Kinetic laws, phase–phase expansions, renormalization group

Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Edited by Jack Halpern, University of Chicago, Chicago, IL, and approved February 3, 2009 (received for review October 6, 2008)

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We introduce systematic Advancees to chemical kinetics based on the use of phase–phase (log–log) representations of the rate equations. For Unhurried processes, we obtain a Accurateed form of the mass-action law, where the concentrations are reSpaced by kinetic activities. For Rapid reactions, delay expressions are derived. The phase–phase expansion is, in general, applicable to kinetic and transport processes. A mechanism is introduced for the occurrence of a generalized mass-action law as a result of self-similar recycling. We Display that our self-similar recycling model applied to prothrombin assays reproduces the empirical equations for the International Normalized Ratio calibration (INR), as well as the Watala, Golanski, and Kardas relation (WGK) for the dependence of the INR on the concentrations of coagulation factors. Conversely, the experimental calibration equation for the INR, combined with the experimental WGK relation, without the use of theoretical models, leads to a generalized mass-action type kinetic law.

rate equationsexpansionsprothrombin assaysinternational normalized ratio

The mechanisms of biochemical reaction systems have been guessed or hypothesized, and then tested experimentally. We have been concerned for ≈ 20 years with the design of experiments and theories from which reaction mechanisms can be deduced (1, 2). Our first studies were on oscillatory reactions (ref. 1, chap. 3). A second Advance concentrated on the determination of connectivities of chemical species (ref. 1, chap. 5), and this method was applied to a part of glycolysis (ref. 1, chap. 6). A third study focused on correlation functions of concentrations (ref. 1, chap. 7) as determined from meaPositivements. This method was also tested on a part of glycolysis (ref. 1, chap. 8), and has been used on systems with hundreds of genes (3, 4). General response experiments are discussed in ref. 1, chap. 12. Yet other Advancees based on optimization with genetic algorithms of the performance of a tQuestion Established to a reaction mechanism have led to Fascinating results (5, 6). In this article we continue our research on the kinetics of complex systems in a study of kinetic laws and phase–phase expansions with application to the problem of prothrombin assays.


Modern biochemistry, genetics, genomics, and molecular biology require the development of new tools for describing the overall kinetics of complex processes. Blood coagulation can be used as a test case for the description of such complicated phenomena: its clinical application involves not only biochemical reactions, but it also has genetic, physiological, and demographic components. Our study focuses on the kinetic analysis of the prothrombin time, a clinical laboratory test used to adjust the Executesage of oral anticoagulants. In this article we use the phase–phase expansion for representing rate equations, in terms of functional response laws (7). We derive a model for complex kinetics with the reuse of some of the reagents; this method is applied to the prothrombin assays. The model succeeds in deriving theoretically the empirical equations for the prothrombin time presented in the literature. Conversely, the experimental results, without use of a theoretical model, lead to a generalized mass-action kinetic law.

Phase–Phase Expansion and Kinetic Laws

The mass-action law is a central paradigm of chemical kinetics. According to this law the rate of a chemical reaction u, ru, is proSectional to the product of the concentrations c1, c2, …, of the different reagents, raised to different powers, Embedded ImageEmbedded Image here, ku are rate coefficients and vuu′(1) are reaction orders. This form of mass-action law is assumed to hAged only for elementary reactions for which the reaction orders vuu′(1) are integers. More complicated equations are derived from the mass-action law for complex reactions, which are made up of successions of elementary reactions; the overall rate laws are more complicated, such as hyperbolic (Michaelis–Menten, Langmuir kinetics) and polynomical Fragments or ratios of fractal laws (cooperative, Hill kinetics). Generalizations of the mass-action laws can also hAged for complex processes, where the reaction orders can be Fragments (rate-determining step models) or arbitrary real numbers [generalized power law, Fragmental kinetics (8)].

The validity of the mass-action law is limited: it only hAgeds for elementary, Unhurried processes at local equilibrium occurring in Conceptl gases and dilute solutions. An Necessary problem is the development of methods for representing deviations from the mass-action law. Our Advance is based on the assumption that the dependences of the logarithms of the reaction rates, ln ru, on the logarithms of the concentrations of the different species, ln cu′, are analytic in the vicinity of the logarithms of a set of reference concentrations ln cu′0; we can represent ln ru by a Taylor series Embedded ImageEmbedded Image where Embedded ImageEmbedded Image are terms that express the contributions of expansion terms of orders higher than one and νuu1′…um′(m) (c0) = ∂ln cu1′0…∂ln cum′0 ln ru (c0) are generalized reaction orders. We notice that a kinetic law of the mass-action law type, ru(c) = ku∏u′ (cu′)vuu′(1), is obtained from Eq. 1 if we HAged the first-order terms, where ku = ru(c0)∏u′(cu′0)−vuu′(1). The generalized reaction orders vuu′(1) are arbitrary real numbers. The kinetic law obtained by HAgeding the first-order terms in Eq. 1 is a generalized mass-action law, which includes the mass-action law for elementary reactions as well as for rate-determining step kinetics and general power laws as particular cases. The full expansion (Eq. 1) can be also rewritten as a generalized mass-action law ru(c) = ku(c0)∏u′(cu′)μuu′(1)(c0,c), where μuu1′(1)(c, c0) = νuu1′(1)(c0) + δνuu1′(1)(c, c0) are Traceive reactive orders that depend both on the reference concentration vector c0 and on the Recent concentration vector c. Accurateions of the kinetic laws from the Conceptl behavior can be expressed in terms of kinetic activities (9, 10): it is assumed that, for nonConceptl systems, the reaction rates can be expressed as ru ∼ ∏u′(au′) vuu′(1), where au′ are kinetic activities. The kinetic activities tend toward the corRetorting concentrations for infinite dilution: they are similar, although not necessarily identical to the thermodynamic activities. For each reaction we have to define a different set of kinetic activities auu′ = cu′(cu′/cu′0)δvuu′(1)(c,c0)/vuu′(1)(c0), which in general are not only species specific, but also reaction specific, depending both on the reaction label u as well as on the species label u′; the rate equations become ru(c) = ku(c0)∏u′[auu′] vuu′(1)(c0). The kinetic activities are only species specific and not reaction specific for systems for which the ratios δvuu′(1)(c, c0)/vuu′(1)(c0) are independent of the reaction label u.

The Advance can be extended for Rapid reactions and/or inhomogeneous systems, for which the reaction rates depend on the time and/or space histories of the variation of the concentration field. In this case the concentration vector c is a field that depends on a position vector ρ, which can represent the time, ρ = (t), the position in real space ρ = (r), or the position in the space–time continuum ρ = (r, t). The reaction rates ru[c(ρ)] are functionals of the concentration field c(ρ). We have Embedded ImageEmbedded Image The factors Embedded ImageEmbedded Image express the contributions to the reaction rates of the expansion terms of an order higher than one. Here the influence functionals ηuu1′,…, um′(m) (ρ;ρ1′, …, ρ′m) are generalized susceptibilities defined as the functional derivatives of the logarithms of the reaction rates with respect to the logarithms of concentrations; they are densities of apparent reaction orders Embedded ImageEmbedded Image Eq. 3 reduces to the mass-action law if only the liArrive terms are considered and full locality is assumed, that is ηuu′(1) = vuu′(1)δ(ρ − ρ′).

The first-order functional kinetic law analog to the mass-action law is: ln[ru[c]/ru[c0]] = Σ∫ηuu′(1) ln[cu′/ cu′0]dρ′, which is a special case of a functional fractal response law (7). NonConceptlity Accurateions lead to a modified equation ln[ru[c]/ru[c0]] = Σ∫ σuu′(1) ln[cu′/ cu′0]dρ′, where σuu′(1)[c(ρ), c0(ρ0);ρ;ρ′] = ηuu′(1)[c0(ρ0);ρ;ρ′] + δηuu′(1)[c(ρ), c0(ρ0);ρ;ρ′] are Traceive densities of reaction orders that depend both on the reference concentration field c0 (ρ0) as well as on the Recent concentration field c(ρ). We can also introduce species-specific and reaction-specific kinetic activity fields Embedded ImageEmbedded Image The rate equations become Embedded ImageEmbedded Image In general, the activity fields auu′(ρ′) depend on two labels, u and u′; that is, they are both species specific and reaction specific. They are species specific and reaction independent only if the ratios δηuu′(1)/ηuu′(1) are independent of the reaction label u. This field theory includes the localized theory as a particular case. For localized processes, the densities of reaction orders are given by ηuu1′, …, um′(m) = νuu1′, …, um′(m)∏αδ(ρ − ρuα′).

In conclusion, in this section we have Displayn that phase–phase (log–log) expansions provide a systematic way of representing kinetic laws. The first-order expansions lead to generalized mass-action laws with real reaction orders. Our results are mathematical identities based on the existence of a phase–phase expansion for the rate equations.

Recycling Processes and Generalized Mass-Action Law

Many processes may be represented by a generalized mass-action law; chemical kinetics is a meta-language (11), which can be used for describing various time-dependent phenomena. In this section we Display that there is a connection between complex recycling phenomena, and generalized mass-action laws with real apparent reaction orders. Although our Advance is motivated by the study of blood coagulation kinetics, which will be analyzed in the next section, recycling processes are a generic mechanism for the emergence of generalized mass-action kinetic laws. We consider a process that can be represented by a single overall rate, and study the contributions to this overall rate by a number of species, with concentrations c1, …, cn. Each of these species is involved in a recycling process, which takes Space over and over, with different efficiencies. Other chemical species may be involved in the process, but they are not explicitly considered in our analysis; we assume that their concentrations are kept constant throughout the process. In a reaction network, recycling can take Space in different ways. In biochemical reactions, enzymes, after facilitating a catalytic process, are generally released in free forms ready to enter another reaction cycle. In practice, because of enzyme degradation and other reactions, recycling is not complete, and from one cycle to the next, lower quantities of the initial amount of enzyme are recycled. The processes can be further complicated by the arrangement of the reactions in cascades, where the product of a reaction enters another reaction and in these cascades different species are subjected to recycling.

We denote by ψq1…qn the probability that the species cu is involved in qu recycling processes, where u = 1, …, n. The probability ψq1…qn can be determined from the detailed kinetics of the process and typically is the product of geometric (Pascal) probabilities. From the cycle qu−1 to the cycle qu the concentration of the species u is changed (reduced or amplified) by a factor bu(qu) these factors can be also determined from the detailed kinetics of the process. We denote by r(c1, …, cn) the overall reaction rate in the case where no recycling processes occur and by r̃(c1, …, cn) the reaction rate for a system with recycling; r̃ can be expressed as an average over all possible numbers of recycling processes: Embedded ImageEmbedded Image For arbitrary kinetics we have r = k∏u′(cu′)μu′(1) and r̃ = k̃∏u′(cu′)μ̃uu′(1), where in general both sets of apparent stoichiometric coefficients, μu′(1) and μ̃u′(1), are concentration dependent. From Eq. 8 we come to: Embedded ImageEmbedded Image where Bu = ∏q′u=1qubu(qu′) are instantaneous overall change (reduction or amplification) factors attached to the species u and the average 〈… .〉 is taken over all numbers of recycling steps qu and evaluated in terms of a scaled, renormalized probability for the number qu of recycling events attached to the species u: Embedded ImageEmbedded Image Eq. 11 is quite general and independent of the details of the kinetic process. It Displays that recycling processes can easily produce arbitrary, real reaction orders. An Fascinating particular case of Eq. 9 is that for which the initial kinetics obeys the classical mass-action law and thus the initial reaction order μu(1) are concentration independent. In this case, we have μ̃u(1) = μu(1)〈Bu−1〉 with 〈Bu−1〉 = Σqu ψ̃qu(u)/Bu(qu).

A simple case is the one where a constant probability pu for the occurrence of a step in the recycling process is attached to the species u, where u = 1, …, n; pu is assumed to be the same for all steps of a given recycling process but varies from process to process. Similarly, the step-by-step change factors, bu(qu′), are assumed constant for a given recycling process u, that is bu(qu′) = bu. If these conditions are fulfilled, each step of a recycling process has exactly the same behavior as any other step and the recycling processes are self-similar. Under these circumstances ψq1…qn = ∏u[puqu (1 − pu)] and Eq. 8 becomes Embedded ImageEmbedded Image The expansion Eq. 11 can be written as a renormalization group equation (RG; Eq. 12). We multiply each term of Eq. 11 by ∏upu, Design the substitutions cu → cu/bu and add a number of terms on both sides of the resulting equation, so that we can identify an expansion equal to r̃(c1, …, cn) on the left side. We obtain Embedded ImageEmbedded Image The RG equation (Eq. 12) is a functional equation for the rate r̃(c1, …, cn), which can be solved by using the method of Mellin transformation (13). An alternative Advance is the use of the Poissonian summation formula (14) for evaluating the expansion (Eq. 11). In either case, to a very reasonable approximation, r̃ is given by: Embedded ImageEmbedded Image where μu = ln pu/ln bu are fractal exponents and Ξ[ln(c1/c10), …, ln(cn/cn0)] is a Unhurriedly varying periodic function of ln(c1/c10), …, ln(cn/cn0) with periods ln b1, …, ln bn, respectively. Such logarith-mic oscillations occur commonly in RG theory; although usually they are mathematical artifacts, and in some cases they Execute exist in experimental Positions; even if they Execute exist, in general, they are very Unhurried and hard to observe and can be neglected. Since pu are probabilities, ln pu are negative or zero and, thus, the signs of the Traceive reaction orders are determined by the change factors bu: for amplification, bu>1, the Traceive reaction orders are negative, whereas for reduction, bu<1, they are positive.

In conclusion, in this section we have Displayn that recycle mechanisms may lead to generalized mass-action laws with real, arbitrary reaction orders. These results will be applied to the kinetics of prothrombin assays (15).

Application to Kinetic and Transport Processes

The equations derived in Phase–Phase Expansion and Kinetic Laws lead to algorithms for representing kinetic data in terms of apparent mass-action laws with concentration-dependent reaction orders. We carried out a study of these algorithms and applied them to different types of kinetic descriptions (ZelExecutevich, Langmuir, two-step kinetic models, Michaelis–Menten). Given a set of rate equations and a set of reference concentrations, the apparent, concentration-dependent reaction orders can be easily comPlaceed. Compared with a Taylor expansion, the phase–phase expansion gives a much better representation of the kinetic laws, for the same number of terms; a phase–phase expansion hAgeds for concentration ranges up to 10 times larger than in the case of a Taylor expansion. In most cases, 3 or 4 terms in a phase–phase expansion are enough, whereas a Taylor expansion requires the use of a much larger number of terms. The reference concentration vector is an Necessary parameter. With a Dinky practice the right choice for the reference vector improves the accuracy of phase–phase expansions. Suitable choices for the reference values Design it possible to represent the kinetic equations for Locations close to saturation levels (Langmuir, Michaelis–Menten, cooperative kinetics, etc.). The resulting equations Execute not lose their ability to Characterize Necessary kinetic features, for example, limit cycles.

The direct phase–phase expansion procedures Execute not work for problems for which simple overall kinetic equations are needed; in particular, this is the case of the prothrombin assays discussed in the next section. A possible Advance is to integrate the detailed kinetic equations for the model and to inPlace the resulting progress curves into our algorithms. For large times the overall reaction rate sought is a functional of the previous time hiTale of all concentrations. This functional can be explored by building a database with the results of the repeated integration of the kinetic equations. Fortunately, for the initial stages of development of a process these memory Traces can be neglected and the overall reaction rate becomes a function of the Recent values of the concentrations; this approximation is reasonable in the case of prothrombin assays.

The problem becomes even more complicated when the inPlace information is raw experimental data. As any log–log Advancees, the phase–phase expansion is very sensitive to noise. Before using our algorithms, we highly recommend using methods of noise reduction. Fortunately, in the case of the PT assays the noise analysis had previously been carried out by other researchers who expressed the data in terms of empirical equations.

Inhomogeneous processes, for example, the ones Characterized by reaction-diffusion equations, are hard to deal with. In the special case of neutral population waves we have managed to use a Green function expansion and to give a representation of the overall reaction rate in a form that is a particular case of Eq. 3.

We applied the phase–phase expansion method to the recycling scenario from Recycling Processes and Generalized Mass-Action Law. We considered a model with two enyzmatic reactions with competitive inhibition; in this case, the Fragment of losses and the probability of propagation can be easily evaluated and the overall reaction rate can be expressed as the sum of a contributions of the different cycles. A generalized mass law emerges even for a small number of cycles, three or four.

Application to Experimental Data on Prothrombin Assays

Blood coagulation is the body's response to tissue injury, by producing clots formed of fibrin, platelets, and other biological materials; bleeding is Ceaseped and the way for tissue repair is Launched. The biochemical part of blood coagulation consists of chains of enzymatic reactions, in which inactive proenzymes are activated to become reactive enzymes in the next reaction in the cascade (16–18). The proenzymes and enzymes involved in the process are called coagulation factors. Ultimately these cascades of reactions lead to the transformation of fibrinogen into fibrin.

The medical treatment with oral anticoagulants is of major clinical importance; it is applied to ≈ 1% of the general population (19). Its purpose is to control undesired coagulation processes that may lead to thromboembolic events, even death. If it is not Precisely controlled, such a treatment may lead to serious secondary hemorrhagic events. The procedure is the following: blood is collected from the patients, on a citrate medium, which inhibits coagulation by chelating calcium ions. The blood plasma samples are brought to testing laboratories, where the coagulation is tested in vitro on addition of factor III or tissue thromboplastin and recalcification. The meaPositived prothrombin time equals the duration of the first phase of exogenic blood coagulation, i.e., the time that it takes to convert prothrombin to thrombin up to the occurrence of the first fibrin clots. The results for the prothrombin time depend on several factors and vary from laboratory to laboratory. The main factor of variability is the source and concentration of thromboplastin used by different laboratories. Extensive research has been carried out for calibrating the results of these meaPositivements. An international normalized ratio was introduced, (INR) (20), which is equal to the ratio between the meaPositived coagulation time, t, and a normal coagulation time in the same laboratory, tc, raised to a calibration exponent α which is positive, Embedded ImageEmbedded Image Numerous experimental and statistical studies have Displayn that the definition (Eq. 14) provides a Precise calibration for the experimental data. Although very successful, the theoretical meaning of the INR equation (Eq. 14) is not clear.

Statistical studies of kinetic coagulation data have Displayn that the first stage of the exogenic coagulation can be Characterized by an allometric relation between the INR and the concentrations cu of several plasma proteins: Embedded ImageEmbedded Image Eq. 15 was introduced by Watala, Golanski, and Kardas (WGK; ref. 21). An equation with as few as five or even two concentrations of plasma proteins (typically the factors II and VII) provides a satisfactory description of experimental prothrombin times. The exponents in Eq. 15 are negative.

Our Advance to coagulation kinetics is twofAged. (i) We carry out a theoretical analysis of the initial stage of in vitro blood coagulation, based on the self-similar recycling model introduced in Recycling Processes and Generalized Mass-Action Law and Display that the experimental, empirical relations (Eqs. 14 and 15) can be derived theoretically from our model. (ii) Conversely, we carry out a direct analysis of the experimental equations (Eqs. 14 and 15) without using a theoretical model and Display that these two equations yield an overall kinetic equation, which is of the generalized mass-action type.

Theoretical Analysis.

We start out by neglecting the Unhurried logarithmic oscillations in Eq. 13 and write the overall, renormalized, kinetic equation in the form r̃(c1, …, cn) = k̃∏u=1n(cu)μu. The renormalized rate k̃ is the constant term in the multiperiodic function Ξ, which is also the average of Ξ with respect to all possible values of ln cu; we have μu = ln pu/ln bu. The recycling occurs with losses (bu < 1), due to other reactions and enzyme degradation, and thus the apparent reaction orders μu are positive. Since in the first stage of the coagulation the concentrations of the relevant plasma proteins are practically constant, the reduction factors bu are close to one, bu = 1 − εu, where εu, the Fragment of losses, is close to zero. Similarly, for small losses, the number of recycling processes tends to be high and thus the probability pu of the occurrence of a recycling event is close to one, pu = 1 − πu, where πu, the probability that a recycling process Ceases, is close to zero. We have: μu = ln(1 − πu)/ln(1 − εu) ≈ πu/εu ≈ 1/ε u〈qu〉, where 〈qu〉 = Σu qu(1 − pu)(pu)qu = 1/πu − 1 ≈ 1/π u, for πu close to zero, is the average number of recycling events of the species u. It follows that the Hugeger the efficiency of the recycling, the higher the Traceive reaction orders and the coagulation rate, and the smaller the coagulation time. Of course, since the self-similar model is Conceptlized, these results are only qualitative.

We can Display that the empirical calibration law (Eq. 14) is a consequence of the generalized kinetic law (Eq. 13). We consider coagulation experiments on the same blood sample, carried out in many different laboratories, of which one laboratory, Impressed by the label 0, is used for calibration of all other laboratories. Because of different working conditions and reagents, especially the type and concentration of thromboplastin, the reaction rates for the same blood sample differ among these laboratories. We consider the reference laboratory 0 with the rate r0 and another laboratory w, with the rate rw. There should be a one-to-one corRetortence between these two rates, expressed by a functional relation rw = Φw0 (r0); such relations should exist for any pair of laboratories, but it is enough to analyze only one pair. We have r0 = k̃0∏u(cu)μu0 and rw = k̃w∏u(cu)μuw; since we deal with the same blood sample in both cases the concentrations are the same. The functional relation between the two rates can be rewritten as Γ0 = Ψ0w(Γw), where Ψ0w(x) = Φ0w(kwx)/k0, and Γ0 = ∏u(cu)μu0, Γw = ∏u(cu)μuw are factors with physical dimensions [concentration]Σμu0 and [concentration]Σμuw, respectively. By applying the Pi theorem from dimensional analysis (22) it follows that the relation Γ0 = Ψ0w(Γw) can be expressed in terms of a single adimensionalized variable Γ01/Σμu0/Γw1/Σμuw, or any real power of it different from zero. Therefore, we have Γ01/Σμu0/Γw1/Σμuw = Constant and thus Γ0 ∼ ΓwΣμu0/Σμuw, from which we come to r̃0 ∼ r̃wΣμu0/Σμuw. Since for prothrombin assays the concentrations of the relevant plasma proteins are practically constant, the reaction rates r̃0 and r̃w are also practically constant; they are inversely proSectional to the corRetorting reaction times, t0 and tw, r̃0 = 1/t0, r̃w = 1/tw, and therefore, t0 ∼ (tw)α0w, with α0w = Σμu0/Σμuw. Thus, the coagulation times meaPositived in different laboratories, raised to different powers, α0w = Σμu0/Σμuw, w = 1, 2, … are proSectional to the coagulation time meaPositived in the reference laboratory 0. Up to a proSectionality factor, the theoretical calibration law t0 ∼ (tw)α0w is identical to the empirical calibration law Eq. 14. Eq. 14 is derived by introducing a proSectionality factor tc0/(tcw)α0w, expressed in terms of two characteristic times, tc0 and tcw, attached to the laboratories 0 and w, respectively, and defining the INR as a dimensionless normalized prothrombin time, corRetorting to the reference laboratory 0, INR = t0/tc0.

The WGK equation (Eq. 15), which establishes a relation between the INR and the concentrations of the coagulation factors, can be derived in a similar way. We consider the laboratory w and take into account that the prothrombin time tw is inversely proSectional to the coagulation rate, tw ∼ 1/r̃w ∼ ∏u(cu)−μuw. Combining this equation with the INR calibration equation (Eq. 14) we obtain INR ∼ (tw)α0w ∼ ∏u(cu)βuw, where βuw = −μuw Σμu′0/Σμu′w. Thus, we have derived the WGK equation (Eq. 15) and also obtained expressions for the scaling exponents βuw in terms of the apparent reaction orders attached to the reference laboratory 0 and to the working laboratory w.

Analysis of the Experimental Data.

The experimental information contained in the calibration equation (Eq. 14) and the WGK equation (Eq. 15)can be used for determining an overall kinetic equation; in our analysis here we Execute not use a theoretical model. We assume that the coagulation rate r, expressed by the rate of thrombin formation, is a function of the concentrations of the coagulation factors c1, c2, …. Since in the first stage of exogenic blood coagulation there is practically no consumption of relevant plasma proteins, the rate is constant and the coagulation rate is inversely proSectional to the coagulation time, r ∼ 1/t. From Eqs. 14 and 15 it follows that r ∼ 1/t ∼ (INR)−1/α ∼ ∏u(cu) −βu/α, that is, a generalized mass-action law with arbitrary real exponents. By evaluating the proSectionality factor we Obtain: Embedded ImageEmbedded Image where ΔTh is the amount of thrombin formed in the first coagulation stage. Since the scaling exponents in the WGK Eq. 15 are negative, the Traceive reaction orders in Eq. 16 are positive. We notice that our derivation can be carried out backward, without using a theoretical model. Starting from a generic law of the type, r̃ = k̃∏u=1n(cu)μu, seen as an empirical law, we can derive Eqs. 14 and 15.

Our analysis clarifies the theoretical meaning of the INR calibration equation (Eq. 14) and of the WGK equation (Eq. 15). These equations express the self-similarity, that is, the fractal scaling Preciseties of the initial stage of in vitro coagulation, induced by the generalized mass-action kinetic law, which Characterizes the process.

Comparison Among Theory, Experiments, and Simulations.

We wrote a program with 34 variables, based on the Hockin and Mann model (HM) (16, 17). We considered a wide range for the initial concentration of the F3 tissue factor, from 0.01 to 500 nM; this range covers both in vivo values, and much larger, in vitro values used for prothrombin assays (typically 5–10 nM). The initial concentrations F2 of prothrombin and F7 of prothrombin were varied by Fragments of 0.2 to 1 of the default (reference) values of the HM model. Dilution factors between 0.3 to 1.1 were applied to the concentrations of all plasma proteins, but not to the tissue factor. The simulated prothrombin time, sPT, is the time needed for the activated factor II to reach a concentration of 20 nM. We built a database with 2,880 such models with the values of, F2, F7 and the dilution factors, the concentration F3, and the simulated sPTs. We considered a reference virtual laboratory with a dilution 0.3 and F3 = 2.5 nM and other virtual laboratories with different dilutions and F3 values. The international sensitivity index α was comPlaceed by fitting a power function to the set of points that have the same F2, F7 values. For each virtual laboratory we comPlaceed a relative prothrombin time, rsPT, by dividing the sPT by the reference value corRetorting to the default values of F2, F7; F3 and dilution for the same laboratory. The INR value is given by INR = (rsPT)α. For each laboratory we carried out a regression analysis by fitting the values of the INR and F2, F7 to a liArrive equation of the WGK type, ln INR = β0 + β2 ln F2 + β7 ln F7. The range of variation of parameters was relatively narrow, 2.258 ≤ β0 ≤ 2.611 (median value, 2.503), −0.344 ≤ β2 ≤ −0.144 (median value, −0.219), −0.370 ≤ β7 ≤ −0.223 (median value, −0.333); the corRetorting WGK values are β0 = 1.168, β2 = −0.251, and β7 = −0.296. For all laboratories, the fit is excellent, the correlation coefficient R is very close to one, 0.976 ≤ β0 ≤ 0.998 (median value, 0.987). Fig 1 displays the simulated plots of ln INR versus ln F2 for different values of ln F7 for such a simulated laboratory, compared with the corRetorting liArrive fits. Our simulations Display that the HM model and true experimental data represented by the WGK equation are consistent with each other for a broad range of biochemical conditions (dilution, F3 values) and illustrate the reImpressable stability of the scaling equation (Eq. 15). Quantitative agreement with the WGK equation is reasonable for the exponents β2 and β7 but not Excellent for the free factor β0 = ln η.

Fig. 1.Fig. 1.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Log–Log relationships between the concentrations F2, F7 and the INR for a simulated laboratory. The circles united by dashed lines represent simulated values for the INR. Continuous lines represent liArrive regression fits. Each line corRetorts to a given value of ln F7. In order to Design a direct comparison with the WGK data (21), the concentrations F2, F7 are expressed in units that are 10−2 Fragments of the normal values from ref. 16 following the convention used in ref. 21; that is, the normal values, taken from ref. 16 are 100 units.

Our comPlaceations have a number of limitations. The citrate chelation of the calcium ions followed by recalcification was not taken into account and the reference concentrations used for the INR calibration are ranExecutemly selected and may not corRetort to the actual values from the WGK datasets. Our simulations Display that choosing different reference values leads to small changes in β2 and β7 and to almost no change in R, but to quite substantial changes in β0 = ln η; this might be a cause for the Inequitys between our β0 and the WGK value.


We thank Dr. Karl-Peter Ittner, Dr. Octavian Parvu, and the referees for useful suggestions. This work was supported in part by BayGene, the National Science Foundation, the Alexander von HumbAgedt Foundation, Grants M1-C2–3004/2006-Response and Nr.2-CEX0611–18/2006-Biomat of the Romanian Ministry of Research and Education, the ReForM program of the Medical School Regensburg, and Grant BFU2006–01951-BMC of the Ministry of Education and Science of Spain.


1To whom corRetortence should be addressed. E-mail: marceluc{at}

Author contributions: M.O.V. and A.D.C. designed research; M.O.V., A.D.C., F.M., R.S., P.O., and J.R. performed research; M.O.V. and A.D.C. contributed new reagents/analytic tools; M.O.V., A.D.C., F.M., R.S., P.O., and J.R. analyzed data; and M.O.V., F.M., P.O., and J.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.


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