Kinetic laws, phase–phase expansions, renormalization group, and INR calibration

Abstract

We introduce systematic approaches to chemical kinetics based on the use of phase–phase (log–log) representations of the rate equations. For slow processes, we obtain a corrected form of the mass-action law, where the concentrations are replaced by kinetic activities. For fast 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 show 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.

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 approach 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 measurements. 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 approaches based on optimization with genetic algorithms of the performance of a task assigned to a reaction mechanism have led to interesting 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.

Background

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 dosage 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, r_u, is proportional to the product of the concentrations c₁, c₂, …, of the different reagents, raised to different powers,

here, k_u are rate coefficients and v_uu′⁽¹⁾ are reaction orders. This form of mass-action law is assumed to hold only for elementary reactions for which the reaction orders v_uu′⁽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 fractions or ratios of fractal laws (cooperative, Hill kinetics). Generalizations of the mass-action laws can also hold for complex processes, where the reaction orders can be fractions (rate-determining step models) or arbitrary real numbers [generalized power law, fractional kinetics (8)].

The validity of the mass-action law is limited: it only holds for elementary, slow processes at local equilibrium occurring in ideal gases and dilute solutions. An important problem is the development of methods for representing deviations from the mass-action law. Our approach is based on the assumption that the dependences of the logarithms of the reaction rates, ln r_u, on the logarithms of the concentrations of the different species, ln c_u′, are analytic in the vicinity of the logarithms of a set of reference concentrations ln c_u′⁰; we can represent ln r_u by a Taylor series

where

are terms that express the contributions of expansion terms of orders higher than one and ν_{uu1′…um′}^(m) (c⁰) = ∂_{ln cu1′⁰}…∂_{ln cum′⁰} ln r_u (c⁰) are generalized reaction orders. We notice that a kinetic law of the mass-action law type, r_u(c) = k_u∏_u′ (c_u′)^v_uu′(1), is obtained from Eq. 1 if we keep the first-order terms, where k_u = r_u(c⁰)∏_u′(c_u′⁰)^{−v_uu′(1)}. The generalized reaction orders v_uu′⁽¹⁾ are arbitrary real numbers. The kinetic law obtained by keeping 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 r_u(c) = k_u(c⁰)∏_u′(c_u′)^{μ_uu′(1)(c0,c)}, where μ_uu1′⁽¹⁾(c, c⁰) = ν_uu1′⁽¹⁾(c⁰) + δν_uu1′⁽¹⁾(c, c⁰) are effective reactive orders that depend both on the reference concentration vector c⁰ and on the current concentration vector c. Corrections of the kinetic laws from the ideal behavior can be expressed in terms of kinetic activities (9, 10): it is assumed that, for nonideal systems, the reaction rates can be expressed as r_u ∼ ∏_u′(a_u′) ^v_uu′(1), where a_u′ are kinetic activities. The kinetic activities tend toward the corresponding 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 a_uu′ = c_u′(c_u′/c_u′⁰)^{δv_uu′(1)(c,c0)/v_uu′(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 r_u(c) = k_u(c⁰)∏_u′[a_uu′] ^{v_uu′(1)(c0)}. The kinetic activities are only species specific and not reaction specific for systems for which the ratios δv_uu′⁽¹⁾(c, c⁰)/v_uu′⁽¹⁾(c⁰) are independent of the reaction label u.

The approach can be extended for fast 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 r_u[c(ρ)] are functionals of the concentration field c(ρ). We have

The factors

express the contributions to the reaction rates of the expansion terms of an order higher than one. Here the influence functionals η_{uu1′,…, 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

Eq. 3 reduces to the mass-action law if only the linear terms are considered and full locality is assumed, that is η_uu′⁽¹⁾ = v_uu′⁽¹⁾δ(ρ − ρ′).

The first-order functional kinetic law analog to the mass-action law is: ln[r_u[c]/r_u[c⁰]] = Σ∫η_uu′⁽¹⁾ ln[c_u′/ c_u′⁰]dρ′, which is a special case of a functional fractal response law (7). Nonideality corrections lead to a modified equation ln[r_u[c]/r_u[c⁰]] = Σ∫ σ_uu′⁽¹⁾ ln[c_u′/ c_u′⁰]dρ′, where σ_uu′⁽¹⁾[c(ρ), c⁰(ρ⁰);ρ;ρ′] = η_uu′⁽¹⁾[c⁰(ρ⁰);ρ;ρ′] + δη_uu′⁽¹⁾[c(ρ), c⁰(ρ⁰);ρ;ρ′] are effective densities of reaction orders that depend both on the reference concentration field c⁰ (ρ⁰) as well as on the current concentration field c(ρ). We can also introduce species-specific and reaction-specific kinetic activity fields

The rate equations become

In general, the activity fields a_uu′(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 δη_uu′⁽¹⁾/η_uu′⁽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 η_{uu1′, …, um′}^(m) = ν_{uu1′, …, um′}^(m)∏_αδ(ρ − ρ_uα′).

In conclusion, in this section we have shown 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 show that there is a connection between complex recycling phenomena, and generalized mass-action laws with real apparent reaction orders. Although our approach 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, …, c_n. Each of these species is involved in a recycling process, which takes place 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 place 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 ψ_q1…qn the probability that the species c_u is involved in q_u recycling processes, where u = 1, …, n. The probability ψ_q1…qn can be determined from the detailed kinetics of the process and typically is the product of geometric (Pascal) probabilities. From the cycle q_u−1 to the cycle q_u the concentration of the species u is changed (reduced or amplified) by a factor b_u^(q_u) these factors can be also determined from the detailed kinetics of the process. We denote by r(c₁, …, c_n) the overall reaction rate in the case where no recycling processes occur and by r̃(c₁, …, c_n) the reaction rate for a system with recycling; r̃ can be expressed as an average over all possible numbers of recycling processes:

For arbitrary kinetics we have r = k∏_u′(c_u′)^μ_u′(1) and r̃ = k˜∏_u′(c_u′)^{μ̃_uu′(1)}, where in general both sets of apparent stoichiometric coefficients, μ_u′⁽¹⁾ and μ̃_u′⁽¹⁾, are concentration dependent. From Eq. 8 we come to:

where B_u = ∏_q′u=1^q_ub_u^(q_u′) are instantaneous overall change (reduction or amplification) factors attached to the species u and the average 〈… .〉 is taken over all numbers of recycling steps q_u and evaluated in terms of a scaled, renormalized probability for the number q_u of recycling events attached to the species u:

Eq. 11 is quite general and independent of the details of the kinetic process. It shows that recycling processes can easily produce arbitrary, real reaction orders. An interesting particular case of Eq. 9 is that for which the initial kinetics obeys the classical mass-action law and thus the initial reaction order μ_u⁽¹⁾ are concentration independent. In this case, we have μ̃_u⁽¹⁾ = μ_u⁽¹⁾〈B_u⁻¹〉 with 〈B_u⁻¹〉 = Σ_qu ψ̃_qu^(u)/B_u^(q_u).

A simple case is the one where a constant probability p_u for the occurrence of a step in the recycling process is attached to the species u, where u = 1, …, n; p_u 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, b_u^(q_u′), are assumed constant for a given recycling process u, that is b_u^(q_u′) = b_u. 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 ψ_q1…qn = ∏_u[p_u^q_u (1 − p_u)] and Eq. 8 becomes

The expansion Eq. 11 can be written as a renormalization group equation (RG; Eq. 12). We multiply each term of Eq. 11 by ∏_up_u, make the substitutions c_u → c_u/b_u and add a number of terms on both sides of the resulting equation, so that we can identify an expansion equal to r̃(c₁, …, c_n) on the left side. We obtain

The RG equation (Eq. 12) is a functional equation for the rate r̃(c₁, …, c_n), which can be solved by using the method of Mellin transformation (13). An alternative approach 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:

where μ_u = ln p_u/ln b_u are fractal exponents and Ξ[ln(c₁/c₁⁰), …, ln(c_n/c_n⁰)] is a slowly varying periodic function of ln(c₁/c₁⁰), …, ln(c_n/c_n⁰) with periods ln b₁, …, ln b_n, respectively. Such logarith-mic oscillations occur commonly in RG theory; although usually they are mathematical artifacts, and in some cases they do exist in experimental situations; even if they do exist, in general, they are very slow and hard to observe and can be neglected. Since p_u are probabilities, ln p_u are negative or zero and, thus, the signs of the effective reaction orders are determined by the change factors b_u: for amplification, b_u>1, the effective reaction orders are negative, whereas for reduction, b_u<1, they are positive.

In conclusion, in this section we have shown 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 (Zeldovich, 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 computed. 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 holds 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 important parameter. With a little practice the right choice for the reference vector improves the accuracy of phase–phase expansions. Suitable choices for the reference values make it possible to represent the kinetic equations for regions close to saturation levels (Langmuir, Michaelis–Menten, cooperative kinetics, etc.). The resulting equations do not lose their ability to describe important kinetic features, for example, limit cycles.

The direct phase–phase expansion procedures do 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 approach is to integrate the detailed kinetic equations for the model and to input the resulting progress curves into our algorithms. For large times the overall reaction rate sought is a functional of the previous time history 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 effects can be neglected and the overall reaction rate becomes a function of the current values of the concentrations; this approximation is reasonable in the case of prothrombin assays.

The problem becomes even more complicated when the input information is raw experimental data. As any log–log approaches, 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 described 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 fraction 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 stopped and the way for tissue repair is opened. 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 properly 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 measured 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 measurements. An international normalized ratio was introduced, (INR) (20), which is equal to the ratio between the measured coagulation time, t, and a normal coagulation time in the same laboratory, t_c, raised to a calibration exponent α which is positive,

Numerous experimental and statistical studies have shown that the definition (Eq. 14) provides a proper 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 shown that the first stage of the exogenic coagulation can be described by an allometric relation between the INR and the concentrations c_u of several plasma proteins:

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 approach to coagulation kinetics is twofold. (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 show 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 show 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 slow logarithmic oscillations in Eq. 13 and write the overall, renormalized, kinetic equation in the form r̃(c₁, …, c_n) = k̃∏_u=1ⁿ(c_u)^μ_u. 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 c_u; we have μ_u = ln p_u/ln b_u. The recycling occurs with losses (b_u < 1), due to other reactions and enzyme degradation, and thus the apparent reaction orders μ_u are positive. Since in the first stage of the coagulation the concentrations of the relevant plasma proteins are practically constant, the reduction factors b_u are close to one, b_u = 1 − ε_u, where ε_u, the fraction of losses, is close to zero. Similarly, for small losses, the number of recycling processes tends to be high and thus the probability p_u of the occurrence of a recycling event is close to one, p_u = 1 − π_u, where π_u, the probability that a recycling process stops, is close to zero. We have: μ_u = ln(1 − π_u)/ln(1 − ε_u) ≈ π_u/ε_u ≈ 1/ε _u〈q_u〉, where 〈q_u〉 = Σ_u q_u(1 − p_u)(p_u)^q_u = 1/π_u − 1 ≈ 1/π _u, for π_u close to zero, is the average number of recycling events of the species u. It follows that the bigger the efficiency of the recycling, the higher the effective reaction orders and the coagulation rate, and the smaller the coagulation time. Of course, since the self-similar model is idealized, these results are only qualitative.

We can show 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, marked 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 r_w. There should be a one-to-one correspondence between these two rates, expressed by a functional relation r_w = Φ_w0 (r₀); such relations should exist for any pair of laboratories, but it is enough to analyze only one pair. We have r₀ = k̃₀∏_u(c_u)^μ_u0 and r_w = k̃_w∏_u(c_u)^μ_uw; 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 = Ψ_0w(Γ_w), where Ψ_0w(x) = Φ_0w(k_wx)/k₀, and Γ₀ = ∏_u(c_u)^μ_u0, Γ_w = ∏_u(c_u)^μ_uw are factors with physical dimensions [concentration]^Σμ_u0 and [concentration]^Σμ_uw, respectively. By applying the Pi theorem from dimensional analysis (22) it follows that the relation Γ₀ = Ψ_0w(Γ_w) can be expressed in terms of a single adimensionalized variable Γ₀^1/Σμ_u0/Γ_w^1/Σμ_uw, or any real power of it different from zero. Therefore, we have Γ₀^1/Σμ_u0/Γ_w^1/Σμ_uw = Constant and thus Γ₀ ∼ Γ_w^{Σμ_u0/Σμ_uw}, from which we come to r̃₀ ∼ r̃_w^{Σμ_u0/Σμ_uw}. Since for prothrombin assays the concentrations of the relevant plasma proteins are practically constant, the reaction rates r̃₀ and r̃_w are also practically constant; they are inversely proportional to the corresponding reaction times, t₀ and t_w, r̃₀ = 1/t₀, r̃_w = 1/t_w, and therefore, t₀ ∼ (t_w)^α_0w, with α_0w = Σμ_u⁰/Σμ_u^w. Thus, the coagulation times measured in different laboratories, raised to different powers, α_0w = Σμ_u⁰/Σμ_u^w, w = 1, 2, … are proportional to the coagulation time measured in the reference laboratory 0. Up to a proportionality factor, the theoretical calibration law t₀ ∼ (t_w)^α_0w is identical to the empirical calibration law Eq. 14. Eq. 14 is derived by introducing a proportionality factor t_c0/(t_cw)^α_0w, expressed in terms of two characteristic times, t_c0 and t_cw, attached to the laboratories 0 and w, respectively, and defining the INR as a dimensionless normalized prothrombin time, corresponding to the reference laboratory 0, INR = t₀/t_c0.

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 t_w is inversely proportional to the coagulation rate, t_w ∼ 1/r̃_w ∼ ∏_u(c_u)^−μ_uw. Combining this equation with the INR calibration equation (Eq. 14) we obtain INR ∼ (t_w)^α_0w ∼ ∏_u(c_u)^β_uw, where β_u^w = −μ_u^w Σμ_u′⁰/Σμ_u′^w. Thus, we have derived the WGK equation (Eq. 15) and also obtained expressions for the scaling exponents β_u^w 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 do 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 proportional to the coagulation time, r ∼ 1/t. From Eqs. 14 and 15 it follows that r ∼ 1/t ∼ (INR)^−1/α ∼ ∏_u(c_u) ^−β_u/α, that is, a generalized mass-action law with arbitrary real exponents. By evaluating the proportionality factor we get:

where ΔT_h is the amount of thrombin formed in the first coagulation stage. Since the scaling exponents in the WGK Eq. 15 are negative, the effective 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(c_u)^μ_u, 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 properties of the initial stage of in vitro coagulation, induced by the generalized mass-action kinetic law, which describes 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 fractions 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 computed by fitting a power function to the set of points that have the same F₂, F₇ values. For each virtual laboratory we computed a relative prothrombin time, rsPT, by dividing the sPT by the reference value corresponding 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 linear 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 corresponding 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 corresponding linear fits. Our simulations show 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 remarkable stability of the scaling equation (Eq. 15). Quantitative agreement with the WGK equation is reasonable for the exponents β₂ and β₇ but not good for the free factor β₀ = ln η.

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 linear regression fits. Each line corresponds to a given value of ln F₇. In order to make a direct comparison with the WGK data (21), the concentrations F₂, F₇ are expressed in units that are 10⁻² fractions 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 computations 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 randomly selected and may not correspond to the actual values from the WGK datasets. Our simulations show 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 differences between our β₀ and the WGK value.