0, then Eq. 3.4 results in sQSSA with the velocity v sQSSA in the original (v, s) space. Similarly, Eq. 3.5 results in tQSSA with the velocity v tQSSA in the (v, p) space which can be written in terms of the original dynamical variables as follows. We denote Eqs. 5 as η -approximations since they are valid only when η → 0. The main flaw of these sQSSA as well as tQSSA methods is that the left-hand sides of Eqs. 5 are still time dependent quantities along with time dependent s and p on the right-hand sides. That is to say, v sQSSA and v tQSSA are tangents of progress curve in (p, t) space at a given time point and not the time averaged velocities as used in most of the steady state experiments. To derive equations which are useful to fit steady state experimental data, one needs to further set ε → 0 in Eqs. 5 . This leads to the stationary reactant assumption s ≅ s 0 − p ≅ s 0 that results in the constant velocity as since lim ε →0 v tQSSA ≅ v sQSSA . In this scenario, one can conduct the average steady state velocity experiments only within the reaction timescales in which these conditions are valid. Upon setting ε → 0 in Eq. 3.6 , one finds that in the pre-steady state regime of (P, S) space. Solving this ODE for the initial condition P = 0 for S = 1, we obtain . Since S ≅ 1 when ( η, ε ) → (0,0), one finds that P ≅ 0 and V ≅ 1 − S in the pre-steady state regime ( p ≅ 0 and v ≅ k 2 ( s 0 − s ) in terms of original variables). Since s ≅ s 0 when s 0 ≫ s 0 , one finds that v ≅ 0 for infinitesimal values of k 2 near the steady state regime. The appropriate working reaction timescales for the average steady state velocity experiments can be derived as follows. Using the transformation rule P = εη 2 U , Eq. 3.1 can be rewritten as the following ordinary perturbation equation with the scaling parameter ϕ = ηε as follows. When ε → 0, then one arrives at the following approximations from Eqs. 6 using the definition of dimensionless velocity V = εη 2 F . We denote Eqs. 7 as ε -approximations. Solving Eqs. 7 for (U, F) in terms of τ and then reverting back to (P, V) and subsequently transforming back in to the original variables (p, v, t), one arrives at the following ε-approximations. Here is the sQSSA with stationary reactant assumption i.e., s ≅ s 0 . When t ≫ t c , then one finds that p ε ≅ v 0 ( t − tc ). From Eqs. 8 one can also obtain s ε ≅ s 0 − v 0 t − v ε (1⁄ k 2 − t c ) which follows from the mass conservation law s ε ≅ s 0 − p ε − v ε ⁄ k 2 . The subscript ε in p ε and s ε corresponds to the ε-approximations. The pre-steady state timescale in Eq. 8 is t c . When t ≫ t c then one finds that s ε ≅ s 0 − v 0 (1⁄ k 2 − t c ) − v 0 t . When t = t c , then the steady state concentration of substrate can be approximated as s εc ≅ s 0 (1 − e 0 ⁄( K M + s 0 )). Clearly, the stationary reactant assumption [ 38 ] s εc ≅ s 0 will be true only when ( e 0 ⁄( K M + s 0 )) ≪ 1 which will be eventually true when ε → 0. When the reaction time t r is such that t r < t c , then p ε ≅ 0 in the pre-steady state regime (infinitesimal quantity especially for ε → 0). When t r ≫ t c , then one finds the linear product evolution regime with reaction time t r as p ε ≅ v 0 ( t r − t c ) along with the constant reaction velocity v ε ≅ v 0 . However, this linear dependence of p ε on reaction timescale will break down towards the end of reaction since the substrate concentration starts to decline towards zero beyond this timescale t m . This approximately occurs when . Since the product concentration evolves linearly with time as v 0 t in the post-steady state regime, we find that . These timescale behaviors are demonstrated using numerically computed integral trajectories of Eqs. 1 in Fig. 1B and 1C . Clearly, one finds that and Further, one can define the timescale separation ratio χ as follows. Here χ is inversely proportional to the perturbation parameter ϕϕ . The successfulness of sQSSA methods strongly depends on extent of timescale separation that is decided by the condition χ ≫ 1 [ 16 , 30 ] which also means that ϕϕ = ηε ≪ (1 + µ ) 2 . The linear dependence of product formation on time will be valid only when the reaction time falls in the range t c ≪ t r < t m where one can conduct average reaction velocity experiments using . However, fitting of steady state data on with v 0 will be biased towards v max over K M since there is not enough datapoints from the pre-steady state regime. When t r < t m , then one can also use Eqs. 8 to conduct average velocity experiments in the following form. Here p ε is the total product formed at the end of reaction time t r as in Eq. 8 and is the average velocity obtained from the steady state experiments. Unlike also considers the nonlinearity of the pre-steady state regime. The conditions for the validity of Eq. 10 are ε → 0 and t r t r ≫ t c , then . Here, directly connects the average reaction velocity with the total substrate concentration s 0 . Clearly, the conditions for the validity of the widely used are viz. ( η, ε ) → (0,0) and t c ≪ t r < t m . Nonlinear least square fitting of the data on , for a given reaction time t r with Eq. 10 can directly yield the parameters (K M , v max and k 1 ). However, such fitting procedures favour accurate estimation of v max over (K M , k 1 ) since they mainly rely on the datapoints from the pre-steady state timescale regime which is actually a transient one compared to the post-steady state timescale. To overcome this issue, we will fix k 1 in the definition of with the median k 1 obtained from the progress curve methods in the later sections. To minimize the error in the estimation of K M one needs to set t r such that it is much lesser than the maximum reaction time t m corresponding to the lowest s 0 value used in the entire experimental setup. When η → 0, then one finds as in Eqs. 5 that . When ε → 0, then one finds that in the post-steady state regime and therefore one obtains Upon solving this nonlinear first order ODE with the initial condition S = 1 at τ = 0, and then reverting back to the original dynamical variables, one finds the following implicit solution. Eqs. 10 is another form of integrated rate equation [ 35 ] that directly connects the average reaction velocity with the enzyme kinetic parameters and reaction time t r . Data on , at a given reaction time t r can be directly used to perform linear least square fitting with Eq. 11 i.e., versus (type I) or (type II) to obtain K M and v max . In the later sections, we will show that type II form gives more accurate estimate of K M than type I. The conditions for the validity of Eq. 11 are ( η, ε ) → (0,0) and t c < t r < t m . One can also integrate given in Eqs. 5 under the conditions ε → 0 for the initial condition s = s 0 at t = 0 and derive the following expression for the average reaction velocity. In this equation, we have used the mass conservation law p = ( s 0 − s − v ⁄ k 2 ) and . When , then one finds from Eqs. 12 that where p csQSSA ≅ ( S 0 − S csQSSA ) which is an alternate form on Eq. 11 . In Eqs.12 , W(x) is the Lambert W function [ 39 , 40 ] which is the solution of y exp(y) = x for y. The conditions of validity of Eqs. 12 is similar to Eqs. 11 . We use Eq. 11 for comparison purposes since Eq. 11 is simple and computationally cheaper than Eq. 12 . K M and v max can be obtained from the steady state datasets using any one of the expressions under appropriate validity conditions and timescales. However, except the condition on ε , all the others conditions (on η , t c , t m ) require a priori knowledge on k 1 , k 2 and K M which are generally not available for unknown enzyme systems. In this scenario, one can perform direct non-linear least square fitting of the time course kinetic data on (p, t) or (s, t) with the system of nonlinear ODEs given in Eqs. 1 to obtain k 1 , k -1 and k 2 . However, accuracy of such fitting procedure strongly depends on the detailed data points on both pre- and post-steady state regimes that warrants progress curve data collection on sub-millisecond timescales apart from the steady state timepoint. Michaelis-Menten kinetics with competitive inhibition When ( ε , η , η, ε ) → (0,0,0,0) and , then earlier detailed theoretical studies revealed [ 22 ] the following approximations for the average velocities of the fully competitive inhibition Scheme B described in Fig. 1 . In fully competitive inhibition, same enzyme will catalyse the conversion of two different substrates (s, i) in to their respective products (p, q) with reaction velocities (v = dp/dt, u = dq/dt). In these equations, various parameters as defined as follows. When the total reaction time is such that t r ≫ t cs and t r ≫ t ci where ( t cs , t ci ) are the pre-steady state timescales, then Eqs. 13.1-2 reduce to the following well-known steady state velocity equations. When δ ≪ 1 or δ ≫ 1, the validity of Eqs. 13.4 will break down since the fully competitive inhibition system can exhibit multiple steady states [ 22 ] under such scenarios. Similarly for pure partial competitive inhibition given Scheme C of Fig. 1 , one can derive the following average velocity expression under the conditions ( ε , χ , ε ) → (0,0,0) where . Similarly, for the pure uncompetitive inhibition given in Scheme C of Fig. 1 one can derive the following average velocity expression. Finally, for the mixed competitive inhibition where both competitive and uncompetitive inhibition scenarios operate as given in Scheme C of Fig. 1 , one can derive the following combined average velocity expression. IC 50 is the inhibitor concentration which is required to decrease 50% of the inhibitor free enzyme activity which can be straightforwardly calculated [ 23 ] as follows for the generalized mixed inhibition scenario by solving the following equation for i 0 . When the inhibition is non-competitive, then the inhibitor binds at the allosteric site of enzyme that is different from the substrate binding active-site. In such scenario, the average steady state velocity equation becomes as follows. Eq. 17 is similar to Eq. 16.1 with K Z = K I and for non-competitive inhibition one finds that IC 50 = K I . For pure competitive inhibition one finds that and for pure uncompetitive inhibition one finds that . Numerical Simulation The stochastic version of nonlinear ODEs given in Eqs. 1.1-1.4 can be numerically integrated using the following Euler type scheme on chemical Langevin equations [ 41 , 42 ]. Here the initial conditions are x = 0, s = s 0 , e = e 0 , and p = 0 at t = 0, ξ q , d where q = (x, p) are delta correlated gaussian white noise terms and ρ is the additional noise control parameter. The time increment parameter Δ t will be adjusted to capture both pre-as well as post-steady state dynamics. Setting ρ = 0, results in the deterministic trajectory of the MMS scheme for the given set of parameters and initial conditions. To check the consistency among the data fitting methods , several stochastic trajectories of Eqs. 1 were generated with fixed t r and e 0 and different s 0 values. Stochastic simulated datasets are posted at ( DOI: 10.5281/zenodo.15094623 ) and the sample code implementations of ML algorithm are given in the Supporting Materials . To obtain the kinetic parameters k 1 , k -2 and k 2 from the progress curve data with initial s 0 and e 0 , we used nonlinear least square fitting [ 43 ] using Marquardt-Levenberg (ML) algorithm [ 44 - 47 ]. The computational workflow of ML scheme starts with minimizing the function .where f ( t | r ) is the numerical solution to Eqs. 1 with the parameter set r = [ k 1 , k −1 , k 2 ], and y is the observed dataset where ( f, y ) = ( p, s ). Upon expanding L in Taylor series around and then truncating with the first order terms, one finds the following approximation. In this equation Δ y i = f ( t i | r u ) − y i . The conditions for the minimum with respect to r will . which can be explicitly written as follows. Eqs. 15 .1-3 can be written in a matrix form [ 43 ] as J T Δ y + ( J T J )Δ r = 0. Definition of various terms in this matrix equation are given below. We compute the partial derivatives with respect to parameters [ k 1 , k −1 , k 2 ] at time point t i as follows where f ( t | r ) is the numerical integrated solution to Eqs. 1 with the parameter set r = [ k 1 , k −1 , k 2 ]. We set α to the lowest possible value depending on the computational accuracy requirement. In Eqs. 16 , J is the Jacobian matrix and explicitly one finds the following. From Eqs. 15 -19 , one obtains the following Marquardt-Levenberg (ML) iterative scheme where λ is the ML parameter and λ = 0 represents the Newton’s method [ 43 ]. The reduced sum of squares of deviation will be computed at the end of u th iteration as . where y u , I = f ( t i | r u ) is the predicted data point values for r u which will be obtained by numerical integration of Eqs. 1 using Euler type scheme given in Eqs. 13 with ρ = 0. Here λ should be adjusted depending on changing pattern of Z u over iterations. When Z u Z u +1 then we set λ → λ⁄δ. Here the δ is the ML tuning parameter [ 44 ]. When | Z u − Z u +1 | < θ where θ is the required tolerance limit, then we exit from iteration and the final parameters will be obtained. For NLS fit of various datasets we used λ = 10 − 2 , δ = 2 and θ = 10 -6 . the final sum of squares error (SSE) and the covariance matrix will be calculated as follows. Upon computing the total sum of squares (SST), one can compute the final nonlinear regression coefficient R 2 as follows. Here where is the final fit parameter values. We use the final SSE and R 2 to compare the goodness of fit across trajectories with different s 0 and replications. Similar workflow was used to obtain K M and v max using as given in Eq. 10 with the median of k 1 obtained from the progress curve analysis across different s 0 over several replications outlined in Eqs. 14 -21 . Linear least square fitting with Eq. 11 [ 43 ] for yields the required parameter estimates along with the errors. Multiple nonlinear regression for competitive inhibition In this section, we will outline the multiple nonlinear regression fitting procedure for the Dixon type steady state average velocity inhibition dataset on , over Eqs. 13-17 . For example, let us consider the case of pure competitive inhibition i.e., as given in Eq. 14 . For simplicity we use the notation ( a, b, c ) = ( K MS , K I , v max ) for the parameters and ( s 0, i , i 0, k ) = ( x i , y k ) for the concentrations of substrate and inhibitor. We denote the substrate levels e 0, i where i = 1 to n and inhibitor levels i 0, k where k = 1 to m and the corresponding average velocities as = product formed in the reaction time t r / total reaction time t r . With these settings one can define the chi-square function to be minimized as follows. Here L needs to be minimized with respect to the parameters (a, b, c) along with two different independent random variables. In this equation, where f i , k = f ( x i , y k | a 0 , b 0 , c 0 ) is the fit function. As in case of single variable nonlinear fitting procedure, we expand L in terms of Taylor’s series around (a 0 , b 0 , c 0 ) and truncate the series with the first order term as follows. Eqs. 24 can be su m arized in the following matrix form. Various terms in Eq. 25 are defined as follows. From Eqs. 25 , one finds the Newton’s iterative scheme for the multiple nonlinear fitting procedure to minimize L as follows. Clearly, before the inversion of J T J matrix, one needs to sum over the index k. Various partial derivative terms can be numerically computed as follows. This idea can be extended to any number of independent random variables apart from (s 0 and i 0 ). As in case of single variable NLS fitting described in Eqs. 20 , Marquardt-Levenberg algorithm can be embedded in the Eq. 26 to speed up the convergence. Sample datasets for competitive inhibitions schemes described by Eqs. 13-17 were generated by adding gaussian white noise terms to the Dixon type steady state velocity datasets with appropriate noise control parameter ρ as where ξ is the random number drawn from the standard normal population N (0, 1). Results and Discussion There are several issues and limitations associated with the currently available steady state as well as progress curve methods to estimate the K M and v max of single substrate MM enzymes. First let us consider sQSSA with stationary reactant assumption which is valid only when t c ≪ t r < t m and ( η, ε ) → (0,0). Since K M values are not generally available for the unknown enzyme systems, to achieve these criteria, one needs to set higher values for s 0 than e 0 which will lead to high background fluctuations in the measured average reaction velocities. For example, let correspond to respectively s 0 = ( s 0,1 , s 0, 2 ). When s 0 → ∞, then one finds the following. Eq. 23 leads to high noise level in the observed average reaction velocities that in turn results in high level of uncertainty in the estimation of K M . Similar effects are prevalent in the progress curve analysis using especially the trajectories on ( p, t ) space since pp ≅ 0 in the pre-steady state regime that leads to high level of background fluctuations in the measured values of p . This in turn results in high uncertainty in the estimation of K M using the data on ( p, t ) which significantly depends on the pre-steady state dynamics. Clearly, the data on ( p, t ) can be used only in the steady state methods which mainly work well in the post-steady state regime. Detailed NLS fitting analysis of the data on ( p, t ) and ( s, t ) spaces in the presence as well as absence of noise suggests the following. Progress curve methods using Marquardt-Levenberg algorithm outlined in the methods section works very well over data on (s, t) as well as (p, t) in the absence of noise i.e. ρ = 0 in Eqs. 13 as demonstrated in Figs. 2A and 2B . NLS fitting over the progress curve data on (p, t) as well as (s, t) can recover the original K M and v max with minimal error. When ρ > 0, then the NLS fit procedure work better with the data on (s, t) than the data on (p, t) as demonstrated in Figs. 2C -F . Particularly, fluctuations in the trajectories of (p, t) in the pre-steady state regime is demonstrated in Fig. 2C . NLS fitting methods can capture k 1 and k 2 very well and the estimate on k -1 is prone to severe uncertainties especially when k -1 ≪ (k 1 , k 2 ) as demonstrated with sample trajectories from Figs. 2C and 2D in Figs. 2E and 2F . This in turn leads to significant amount error in the estimation of K M although v max is not much affected by this issue. The distribution of NLS fit parameters obtained at different noise levels are shown in Figs. 3 for both (s, t) and (p, t) time course datasets. These results clearly suggest that K M values obtained from the NLS fit of time course dataset on (p, t) space are unreliable and prone to significant fluctuations in the presence of high noise levels. When the noise level increases, then the distribution of fit parameters corresponding to (p, t) space dataset exhibits bimodality as shown in ( Figs. 3D, 3F ). Whereas, NLS fit over time course dataset on (s, t) space can yield reliable median estimates of K M even at high noise levels. Particularly, monomodal type distribution of fit values of K M and v max are observed for the dataset on (s, t) space as demonstrated in Figs. 3A, 3C, 3E . Remarkably both time course datasets on (s, t) and (p, t) spaces can yield reliable NLS fit median estimates of v max across various noise levels as demonstrated in Fig. 3 . This observation is reasonable since v max mostly relies on the post-steady state dynamics and therefore the fluctuations in the pre steady state regime of (p, t) space will not influence the error in the estimation of v max . Download figure Open in new tab FIGURE 2. Progress curve analysis of data on (p, t) and (s, t). Co m on simulation settings for the numerical integration scheme in Eqs. 13 are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 10 -3 s, e 0 = 0.85 µM, s 0 = 50 µM so that K M = 0.313725 µM, t c ≈ 0.0039 s, t m ≈ 118.38 s, total simulation time = 10 s and v max = 0.935 µM s -1 . A . Simulation of Eqs. 13 with ρ = 0 and NLS fit of the simulation trajectory with Eqs. 1 using ML algorithm yielded the parameters k 1 = 5.1 ± 1.25 x 10 -5 , k -1 = 0.499 ± 7 x 10 -6 , k 2 = 1.1 ± 4.5 x 10 -8 for the data on (p, t) with K M = 0.314 at 95% confidence level. B . NLS fit using ML algorithm over Eqs. 1 yielded k 1 = 5.1 ± 6 x 10 -6 , k-1 = 0.499 ± 9 x 10 -6 , k 2 = 1.1 ± 4.5 x 10 -8 for data on (s, t) generated using ρ = 0 in Eqs. 13 with K M = 0.313639 at 95% confidence level. C . stochastic trajectories (1000 numbers) of Eqs. 13 with ρ = 10 -2 demonstrating the fluctuations in the pre-steady state regime of (p, t) space. D . stochastic trajectories of Eqs. 13 with ρ = 10 -2 demonstrating less fluctuations in the pre-steady state regime of (s, t) space. E . NLS fit of sample (p, t) trajectory with ρ = 10 -2 yielded k 1 = 2.29± 0.09, k -1 = 1.53 ± 0.1, k 2 = 1.12 ± 2 x 10 -3 with K M = 1.18 and v max = 0.95 at 95% confidence level. F . NLS fit of sample (s, t) trajectory with ρ = 10 -2 over Eqs. 1 yielded k 1 = 5.1 ± 0.07, k -1 = ± 0.1, k 2 = 1.1 ± 5 x 10 -4 with K M = 0.47 and v max = 0.94 at 95% confidence level. Download figure Open in new tab FIGURE 3. Distribution of nonlinear least squares (NLS) fit parameters across various noise levels. Co m on simulation settings for the numerical integration scheme in Eqs. 13 are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 10 -3 s, total simulation time = 10 s, e 0 = 0.85 µM so that K M = 0.313725 µM and v max = 0.935 µM s -1 , s 0 = 50 µM for which one finds that t c ≈ 0.0039 s and t m ≈ 118.38 s. NLS fit was done using Marquardt-Levenberg algorithm as outlined in the methods section. In A, C and E, NLS fittings were done using data on (s, t). Whereas, in B, D and F the fittings were done using data on (p, t). Upon obtaining k 1 , k -1 and k 2 , K M and v max will be calculated. Noise levels in Eqs. 13 were set as follows. A, B . ρ = 10 -3 . C, D . ρ = 10 -2 . E, F . ρ = 10 -1 . The median values of K M and v max were computed across 10 3 replications. A . the median values of (k 1 , k -1 , k 2 , K M , v max ) = (5.1005, 0.5009, 1.100, 0.3140, 0.9350). B . the median values of (k 1 , k -1 , k 2 , K M , v max ) = (4.9188, 0.3849, 1.1001, 0.3163, 0.9351). C . the median values of (k 1 , k -1 , k 2 , K M , v max ) = (5.2031, 0.5499, 1.1021, 0.3197, 0.9368). D . the median values of (k 1 , k -1 , k 2 , K M , v max ) = (4.7469, 1.1193, 1.0981, 1.5888, 0.9334). E . the median values of (k 1 , k -1 , k 2 , K M , v max ) = (6.5060, 0.4776, 1.1348, 0.3727, 0.9646). F . the median values of (k 1 , k -1 , k 2 , K M , v max ) = (16.0345, 0.7332, 1.1380, 3.1860, 0.9673). These observations suggest that neither progress curve analysis alone nor steady state methods alone can provide consistent estimates of K M . Multiple ways of analyzing the same dataset of (p, t) and (s, t) is clearly required to check for the consistency across various steady state and progress curve methods and obtain reliable estimates. In this context, one can use the median value of k 1 obtained from progress curve methods into the NLS fit on as given in Eq. 10 to obtain consistent estimates of K M and v max across progress curve analysis and steady state methods. We consider the following methods I, II, III and IV using the same main time course datasets on (s, p, t). For Methods II, III and IV, sub-dataset on average reaction velocity versus s 0 at different t r will be generated from the main dataset. Here the average velocity is defined as (total product formed at the end of t r ) / t r where t r will be iterated approximately from the steady state timescale t c to t m . Various NLS fit methods I, II, III and IV are su m arized as follows. Method I is the direct NLS fitting of the main progress curve dataset on (s, t) with fixed e 0 and different s 0 across several replications with the original rate equations Eqs. 1.1 - 1.4 using Marquardt-Levenberg algorithm as outlined in the methods section. K M values will be calculated from the fit values of (k 1 , k -1 , k 2 ) and the median of (k 1 , k -1 , k 2 , K M ) values will also be obtained. In Method II , the parameter k 1 in the definition of of Eqs. 10 will be fixed with the median of k 1 values obtained from Method I and the sub-dataset on , computed from the main time course dataset on (p, t) at different t r will be used for NLS fitting with fixed k 1 to obtain K M and v max . In Method III , as defined in Eq. 11 will be used to fit the sub-dataset on computed from the main dataset on (p, t) at different t r and s 0 . One can also use that is computationally costlier than In Method IV that is widely used across various literature, will be used to fit the sub-dataset on computed from the main dataset on (p, t) at different reaction time t r and s 0 . In Methods II, III and IV, the K M value corresponding to t r with minimum sum of square error (SSE) as given in Eq. 21 will be computed as demonstrated in Fig. 4 . There exists a critical reaction time t r at which the sum of square error in the estimation of K M attains the minimum which is clearly observed in the absence of noise in the dataset (ρ = 0, Fig. 4A ). Remarkably, Fig. 4B suggest that the error in the estimation of K M strongly depends on the reaction time t r in steady state Methods II, III and IV. In the presence of noise in the dataset, the median of SSE corresponding to K M decreases monotonically and the fit parameter value converges to the original K M as t r is increased towards t m . The speed of convergence of fit K M towards the original K M is much faster in Method II than in Methods III and IV as demonstrated in Fig. 4B . Sum of square errors of various approximations can be defined as where t n = t r and is the numerically simulated average velocity at time t = t i , and are the approximations where the subscripts G = sQSSA, csQSSA, ε, isQSSA (I, II) associated with . The computed SSE seems to be strongly dependent on t r and range of iteration of s 0 as demonstrated in Fig. 4C and 4D . Clearly, SSE of is minimum near t c and the SSEs corresponding to attain minimum approximately near t m . The timescale t Q at which the SSE of attains minimum can be computed by solving for t as shown in Fig. 4E . This is the critical time point at which both pre- and post-steady state approximations start to deviate from the original simulated average reaction velocity values. Download figure Open in new tab FIGURE 4. Variation of the fitted K M values with respect to reaction time t r across different fitting methods. Co m on simulation settings for the numerical integration scheme in Eqs. 13 are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 10 -3 s, total simulation time = 10 s, e 0 = 0.85 µM so that K M = 0.313725 µM and v max = 0.935 µM s -1 , s 0 was iterated from 50 to 250 µM with interval of 25 µM. For s 0 = 50 µM one finds that t c ≈ 0.0039 s and t m ≈ 118.38 s. The reaction time t r was iterated from 0.1 to 10 s. The sub datasets on , at different t r were generated from the main progress curve dataset on (p, t) and used to fit Eqs. 10 (ε-approximation), Eq. 11 (isQSSA) and v 0 (sQSSA) and the median values of K M were obtained across 10 3 replications. A . ρ = 0 in Eqs. 13. B . ρ = 10 -2 in Eqs. 13. C . Δ t = 10 -5 s, s 0 was iterated from 50 to 250 µM with interval of 25 µM. For p csQSSA / t r , s 0 was iterated from 100 to 200 µM with interval of 10 µM. D . For all the steady state methods, Δ t = 10 -5 s, s 0 was iterated from 100 to 200 µM with interval of 10 µM. E . s 0 = 100 µM and Δ t = 10 -5 s for which t Q ≈ 10.49 s. When t r = t Q , then the reaction velocity will be leading to minimum amount of error in the approximation Remarkably, all the fitting methods approximately perform well near t Q , at which the variation of the fit parameters across these methods will be at minimum. The entire procedure will be repeated with different e 0 until obtaining consistent median estimates of K M and v max with minimal threshold variation across methods I to IV. We consider the coefficient of variation (CV = standard deviation / mean) of the median K M values obtained from methods I-IV as the reliability index . Figs. 4C, 4D and 4E suggest that the reliability index will be at minimum when the reaction time is set as t r = t Q . In the absence of a priori knowledge on enzyme kinetic parameters, one can use the reliability index to obtain best fit values of K M which occurs near the optimum reaction time t Q . Upon obtaining such consistent median estimates of K M corresponding to a given threshold reliability index , all the K M values obtained from the same dataset using different methods will be then pooled and the overall median of K M and v max will be computed as depicted in Fig. 5 . Download figure Open in new tab FIGURE 5. Computational algorithm to obtain consistent estimate of K M from the progress curve dataset. Here, the progress curve data on (p, s, t) will be the main input for Methods I, II, III and IV. Here t max is the maximum data collection time. Method I is the direct nonlinear least square fitting of the dataset on (s, t) with fixed e 0 and different s 0 across replications with the original Eqs. 1.1-1.4 using Marquardt-Levenberg algorithm as outlined in the methods section. The parameter k 1 in of Eqs. 10 of Method II will be fixed with the median of k 1 values obtained from Method I. For Methods II-IV, sub-dataset on average velocity ((totalproduct formed at the end of t r ) / t r , where t r will be iterated from t c towards t m ) versus s 0 at different t r will be generated from the main dataset. NLS fittings over , were done using models corresponding to Methods II-IV at different t r . The K M value corresponding to t r with minimum sum of square error (SSE) will be considered. Upon obtaining consistency on the estimates of K M across Methods I-IV corresponding to the given threshold reliability index , all the K M values obtained from the same dataset using different methods will be then pooled and the overall median K M will be computed. Here reliability index is the coefficient of variation of median K M values obtained from Methods I-IV. The distributions of various fit parameters obtained from Methods I to IV over several replications are shown in Fig. 6 along with the overall pooled distribution of K M and v max . The distribution of fit parameters (k 1 , k -1 , k 2 ) show skewness towards lower values ( Figs. 6A 1-3 ). Particularly, the distribution of k -1 exhibits a bimodal type with zero spike. As result, the distribution of K M also exhibits spike near zero and v max exhibits skewed distribution ( Figs. 6B 1-2 ). Since the median of k 1 obtained from Method I is used for the NLS fitting calculations in Method II, the distribution of K M values obtained from Method II also exhibits a spike near zero as shown in Fig. 6B -3 . Interestingly, both K M and v max obtained from Method III exhibit sy m etric type distribution ( Figs. 6C -1, 6D-1 ). However, the distribution of K M values obtained using Method III span towards negative values ( Fig. 6C -1 ). The distribution of K M values obtained from Method IV also exhibit a spike near zero ( Fig. 6C -2 ). Clearly, the inconsistency among the K M values obtained from different methods originates mainly from the type of distribution of fit parameter in the presence of noise. Download figure Open in new tab FIGURE 6. Distribution of fit parameters across various methods I-IV. Co m on simulation settings for the numerical integration scheme in Eqs. 13 are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 10 -3 s, total simulation time = 10 s, e 0 = 0.85 µM so that K M = 0.313725 µM and v max = 0.935 µM s -1 , s 0 was iterated from 50 to 250 µM with interval of 25 µM. For s 0 = 50 µM one finds that t c ≈ 0.0039 s and t m ≈ 118.38 s. The median values corresponding to progress curve fit over 10 3 trajectories on (s, t) dataset were (k 1 , k -1 , k 2 , K M , v max ) = (5.164, 0.587, 1.102, 0.331, 0.937). A1-3, B1-2 . NLS fit method I using Marquardt-Levenberg algorithm. Upon obtaining k 1 , k -1 and k 2 , K M and v max will be calculated as in B1, B2. B3, C3 . Method II. Here the obtained median values from datasets on (p, t) were (K M , v max ) = (0.338, 0.935). C1, D1 . Method III. Here the obtained median values from datasets on (p, t) were (K M , v max ) = (0.332, 0.935). C2, D2 . Method IV. Here the obtained median values from datasets on (p, t) were (K M , v max ) = (0.366, 0.935). D3 . Overall distribution of pooled K M values across all the methods I-IV with overall median of K M = 0.3418 µM which correspond to the coefficient of variation of median K M values obtained from Methods I, II, III and IV as 0.048. Efficient progress curve fitting also seems to be dependent on the detailed data collection around the transition regions in (s, t) and (p, t) spaces with maximum change in curvature [ 37 ]. Clearly, there exists at least three different timescale regimes in the trajectory on (p, t) space with significant change in curvature viz. exponential increasing phase in the pre-steady state regime, linear increasing phase in the steady state region and reaction-ending phase. Similar timescale regimes also exist in (s, t) space. Significant change in the curvature occurs at the interface of these timescale regimes approximately near t c and t m . Accurate values of these interface timescales can be calculated from e ε and e csQSSA for the pre- and post-steady state regimes respectively. The approximate curvature ζ d associated with e ε of Eq. 8 can be defined as follows [ 48 ]. Here p ε and v ε are defined as in Eqs. 8 . Upon solving for t, one finds the time scale associated with the maximum change in curvature at the interface of pre-steady to steady steady-state transition t R as follows. Similarly noting the definition of e csQSSA as given in Eqs. 12 , the approximate curvature ψ t in the steady state to post-steady state transition regime of (s, t) space can be defined as follows. Upon solving for t, one finds the timescale associated with the maximum change in curvature at the interface of steady state to reaction ending regime t U as follows. Since the enzyme system evolves with constant velocity v 0 in the timescale region t R < t < t U , progress curve fitting methods can yield reliable estimate of K M only when the experimental data collection is done around the timescales (t R , t U ) with maximum change in the curvature of (s, t) trajectory. Whereas, steady state methods work very well in the constant velocity regime with minimal change in the curvature of the trajectory of (s, t). Change in curvature of the trajectory in (s, t) space with respect to time is shown in Fig. 7A along with the approximations given in Eqs. 25 -27 which are in good agreement with the numerical simulation results. Further, Eqs. 25 and 27 suggest that (t R , t U ) will be close to (t c , t m ) and for practical purposes one can use (t c , t m ) for checking the validity conditions of various model fitting methods. The simulated average reaction velocity (p / t) along with the approximations are shown in Fig. 7B . These suggest that the approximation works well in the pre-steady to steady state regime and work well in the post steady state regime and work only in narrow range of timescales with constant reaction velocity. Clearly, the reaction time t r plays important role in deciding the reliability index among Methods I to IV. The error in the estimation of K M using proportionally increases when t r increases well beyond the steady state. Conversely, inclusion of data points from the pre-steady state regime increases the error in or These means that the error in the estimation of KM using the widely used will be always higher than the error associated with . Reliability index can be used as a check point to optimize the appropriate t r to obtain accurate estimate of K M values from progress curve and various steady state methods. Download figure Open in new tab FIGURE 7. Change in the curvature of trajectory on (s, t) space and average reaction velocity. Co m on simulation settings for the numerical integration scheme in Eqs. 13 are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 0.5 x 10 -4 s, total simulation time = 1000 s, noise level set as ρ = 0, e 0 = 0.85 µM so that K M = 0.313725 µM and v max = 0.935 µM s -1 , and s 0 = 100 µM. For these settings, one finds that t c ≈ 0.002 s, t m ≈ 91.2 s and t Q ≈ 10.49 s. A . The transition timescales (t R , t U ) at which maximum change in the curvature occurs can be calculated using Eqs. 25 -27 as t R = 0.0126 s and t U = 92 s. The time dependent curvatures associated with the trajectory in (s, t) space ζ d and ψψ d corresponding to the pre- and post-steady state regimes respectively are defined as in Eqs. 24 -27 . Progress curve Method I works very well in the regions with maximum change in the curvature. Whereas, steady state methods II, III and IV work very well in the regions with minimal change in the curvature. B . Change in the average velocity (p / t) with time. Method II works well from the pre-steady to steady state regime. Whereas, Method III works well in the steady to post-steady state regime. Method IV works well only in a narrow timescale regime with constant average velocity. Software packages such as global kinetic explorer and dynafit [ 49 - 51 ] were developed to fit the data obtained from progress curve and steady state experiments. Several comparative studies were done earlier over steady state and progress curve methods to optimize the analysis workflow towards minimizing the estimation error [ 52 , 53 ]. Selective removal of data points [ 54 ] over certain progress curve regime were also proposed. However, all these methods are solely either progress curve or steady state type and there is no way to check the consistency of fit parameters across various methods since each method requires different type of input dataset. Accurate estimation of kinetic parameters depends on strict impose of validity conditions of each method as given in Table 1 . Unfortunately, these validity conditions require a priori knowledge on various enzyme kinetic parameters which are not generally available for unknown systems. Since the conditions of validity for each method is different, a global analysis workflow with a comon input dataset is required to obtain consistent estimates. In this context, our proposed computational workflow depicted in Fig. 5 can yield reliable estimates of K M from the comon time course dataset at different initial substrate concentrations s 0 and replications. The reliability index which is the coefficient of variation of median K M values obtained from Methods I-IV can act as a check point against unreliable estimates of K M . When the computed reliability index is more than the given threshold value, then one needs to iterate the reaction time until obtaining the desired threshold level. Clearly, the proposed methods do not require a priori knowledge on K M values unlike the currently available sQSSA methods. View this table: View inline View popup Download powerpoint TABLE 1. Conditions of validity of various fitting methods to obtain K M and v max Identification of the correct inhibition mode using the experimental dataset on viz. competitive, uncompetitive, noncompetitive and mixed is essential to design drug molecules for the target enzyme and obtain their IC 50 values for the comparison purposes. Dixon type graphical methods [ 26 ] used to identify the type of inhibition are not accurate in the presence of noise in the dataset on . It will be difficult to visually identify and measure the intersection points in the Dixon plot. Sequential and simultaneous nonlinear regression methods [ 27 , 28 ] were proposed to overcome such issues. In this context, we have developed the multiple nonlinear regression (MNLS) method to evaluate various inhibition parameters directly by fitting dataset on over the respective inhibition model equations as shown in Fig. 8 . Using MNLS method, one can directly obtain the parameters fit values (K M , K I , K Z , v max ) along with their standard errors from the Dixon type datasets via minimizing the overall sum of square deviations as defined in Eq. 28 . Fit parameter values for sample datasets in the presence and absence of noise are given in Tables 2 and 3 . MNLS method can recover the exact parameter values in the absence of noise in the datasets as demonstrated in Figs. 8A, 8C, 8E, 8F and Table 2 . Further, an algorithm based on the sum of square error (SSE) values of various multiple nonlinear fit functions and standard errors of fit parameters is also developed as shown in Fig. 9 . In this algorithm, the given enzyme inhibition dataset on will be simultaneously fitted with competitive, uncompetitive, noncompetitive and mixed inhibition model equations as given in Eqs. 14 -17 . The overall sum of square errors computed with respect to each of these model functions were compared and the model function showing the lowest overall SSE will be chosen as the best fit model. When the chosen model is a mixed-competitive inhibition model, then further investigations will be carried out as follows to avoid ambiguities arising due to the generalized nature of mixed inhibition model equation. View this table: View inline View popup Download powerpoint TABLE 2. Multiple nonlinear regression of enzyme inhibition data in the absence of noise View this table: View inline View popup Download powerpoint TABLE 3. Multiple nonlinear regression of enzyme inhibition data in the presence of noise Download figure Open in new tab FIGURE 8. Multiple nonlinear regression fitting of Dixon type enzyme inhibition dataset on , in the presence and absence of noise. A, C, E and G are in the absence of noise. B, D, F and H are in the presence of noise with strength ρ = 10 -3 . The preset values of parameters are K M = 2.80952 µM, K I = 0.354839 µM, K Z = 2.12903 µM and v max = 4.62 µM/s. s 0 was iterated from 10 to 100 µM with an increment of 10 µM and i 0 was iterated in 10 to 110 µM with an increment of 10 µM. Eqs. 14 -17 were used to generate datasets for (1) competitive, (2) uncompetitive, (3) noncompetitive, (4) mixed-competitive modes of enzyme inhibition and the multiple nonlinear fitting were done as outline in the methods section. The fit results are su m arized Tables 2 and 3. A, B . Competitive inhibition data fitted with (1)-(4) modes of inhibition. C, D . Uncompetitive inhibition data fitted with (1)-(4) modes of inhibition. E, F . Noncompetitive inhibition data fitted with (1)-(4) modes of inhibition. G, H . Mixed-competitive inhibition data fitted with (1)-(4) modes of inhibition. Download figure Open in new tab FIGURE 9. Classification algorithm for the type of inhibition from Dixon type dataset. Here the given dataset on was fitted with competitive, uncompetitive, noncompetitive and mixed inhibition models and the sum of square errors were compared and model showing lowest overall SSE will be chosen. If the model is a mixed-competitive inhibition model, then the following investigations should be carried out. (1) when K I = K Z , then it will the noncompetitive inhibition model. (2) when the parameter error corresponding to K I or K Z is higher than the next lowest SSE, then the next lowest SSE model will be the best fit model. (3) when K I >> K Z , then it will be the uncompetitive inhibition model. (4) when K I << K Z , then it will be the competitive inhibition model. When K I = K Z , then it will be interpreted as the noncompetitive inhibition model. This scenario occurs when one tries to fit noninhibition dataset over mixed inhibition model function. When the parameter error corresponding to K I or K Z is higher than the next lowest SSE, then the next lowest SSE model will be the best fit model. When K I >> K Z , then it will be interpreted as the uncompetitive inhibition model. When the dataset on uncompetitive inhibition is fitted with mixed inhibition model equation, then one observes K I >> K Z in the fit results as shown in Tables 2 and 3 . When K I << K Z , then it will be interpreted as the competitive inhibition model. When the dataset on competitive inhibition is fitted with mixed inhibition model equation, then one observes K I << K Z in the fit results as shown in Table 2 and 3 . Performance of the inhibition type classification algorithm in the presence of noise is shown in Fig. 10 . Clearly, our inhibition type classification algorithm works very well especially at low noise levels. Further, fitting of competitive inhibition dataset may be wrongly shown as mixed inhibition type at moderate noise levels as demonstrated in Fig. 10A . At high noise levels, uncompetitive inhibition dataset may be wrongly shown as ( Fig. 10B ) noncompetitive inhibition and noncompetitive inhibition dataset may be wrongly detected as mixed, competitive and uncompetitive inhibition ( Fig. 10C ). These results clearly suggests that repeated data collection and MNLS analysis is required to obtain consistent classification of inhibition schemes and the respective parameters and IC 50 . Download figure Open in new tab FIGURE 10. Performance of the inhibition type classification algorithm. The preset values of parameters are K M = 2.80952 µM, K I = 0.354839 µM, K Z = 2.12903 µM and v max = 4.62 µM/s. s 0 was iterated from 10 to 100 µM with an increment of 10 µM and i 0 was iterated in 10 to 110 µM with an increment of 10 µM. Eqs. 14 -17 were used to generate datasets for (1) competitive, (2) uncompetitive, (3) noncompetitive, (4) mixed-competitive modes of enzyme inhibition and the multiple nonlinear fitting were done as outlined in the methods section. Gaussian white noise with zero mean and unit variance was added through the control parameter ρ that was iterated from 10 -6 to 1 as where ξ is the random number drawn from the standard normal population N (0, 1). Fraction fit results were computed over 1000 randomly generated datasets. A . Competitive inhibition datasets were fitted with model equations corresponding to (1)-(4) modes of inhibition. B . Uncompetitive inhibition datasets were fitted with model equations corresponding to (1)-(4) modes of inhibition. C . Noncompetitive inhibition datasets were fitted with model equations corresponding to (1)-(4) modes of inhibition. D . Mixed-competitive inhibition datasets were fitted with the model equations corresponding to (1)-(4) modes of inhibition. Conclusion Accurate estimation of enzyme kinetic parameters such as K M and v max from the experimental datasets is critical to characterize any enzyme. The validity of the widely used standard quasi steady state method over decades relies on the a priori information of K M and other kinetic parameters which are generally not available for newly identified enzyme systems. Progress curve methods were proposed to alleviate such issues. However, nonlinear least square fitting of the time course dataset with the original differential rate equations works well only in the timescale regimes with maximum change in the curvature of the trajectory of substrate evolution. Identifying these timescales requires the knowledge on enzyme kinetic parameters. Though experimental trial and error methods can be used to identify such timescales, they are generally cost ineffective. Further, progress curve methods work better with the time course data on substrate than the time course data on product since the product concentration is close to zero in the pre-steady state regime. Since steady state and progress curve analysis use different types of datasets in the current analysis set up, the estimated K M values will be inconsistent among these methods. This leads to the dilema of identifying correct K M values among various steady state and progress curve analyses. Here we have shown that the error in the estimation of K M using steady state methods is strongly dependent on the total reaction time. We have further shown that the steady state methods work well over the reaction timescales with minimum change in the curvature of the trajectory of product evolution with constant reaction velocity. Considering these theoretical factors, we have developed an integrated method of analysis which comprises of both progress curve and steady state analysis using the same time course datasets that also consider the total reaction time. We considered three different methods of steady state analyses that involve fitting of the average reaction velocity with the total substrate concertation. We have defined the reliability index which is the coefficient of variation of the median K M values obtained from progress curve and other steady state fitting methods using the same time course dataset at different substrate concentrations and replications. We further demonstrate that reliable estimate of K M can be obtained by iterating the total reaction time to minimize the reliability index across time course and steady state curve fitting methods. We have developed a multiple nonlinear fitting procedure-based algorithm to classify the average velocity, total substrate and inhibitor concentrations dataset over competitive, uncompetitive, noncompetitive and mixed types. This method works very well at low to medium noise levels. However, at high noise levels the competitive inhibition dataset may be wrongly shown as mixed inhibition type, uncompetitive inhibition dataset may be wrongly shown as noncompetitive inhibition and noncompetitive inhibition dataset may be wrongly detected as mixed, competitive and uncompetitive inhibition types. Repeated data collection and multiple nonlinear fitting analysis is required to obtain consistent estimates of inhibition parameters and IC 50 . Data availability statement The datasets generated during and/or analysed during the current study are available in the Zenodo. DOI: 10.5281/zenodo.15094623 SUPPORTING MATERIALS Datasets for the global analysis DOI: 10.5281/zenodo.15094623 FIGURES 2C, 2D sampleptr.txt and samplestr.txt Simulation settings (Eqs. 13 ) are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 10 -3 s, e 0 = 0.85 µM, s 0 = 50 µM and ρ = 10 -2 . Totally there are 1000 replications. First column is the time in seconds, and subsequent columns are concentrations of product (sampleptr.txt) and substrate (samplestr.txt) respectively. FIGURE 3 Simulation settings (Eqs. 13) are k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , Δ t = 10 -3 s, e 0 = 0.85 µM and s 0 = 50 µM. Totally there are 1000 replications. First column is the time in seconds, and subsequent columns are replications of concentrations of product (sampleptr.txt) and substrate (samplestr.txt) respectively. sampledat_ptpm1.txt, sampledat_ptpm2.txt, sampledat_ptpm3.txt respectively with ρ = 10 -1 , 10 -2 and 10 -3 for product evolution. sampledat_stpm1.txt, sampledat_stpm2.txt, sampledat_stpm3.txt respectively with ρ = 10 -1 , 10 -2 and 10 -3 for substrate evolution. FIGURES 2E, 2F , 4 and 6 Datasets were generated by numerical integration of Eqs. 13 as given in main text with the following settings k 1 = 5.1 µM -1 s -1 , k -1 = 0.5 s -1 , k 2 = 1.1 s -1 , ρ = 10 -2 , Δ t = 10 -3 s, e 0 = 0.85 µM, s 0 was iterated from 50 to 250 with interval 25 µM. Totally 1000 replications were generated for the global analysis as in Fig. 5 of the main text. The data files are named as “sample_enzkin_ w t_noise_data_R_ N .dat” where N is the replication number taking values from 0 to 999 and w = s or p depending on the trajectory of substrate or product. The first column is time and columns 2 to 10 represents the data corresponding to different initial substrate concentrations s 0 viz. 50, 75, 100…250 µM. Time course sample datasets on the trajectories of substrate depletion were using in direct nonlinear least square fit with Eqs. 1 to obtain rate constants [ k 1 , k −1 , k 2 ] using Marquardt-Levenberg algorithm as depicted in methods section. Using these fit values K M and v max were calculated as K M = ( k −1 + k 2 )⁄ k 1 and v max = k 2 e 0 . Analysis over 1000 such time course datasets, one obtains 1000 such values of [ k 1 , k −1 , k 2 , K M , v max ] which were used to construct the histogram as depicted in Figs. 6 . Trajectories on the product evolution were used for the steady state analysis by computing average velocities (p / t) at a given reaction time across various substate concentrations and replications. Fig. 2E : “sample_enzkin_ p t_noise_data_R_ 0 .dat”, first and second columns were used as time course data (s 0 = 50 µM) to NLS fit of (p, t). Fig. 2F : “sample_enzkin_ s t_noise_data_R_ 0 .dat”, first and second columns were usedas time course data (s 0 = 50 µM) to NLS fit of (s, t). Sample MATLAB code for NLS fit of time course (p, t) data to Eqs. 1 of main text Download figure Open in new tab Download figure Open in new tab Sample MATLAB code for NLS fit of time course (s, t) data to Eqs. 1 of main text Download figure Open in new tab Download figure Open in new tab Footnotes added multiple nonlinear regression (MNLS) fitting of competitive, uncompetitive, noncompetitive and mixed enzyme inhibition. added an algorithm to correctly classify the type of inhibition. https://doi.org/10.5281/zenodo.15094623 References 1. ↵ Alberts B. Molecular biology of the cell . New York : Garland Science ; 2002 . 2. Stryer L. Biochemistry . New York: W.H. Freeman ; 1988 . 3. ↵ Voet D , Voet JG . Biochemistry . New York: J . Wiley & Sons ; 1995 . 4. ↵ Michaelis L , Menten ML . {Die kinetik der invertinwirkung} . Biochem Z . 1913 ; 49 ( 333-369 ): 352 . doi: citeulike-article-id:5936552. OpenUrl CrossRef 5. ↵ Briggs GE , Haldane JBS . A Note on the Kinetics of Enzyme Action . 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Share Theory on the accurate estimation of Michaelis-Menten enzyme kinetic parameters from steady state and progress curve datasets Rajamanickam Murugan bioRxiv 2025.04.02.646753; doi: https://doi.org/10.1101/2025.04.02.646753 Share This Article: Copy Citation Tools Theory on the accurate estimation of Michaelis-Menten enzyme kinetic parameters from steady state and progress curve datasets Rajamanickam Murugan bioRxiv 2025.04.02.646753; doi: https://doi.org/10.1101/2025.04.02.646753 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Biochemistry Subject Areas All Articles Animal Behavior and Cognition (7629) Biochemistry (17660) Bioengineering (13881) Bioinformatics (41913) Biophysics (21436) Cancer Biology (18578) Cell Biology (25482) Clinical Trials (138) Developmental Biology (13372) Ecology (19889) Epidemiology (2067) Evolutionary Biology (24302) Genetics (15599) Genomics (22483) Immunology (17728) Microbiology (40365) Molecular Biology (17163) Neuroscience (88540) Paleontology (666) Pathology (2830) Pharmacology and Toxicology (4821) Physiology (7637) Plant Biology (15130) Scientific Communication and Education (2045) Synthetic Biology (4290) Systems Biology (9818) Zoology (2269)