Mathematical model for the homeostasis of alpha-macroglobulins in the rat

Mathematical Biosciences 234 (2011) 17–24

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Mathematical model for the homeostasis of alpha-macroglobulins in the rat M.C. Aguirre, M. Armendariz, M. Lupo, A. Rigalli ⇑ Bone Biology Laboratory, Rosario National University, Argentina

a r t i c l e

i n f o

Article history: Received 19 September 2010 Received in revised form 4 August 2011 Accepted 5 August 2011 Available online 12 August 2011 Keywords: Alpha-macroglobulin Homeostasis Rat Monofluorophosphate

a b s t r a c t Alpha-macroglobulins (AM) are proteins that inactivate proteinases. Sodium monofluorophosphate (MFP) binds to AM and transiently changes AM plasma levels. As a consequence MFP is useful to modify AM homeostasis. A mathematical model to study the homeostasis of AM is proposed in this paper. The model describes changes in plasma concentration of AM, MFP concentration in the gastrointestinal tract, MFP plasma concentration, plasma concentration of AMMFP and includes rate constants of the processes involved in AM homeostasis. Estimation of the rate constants values was achieved using experimental and mathematical resources. The homeostasis of AM after an oral dose of 80 lmol of MFP was analyzed with a simulation tool. Experimental conditions that modify the homeostasis of AM had been simulated and validated using specific drugs that change some parameter of the system. The mathematical model describes accurately the behavior of the biological model. The results allow concluding that the simplifications made did not underestimate the main processes involved in the homeostasis and, also that the assumptions made were correct. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Alpha-macroglobulins (AM) are proteins whose cardinal function is the inactivation of proteinases. All of them have similar amino acid sequence, molecular mass and quaternary structure [1]. The binding of AM to proteinases induces elimination of the complex by receptors located mainly in the liver. AM are associated to a large number of important clinical entities such as, Alzheimer’s disease [2], blood coagulation system [3], blood viscosity [4], response to snake venom [5], resistance to radiation [6], Chagas’ disease [7] and pancreatitis [8,9]. The knowledge of AM homeostasis might contribute to understand its function in these diseases. There are few studies of the homeostasis of AM, although there is a great knowledge of its gene expression, receptor and protein expression [10]. The only information related to our work, was about plasma concentration of AM on burned people [11]. In addition, there is no information of mathematical models for the study of the homeostasis of AM. Sodium monofluorophosphate (MFP) is used in the treatment of osteoporosis, since fluoride acts as a mitogen of bone cells [12]. After an oral administration of MFP, one part is hydrolyzed to fluoride by alkaline phosphatase [13] and a fraction is absorbed in the stomach and in the intestine [14]. In plasma, MFP binds to AM in an equimolar ratio and forms a so-called AMMFP complex [15]. Subsequently after binding, AM losses its antiproteasic activity and is cleared by receptors located in liver and bone [16]. After ⇑ Corresponding author. Tel.: +54 341 155786840. E-mail address: [email protected] (A. Rigalli). 0025-5564/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2011.08.002

these findings we developed a novel mathematical model that allows studying the regulation of AM, using mathematical functions that relate the variables and parameters involved in the process. As MFP binds to AM, which modifies its concentration, it could be used as an experimental resource that transiently changes AM plasma levels. Moreover, MFP is a drug without serious secondary effects; hence it could be administered orally to humans and intravenously to rats. In this work it is proposed a mathematical model to study the homeostasis of AM, including the inactivation processes, when AM binds to MFP, the restoration of AM to blood stream and the uptake of AMMFP. Additionally, a quantitative and qualitative analysis has been done to verify the aforementioned model. The values of the rate constants for the homeostatic processes were obtained using experimental data of AM and AMMFP concentration. Finally, the validation of the model was achieved using substances that have a known effect over AM homeostasis. 2. Materials and methods Experiments were carried out in female IIM/Fm sub strain ‘m’ [17] rats of 200 ± 20 g body weight. When necessary, urethane (120 mg/100 g body weight, by intraperitoneal administration) was used as anesthetic. If necessary, the femoral artery was catheterized in order to inject test solutions and obtain blood samples. Heparin was used as anticoagulant. In addition, when the animals were not under the anesthetic effect, blood samples were obtained from the vein of the tail and substances were administered by orogastric tube.

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Blood samples were centrifuged and plasma were saved at 20 °C to measure AM and AMMFP concentrations. Dot-Blot assay was used to measure total concentration of AM (not bound AM + AMMFP) [9]. AMMFP concentration was measured by the difference between total plasma fluorine concentrations and its ultrafiltered through a 30 kDa cut-off ultrafiltration membrane [18]. 2.1. Determination of the reaction orders of AM inactivation and AMMFP uptake processes The femoral arteries of 8 rats were catheterized in order to administer 1 lmol of MFP (1 ml MFP 1 mmol/l + 5 mg heparin/ 100 g body weight) and collect blood samples at 0, 2.5, 5, 10, 20, 30, 50, 60 and 70 min. Plasma concentration of AMMFP and AM were obtained. The amount of MFP to be injected was calculated in order that MFP concentration (theoretical) reached in plasma after the injection was 10 times higher than AM concentration. Results obtained from this experiment were used to determine the reaction order and kinetics constants. 2.2. Determining and optimizing parameters The experiment comprised the administration of 80 lmol of MFP to 8 rats using an orogastric tube (1 ml of MFP 80 mmol/l). Blood samples were obtained at 0, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110 and 120 min thus AM and AMMFP concentration were determined and used to estimate the rate constants values and parameters of the model. 2.3. Experiments to validate the model Experiments that modify the intestinal hydrolysis of MFP and AMMFP uptake were carried out to validate the model. MFP intestinal hydrolysis was diminished by the coadministration of MPF and calcium [19], and AMMFP uptake was inhibited by the intravenous administration of 50 -Polyinosinic acid (Poly-I) [20]. 2.3.1. Experiments to validate the model using 50 -Polyinosinic acid and orogastric administration of MFP Rats received 80 lmol of MFP (1 ml MFP 80 mmol/l) through an orogastric tube (n = 4) and 1 ml of physiological solution containing 5 mg of Poly-I by cannulation of the femoral artery (treated rats). Control animals (n = 4) received the same MFP dose and

1 ml of physiological solution by a catheter introduced in the femoral artery. Poly-I was injected 15 min before maximum plasma concentration of AMMFP was reached. This time was established accurately based in previous works [13]. Blood samples were obtained before and 5, 15, 30, 40, 50, 60, 70, 80, 90, 105, 120, 130, 150 and 160 min after MFP administration, using the femoral catheter. Plasma concentration of AMMFP was determined as stated above. 2.3.2. Coadministration of MFP and calcium to validate the model Calcium was used in order to inhibit MFP hydrolysis by the intestinal alkaline phosphatase. The control group (n = 4) received 80 lmol of MFP through an orogastric tube (1 ml of 80 mmol/l of MFP solution). The treated group (n = 4) received MFP simultaneously with calcium (1 ml MFP 80 mmol/l + CaCl2 50 mmol/l). Blood samples were obtained from the vein of the tail before and 30, 60, 90, 120, 150, 180, 210, 240 and 270 min after the administration. Plasma was used to measure the concentration of AMMFP. 3. Results It is known that, after an oral dose of MFP a fraction is absorbed in the stomach [13] and bound to AM [14,15]. The MFP fraction that goes into the intestine is hydrolyzed by the intestinal alkaline phosphatase or is absorbed by the intestinal mucosa and bound to AM to form AMMFP, which is cleared by tissue receptors. These processes are represented by the biological model shown in Illustration 1. The model contains four variables that represent: plasma concentration of AM (AM), MFP concentration in the gastrointestinal tract (MFPd), MFP plasma concentration (MFP) and plasma concentration of AMMFP (AMMFP). In the model there are eight rates defined as a function of the four variables mentioned:

vS: AM secretion rate. vI: AM inactivation rate independent of MFP. vi: AM inactivation rate by plasma MFP. va: MFP absorption rate in the gastrointestinal tract. vh: MFP hydrolysis rate in the intestinal lumen. ve: MFP elimination rate from bloodstream (without including AM binding). vc: AMMFP uptake rate by tissue receptors.

Illustration 1. Biological model of the homeostasis of AM after an oral dose of MFP. See the text for explanation of rate and variables.

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The dynamic of these processes is represented by the following differential equations:

8 dAM ¼ v S  v I  v i; > dt > > > < dMFP ¼ v a  v i  v e ; dt dAMMFP > ¼ vi  vc; > dt > > : dMFPd ¼ v a  v h : dt

ð1Þ

The first equation of system (1) describes the changes of AM plasma concentration. The second equation describes plasma MFP rate of variation, which is the balance between the rates of absorption, elimination and inhibition of AM. The third equation consists in AMMFP rate of variation that is the difference between the formation and uptake rates of the complex. Finally, the fourth equation describes the changes in MFP concentration in the gastrointestinal tract, which is equal to the sum of the absorption and hydrolysis rates. The reaction orders of each chemical reaction were studied in order to find the equations for the aforementioned rates.

The concentration of AMMFP from 0 to 10 min represents the AM concentration that is transformed into AMMFP. For this reason, the difference between this value and the total AM concentration results in the amount of AM not bound to MFP. The logarithm of AM was adjusted by a linear function of time, as a consequence it can be considered as a first order process and its rate constant is: ki = 0.083583 ± 0.01998 min1. Previous results indicated a 1:1 M relation between MFP and AM. Consequently, it is assumed that the reaction is a first order kinetic respect to MFP. Thus, applying the mass action law vi could be written as:

v i ¼ ki AM:MFP: Taking into consideration AMMFP concentration values measured after MFP intravenous injection, the complex uptake rate constant kc value was obtained. The analysis of the process was done at different times after maximum complex concentration. As a consequence, it was proved that AMMFP uptake process follows a first order kinetic and the uptake rate constant value is kc = 0.062253 ± 0.049367 min1. In addition,

v c ¼ kc AMMFP: 3.1. Calculation of the reaction orders and rate constants of AM inactivation and AMMFP uptake processes Illustration 2 shows the change of AM and AMMFP concentration after the intravenous administration of MFP. AMMFP concentration rapidly increases immediately after MFP intravenous injection. The AMMFP formation reaction, or AM inactivation reaction, is represented by the next chemical equation:

AM þ MFP ! AMMFP: Knowing that AMMFP concentration initially is null, the reaction could be considered an isolated reaction without reverse rate. Gel filtration in vitro experiments that studied the reversibility of this process endorsed this consideration, for times involved in the model [15]. Immediately after the injection, it might be consider that the reaction order is pseudo order cero respect to MFP, as MFP concentration is 10-fold AM concentration. In this case the inactivation rate of AM vi could mostly be dependent on AM concentration. In this situation the reaction order respect to AM could be obtained from the relation between AM and time.

As it was demonstrated, MFP absorption rate va in the gastrointestinal tract (stomach and intestine) is a first order reaction [14].

v a ¼ ka MFPd : The rate of hydrolysis vh in the duodenum is also a first order kinetic respect to MFP concentration [14].

v h ¼ kh MFPd : The processes of AM secretion and inactivation independent of MFP binding are unknown. Therefore vS and vI were represented through linear functions with respect to the concentration of AM. Hence, vS(AM) = hSAM + AS and vI(AM) = hIAM + AI where hS, hI, AS and AI are real constants. Resulting in

v S ðAMÞ  v I ðAMÞ ¼ kAM þ A; where

k ¼ hI  hS and

A ¼ AS  AI : The clearance of MFP from plasma due to processes that do not involved reactions with AM is assumed as a first order process depending on MFP, therefore

v e ¼ ke MFP: 3.2. Mathematical model formulation Equations found in previous analyses are replaced in system (1) generating system (2) that models the changes in plasma concentration of AM, MFP and AMMFP plasma concentration, after an oral dose of MFP

8 dAM ðtÞ ¼ kAMðtÞ þ A  ki MFPðtÞAMðtÞ; > dt > > > < dMFP ðtÞ ¼ ka MFPd ðtÞ  ki MFPðtÞAMðtÞ  ke MFPðtÞ; dt dAMMFP > > dt ðtÞ ¼ ki MFPðtÞAMðtÞ  kc AMMFPðtÞ; > > : dMFPd ðtÞ ¼ ðka þ kh ÞMFPd ðtÞ: dt

ð2Þ

System (2) is subjected to initial conditions Illustration 2. Changes in AM and AMMFP concentrations after an intravenous injection of MFP. Data are shown as mean ± sd.

AMð0Þ ¼ AM0

MFPð0Þ ¼ 0 AMMFPð0Þ ¼ 0 MFPd ð0Þ ¼ MFP d0 :

Parameters in this model are:

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ke is the sum of all rate constants of MFP elimination (renal elimination and plasma hydrolysis), not including MFP bound to AM. ki is the rate constant of inactivation of AM, by binding to MFP. kc is the AMMFP plasma elimination rate constant. ka is the MFP absorption rate constant in the gastrointestinal tract. kh is the MFP intestinal rate constant of hydrolysis. k and A are linear function parameters that approximate the balance between AM inactivation and secretion by physiological processes. 3.3. Qualitative analysis of the system

The X⁄ equilibrium point is of biological interest if and only if x1 min 6 Ak 6 x1 max . In addition, it was studied the local asymptotic stability of equi2A 3 k

607 7 librium state X ¼ 6 405 . 0 The X⁄ stationary point stability is governed by the eigenvalues of the matrix.

0

k ki Ak B0 ki Ak  ke B B @0 ki Ak 0

It is convenient to adopt the following notation: x1 = AM, x2 = MFP, x3 = AMMFP and x4 = MFPd. System (2) could be written as:

8 x_ 1 > > > < x_ 2 > > x_ 3 > : x_ 4

¼ kx1 þ A  ki x1 x2 ; ¼ ka x4  ki x1 x2  ke x2 ;

ð3Þ

¼ ki x1 x2  kc x3 ; ¼ ðka þ kh Þx4

and the next initial conditions are satisfied

x1 ð0Þ ¼ x10 ;

x2 ð0Þ ¼ x20 ¼ 0;

x3 ð0Þ ¼ x30 ¼ 0 and x4 ð0Þ ¼ x40 :

The definition domain with biological meaning is the set: ( ) T ½x1 ; x2 ; x3 ; x4  2 R4 = X¼ : x1 min 6 x1 6 x1 max ; 0 6 x2 6 x20 ; 0 6 x3 6 x3 max ; 0 6 x4 6 x40 Maximum and minimum AM concentrations compatible with life are represented by x1min and x1max, respectively. Additionally, x3max is the maximum concentration of AMMFP. System (3) is an autonomous dynamic first order system. The fundamental existence and uniqueness theorem assures that the problem with initial values has a unique defined solution at t P 0. The equilibrium solutions of system (3) must satisfy the following algebraic equations:

8 kx1 þ A  ki x1 x2 ¼ 0; > > > < ka x4  ki x1 x2  ke x2 ¼ 0; > > ki x1 x2  kc x3 ¼ 0; > : ðka þ kh Þx4 ¼ 0:

ð4Þ

From the forth equation of (4) results that x4 = 0. Replacing this value in the second equation results that

x2 ðki x1 þ ke Þ ¼ 0: The two possible solutions are x2 = 0 or x1 ¼  kkei If x2 = 0, the first equation of (4) results in A = 0 when k = 0, 2 3 x1 607 7 therefore there are infinite stationary points X ¼ 6 4 0 5. However, 0 this situation is not of biological interest. When x2 = 0 and k – 0 one equilibrium state is obtained 2A 3 k

607 7 X ¼ 6 4 0 5. 2 k 3  ke 0 6 x i 7  ke 6 If x1 ¼  ki then X ¼ 4 2 7 is the other point of equilibrium. x3 5 0 The second critical point obtained is not part of X as AM equilibrium value is negative. Consequently, it studies lacks of biological interest.

0

0

0

0

ka

kc

0

0

ðkh þ ka Þ

1 C C C; A

ð5Þ

which clearly are k1 ¼ k; k2 ¼ ðki Ak þ ke Þ; k3 ¼ kc ; k4 ¼ ðkh þ ka Þ All the eigenvalues have a non-zero real part, therefore X⁄ is an asymptotically stable hyperbolic equilibrium point if and only if k > 0 and A > 0. Main results obtained are below:  T The system has an only equilibrium point X ¼ Ak ; 0; 0; 0 included in the biological defined domain X if and only if x1 min 6 Ak 6 x1 max . If k < 0 and A < 0 as a consequence X⁄ is the asymptotically unstable equilibrium point in X, more precisely a saddle point. If k > 0 and A > 0, this model allows a unique equilibrium point X asymptotically stable in X if x1 min 6 Ak 6 x1 max . Three different situations are distinguished: (a) If 0 < x10 < Ak the model predicts that the system reaches an equilibrium point in which the AM basal plasma level is higher than the measured value at time 0. This relation determines that the AM secreted by the liver is higher than the eliminated by binding to proteinases, in the other tissues. (b) The system returns to the conditions existing before the perturbation when x10 ¼ Ak. This situation responds to a metabolic state where hepatic production and proteinases inactivation are in equilibrium. (c) When Ak < x10 the model predicts that the system reaches equilibrium at a basal value of AM concentration lower than the value measured before the perturbation. In this situation, the liver produces less AM than the needed to inactivate proteinases. In order to know the system behavior a simulator was done using the Simulink, Matlab auxiliary tool. The system evolution was analyzed after an oral dose of 80 lmol of MFP. Basal AM concentration was fixed at the mean value of 10 lmol/l. Consequently, initial conditions were established in x10 ¼ 10; x20 ¼ 0; x30 ¼ 0; x40 ¼ 80. The values of the rate constants ki and kc, previously obtained, were used. In addition kh = 0.054 min1 and ka = 0.0078 min1 were obtained from previous reports [13]. Finally, it was proposed that ke = 0.05 min1, k = 0.04 and A = 0.5. The simulation with these values produces the graphs showed in Illustration 3 (solid line). An exponential decay of MFPd is observed (Illustration 3a). The diminished of AM is then restored and reaches a basal concentration value higher than the initial one since Ak ¼ 12:4 (Illustration 3b). The complex is formed immediately reaching a maximum value and then decays along the time (Illustration 3c). Plasma MFP reaches a maximum value and decays along the time (Illustration 3d). The simulator was used to test the system behavior with different values of the parameters. In each situation, the value of each parameter was changed in a factor of 101 and 101, respectively.

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Illustration 3. In solid lines functions obtained by simulation of MFP in the gastrointestinal tract (MFPd, panel a), plasma levels of AM (AM, panel b), AMMFP complex (panel c) after an oral dose of 80 umol of MFP and MFP in plasma (MFP, panel d) are shown. In dashed lines are shown changes in the mentioned variables after a 10-fold decrease in ki.

As an example, it is showed that when ki is 10 times lower (ki = 0.008 min1) the variables behavior are modified compared to the aforementioned situation (Illustration 3 A, B, C and D (dashdot line)). As it was predicted by the biological model, no change in MFPd, less inactivation of AM, less formation of AMMFP and higher concentration of MFP are observed. 3.4. Parameters estimation Experimental data obtained after the oral administration of MFP to 8 rats are shown in Table 1. The simulator was used to reach a first approximation of the system parameters needed to initiate the optimization process. Apart from that, the optimization was done with the Edsberg and Wedin DIFFPAR (DIFFerential equations with unknown PARameters) pack [21]. The toolbox DIFFPAR is based on formulating the parameter estimation problem as a non-linear weighted least squares problem. A Gauss–Newton type algorithm with local regularization is used for minimizing an objective function. The authors consider that this toolbox was useful to solve this problem due to its facilities for iteration in logarithmated values of the parameters (implicit positivity constraint), selection of which parameters that should be estimated and giving different weights for the measurements. The graphical user interface is based on Matlab’s handle graphics, and is mainly coded as one

Table 1 AM and AMMFP plasma concentration after an oral dose of MFP (mean ± sd, n = 8). Time (min.)

AM (lmol/L)

AMMFP (lmol/L)

0 5 10 15 20 30 40 50 60 70 80 90 100 110 120

10.00 ± 2.50 9.92 ± 3.51 10.22 ± 3.21 10.19 ± 2.33 10.76 ± 2.10 10.00 ± 3.22 11.53 ± 2.23 11.76 ± 4.01 14.44 ± 3.78 11.94 ± 4.16 11.87 ± 5.00 9.81 ± 4.90 11.44 ± 3.88 11.08 ± 2.79 11.30 ± 3.12

0.00 4.54 ± 1.17 5.83 ± 2.00 11.38 ± 1.10 30.89 ± 5.49 36.21 ± 3.69 54.34 ± 8.67 58.50 ± 6.78 85.24 ± 8.83 60.00 ± 10.09 56.32 ± 7.93 30.00 ± 5.67 27.00 ± 6.79 26.65 ± 3.08 20.76 ± 2.79

function with separate switches or cases that create user interface elements. The optimization process was initiated using the initial parameters vector b0 = [ki; ke; kc; k; A; ka; kh] = [0.02; 0.0015; 0.076; 0.22; 2.5; 0.078; 0.0054] and the initial conditions vector X0 = [10; 0; 0; 80]T. A weight of 1 was assigned to the variables AM and AMMFP. Additionally, a weight of 0 was assigned to the

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Table 2 Estimated values of parameters and confidence intervals at 95% of the estimated parameters using AM and AMMFP measurements. Parameters

Inferior end

Estimated parameters

Superior end

ki ke kc k A ka kh

1.6600 3 103 8.3418  102 4.6243  105 1.0414 10.783 2.1089  102 6.0210  102

8.1866  103 3.5416  104 1.0225  102 3.9580  101 5.0676 7.3067  102 6.3436  104

1.8033  102 8.4126  102 2.0403  102 1.8330 20.918 1.2505  101 6.1479  102

It is observed that plasma MFP elimination and AMMFP uptake constants values are modified. The model predicts the expected change in the complex uptake constant. Modification of the rate constants of plasma MFP elimination and MFP gastrointestinal hydrolysis were not expected. Unknown Poly-I effects could be responsible of these changes. 4. Discussion

variables that correspond to gastrointestinal MFP and MFP concentration. Estimated values of parameters and 95% confidence intervals are shown in Table 2. The equilibrium point of AM was calculated: Ak ¼ 12:80 lmol/l, this is a higher value than the one measured at initial time. The correlation matrix of Table 3 shows a low correlation of ki with ke and kc, on the contrary it is observed a strong relation between k and A due to the fact that the proportionality between them is the equilibrium point. Additionally, a strong correlation between ka and kh is evident and they also correlate with ke. This implies that the model reacts to the perturbation induced by MFP, modifying the elimination constant depending on the gastrointestinal absorption and capacity of hydrolysis. Strong negative correlation between ka and kh is expected, as an increased in MFP hydrolysis will be reflected in a lower gastrointestinal absorption. The same conclusion could be reached with the correlation between kh and ke. A lower hydrolysis will conduct to a higher gastrointestinal MFP absorption and, consequently to a higher MFP elimination. Illustration 4 shows the adjusted curves. 3.5. Model validation The model was tested using the effects of drugs that perturbed MFP gastrointestinal absorption and the AMMFP uptake. In each case the parameters values of the system were estimated. The values shown in Table 2 are used as initial values. Results obtained, when MFP is coadministered with calcium, are shown in Table 4. Data was adjusted using the DIFFPAR interface. Table 5 shows optimal values obtained with the adjustment. An increased in the absorption constant is observed whereas the hydrolysis kinetic constant decreased markedly. On the other hand, the higher MFP absorption could modify the kinetic of inactivation of AM, this is reflected in the increased of ki. The behavior of the model when the system is perturbed with a 50 -Polyinosinic acid (Poly-I) dose is showed below. Table 6 shows the measured AMMFP concentration values after the oral administration of MFP and Poly-I. The control group (n = 4) received only MFP. The estimating parameters are shown in Table 7.

A mathematical model to investigate alpha-macroglobulin (AM) homeostasis is shown in this work. The developed model could be applied in the rat, specie in which it was validated. This model includes internal disturbing factors, and MFP as an external one. The variables of the system are known and the parameters have physiological meaning. The system qualitative analysis allowed the determination of the unique equilibrium point. Oral MFP administration allows the modification of plasma levels of AM and AMMFP complex. Estimation of the rate constants values was achieved using experimental and mathematical resources in order to be independent of some variables and to reduce the number of unknown parameters. Between these resources it could be mentioned data analysis at times intervals, where some homeostatic processes could be consider negligible. These resources have been applied in vivo and guarantee the reliance of the values of the constants obtained. Through these data it is possible to estimate the model rate constants using iterative numerical methods for estimation of parameters. Specifically, the DIFFPAR software was adapted to the developed system. This system has a suitable response to experimental situations with predictable results obtained in previous studies, such as AMMFP complex elimination receptors or intestinal MFP hydrolysis. It was assumed that AM inactivation and secretion rates have a linear behavior respect to AM concentration. When MFP is administered to rats and humans it is observed the presence of fluoride bound to proteins. It has been demonstrated that this fluoride is originated by MFP gastrointestinal absorption and it is bound to AM [14]. The MFP joined to AM perturbs AM homeostasis generating changes in its plasma concentration. This is reflected in modifications in pancreatitis course [9] and blood viscosity [4]. Although, not all the animals involved in the study have the same modifications of neither AM concentration nor the desirable or non-desirable MFP effects. This makes us assume that AM homeostasis is not perturbed in the same way in each case. The model predicts that not always MFP administration will have the same effects over AM concentration. As it has been demonstrated, the model also predicts that after a MFP dose, AM concentration could reach a new equilibrium value equal, lower or higher than the basal initial one, depending on the relation between the parameters involved in the control of the secretion. The application of the model will permit the selection of those animals that behave in a similar way. In the case of pancreatitis, where AM activity is needed to inhibit proteinases released by the damaged pancreas, it allows the selection of those animals in

Table 3 Correlation matrix of the rate constant obtained from the optimization process.

ki ke kc k A ka kh

ki

ke

kc

k

A

ka

kh

1 0.087131 0.084813 0.59286 0.68828 0.093618 0.073208

0.087131 1 0.70253 0.17312 0.18922 0.85530 0.92283

0.084813 0.70253 1 0.14701 0.16578 0.51452 0.41684

0.59286 0.17312 0.14701 1 0.98455 0.14475 0.12707

0.68828 0.18922 0.16578 0.98455 1 0.15701 0.13703

0.093618 0.85530 0.51452 0.14475 0.15701 1 0.84473

0.073208 0.92283 0.41684 0.12707 0.13703 0.84473 1

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Illustration 4. Experimental values of AM and AMMFP complex after an oral dose of MFP are shown. Lines represent values predicted by the model.

Table 4 AMMFP levels after the oral administration of MFP (control) and MFP with calcium (treated). Results are expressed in lmol/L, mean ± sd.

Table 6 AMMFP concentration after an oral dose of MFP by an orogastric tube with (treated) and without (control) the intravenous administration of Poly-I. (mean ± sd).

Time (min)

Control (n = 4)

Treated (n = 4)

Time (min)

Control (n = 4)

Treated (n = 4)

0 30 60 90 120 150 180 210 240 270

0 1.3 ± 1.0 2.7 ± 1.0 8.1 ± 2.1 10.6 ± 2.2 9.1 ± 3.1 6.2 ± 1.3 5.5 ± 1.2 4.2 ± 2.1 3.6 ± 1.1

0 30.1 ± 5.1 96.2 ± 7.2 64.1 ± 11.3 47.8 ± 11.2 32.3 ± 8.5 14.2 ± 7.2 22.4 ± 7.8 6.2 ± 4.1 4.8 ± 1.0

0 5 15 30 40 50 60 70 80 90 105 120 130 150 160

0 14.7 ± 5.0 15.4 ± 4.9 93.8 ± 12.8 94.2 ± 11.0 110.2 ± 17.0 128.1 ± 29.4 130.2 ± 30.9 134.2 ± 37.8 135.4 ± 42.3 126.5 ± 37.5 127.8 ± 34.7 94.2 ± 21.0 61.0 ± 13.6 40.7 ± 10.0

0 12.2 ± 5.2 30.1 ± 12.4 96.2 ± 23.7 94.2 ± 22.0 110.2 ± 29.7 104.0 ± 28.9 130.2 ± 35.3 134.2 ± 30.5 144.8 ± 25.6 143.8 ± 23.8 209.0 ± 28.7 200.1 ± 39.8 195.5 ± 41.4 160.3 ± 33.8

Table 5 Estimated parameters with (treated) and without (control) calcium co administered with MFP. Parameters

Control (n = 4)

Treated (n = 4)

ki ke kc k A ka kh

8.1866  103 3.5416  104 1.61184  102 1 2.1218 6.1194  103 9.4378  103

1.1002  101 3.5416  104 7.0700  103 1 2.0797 8.7239  102 6.3436  107

which MFP will produce an increase in AM and will be more protected from the disease. Moreover, the model will allow selecting in which animal the administration of MFP will produce an increase or decrease in blood viscosity. The mathematical model was used to simulate metabolic processes. In order to do this, it was used the Simulink mathematical tool from Matlab to construct a simulator that allows in silica studies of the biological model changes. This tool permits the simulation of the variables not possible to measure in living animals, namely plasma and gastrointestinal MFP. There is no experimental data to be compared with the latter nonetheless the effects found are reasonable and convincing. Experimental conditions that modify the homeostasis of AM have been simulated and validated using specific drugs that changes some parameters of the system. In the case of orally coadministered calcium and MFP, the model predicts 10000 times de-

Table 7 Estimated parameters with (treated) and without (control) Poly-I. Parameter

Control (n = 4)

Treated (n = 4)

ki ke kc k A ka kh

8.1866  103 3.5416  104 1.4883  104 1 13.848 1.3824  101 6.3436  104

1.3204  103 2.9209  109 1.1097  106 1 70.609 4.3740  101 4.3264  105

crease of MFP gastrointestinal hydrolysis rate constant with a 10 times augment of MFP intestinal absorption constant. This result is completely compatible with the increase of plasma proteins bound to fluoride, observed in rats that received coadministration of MFP and calcium [19]. It is interesting to observe that ki, inactivation rate constant, is another parameter that increases 10 times. When 50 -polyinosinic acid was administered to rats after an oral dose of MFP, as it was expected a 100 times lower complex plasma elimination rate constant kc was obtained. Moreover, a change in kh value was observed that would not have any explanation. It is unknown if 50 -polyinosinic acid has effects that may influence over homeostatic processes related to this rate constant.

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The mathematical model achieved describes accurately the behavior of the biological model studied. The results obtained allow concluding that the simplifications made did not underestimate the main processes involved in the homeostasis and, also that the assumptions made (that have not been proved) were correct. Acknowledgements This work was founded by CONICET GRANT PIP 112-20080100462. Authors thank Ariana Foresto for the revision of the manuscript. References [1] F. Van Leuven, L. Umans, K. Lorent, C. Hilliker, L. Serneels, L. Overbergh, L. Stas, L. Raymakers, Molecular analysis of the human and mouse alpha-2macroglobulin family, in: W. Borth, R. Feinman, S.L. Gonias, J. Quigley, D. Strickland (Eds.), Biology of Alpha-2-macroglobulin. Its receptor, and related proteins, Ann. N Y Acad. Sci., New York, 1994, vol. 737, p. 163. [2] I. Tooyama, T. Kawamatta, H. Akiayama, S.K. Moestrup, J. Gliemann, L. McGeer, Immunohistochemical study of alpha-2-macroglobulin receptor in Alzheimer and control postmortem brain, Mol. Chem. Neuropathol. 18 (1993) 153. [3] J.M. Stassen, J. Arnout, H. Deckmyn, The hemostatic system, Curr. Med. Chem. 11 (2004) 2245. [4] V. Di Loreto, A. Rigalli, L. Cinara, G. Hernández, Effect of disodium monofluorphosphate on plasma and blood viscosity in the rat, Clin. Hemorheol. Microcirc. 40 (4) (2008) 259. [5] Y.L. Tseng, W.B. Wu, C.C. Hsu, H.C. Peng, T.F. Huang, Inhibitory effects of human alpha-2-macroglobulin and mouse serum on the PSGL-1 and glycoprotein Ib proteolysis by a snake venom metalloproteinase, triflamp, Toxicon 43 (2004) 769. [6] L. Sevaljevic, S. Dobric, D. Bogojevic, M. Petrovic, G. Koricanac, M. Vulovic, D. Danzir, N. Ribarac-Stepic, The radioprotective activities of turpentine-induced inflammation and alpha-2-macroglobulin: the effect of dexamethasone on the radioprotective efficacy of the inflammation, J. Radiat. Res. 44 (2003) 59.

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