Model of α-linolenic acid metabolism

Transcription

1 Model of α-linoleni aid metabolism N.Kokulan, C.-H. Lai Shool of Computing and Mathematial Sienes University of Greenwih London, UK RAE2012 Competitive Grant with Shool of Siene Projet progress meeting on Friday 14/12/2012

2 OVERVIEW Introdution ompartmental model of α-linoleni aid metabolism The set of differential equations Parameters estimation Numerial Results Experiments using data from Shool of Siene

3 Introdution A mathematial model of α-linoleni aid metabolism is derived from the experimental data of plasma onentration [1] at different times for 18:3n-3, 20:5n-3, 22:5n-3 and 22:6n- 3 in 95 healthy volunteers who have been ingested with 1g of isotope traer of α-linoleni aid. Mathematial methods based on inverse problems for the identifiation of unknown kinetis parameters in the system of ODEs using time series measurements. The goal of this model is to obtained kinetis parameters whih an be used to quantify the biosynthesis of long hain n-3 PUFA beginning with α-linoleni aid.

4 Compartmental model of α-linoleni aid metabolism A physiologial ompartmental model of α-linoleni aid metabolism is employed with well-known onsideration of metaboli pathway for α-linoleni. Assumptions The liver is the main site for the biosynthesis of fatty aid. As a liver biopsy was not possible in these subjets therefore in this study rate onstants determined from the model represents kinetis of fatty aid from their plasma pool onentrations. The model presumes that eah rate onstant reflets several steps of the metabolism that our within the liver [1] It is assumed that this kineti funtion is similar to the appearane of all the fatty aids measured in the plasma.

5 (1) 18:3n:3 (stomah) (2) 18:3n:3 (3) 20:5n:3 (4) 22:5n:3 (5) 22:6n:3 Compartmental model for the metabolism of α-linoleni aid K 2,1 K 3,2 K 4,3 k 5,4 K 0,2 K 0,3 K 0,4 K 0,5 Fig 1: Compartmental model for the metabolism of α-linoleni aid. Not Returned (oxidized) The struture of the model onsists of five ompartments. The retangle represents eah metabolite of the fatty aid in the metabolism proess. Compartment 1 represents administration of the isotope (1g) via the gastro-intestinal trat. Compartments 2 to 5 represent n-3 fatty aid in the plasma following on suessive steps in desaturation and elongation of the label fatty aid. The frational transfer rate Kij is the fration of the substrate that is transferred from substrate ompartment j to produt ompartment i. The arrows indiating K0j is the fration of the substrate that is irreversibly lost by eah ompartment j.

6 The system of differential equations d dt d dt 5 = k5,4 4 k0,5 5 4 = k 4,3 3 k 0,4 4 k 5,4 4 (1) (2) d dt 3 = k 3,2 2 k 0,3 3 k 4,3 3 (3) d dt 2 = k2,11 k 0,22 k3,2 2 (4) d 1 dd = k 21 1 (5) The parameters k 5,4, k 4,3, k 3,2, k 2,1, k 0,5, k 0,4, k 0,3 and k 0,2 of the differential equations are to be estimated by means of linear least squares method using measured data.

7 Conentration assumption in the stomah Assuming the onentration of 18:3n-6 in stomah follows a dereasing quadrati funtion wrt time: 2 at + bt + Here the parameters a and b an be inluded in the inverse problem formulation and is known from the initial ondition of the onentration in the stomah.

8 Results & experimental data Estimated parameters Time series experimental data Kinetis Estimated values Parameters k k k k k k k k Time Subjet 18 was used in the above experiment to determine the respetive values of the kinetis parameters. The parameters obtained are lose to those obtained in the referene paper [1] in whih their results were alulated by means of the software WIMSAM.

9 Fatty aids omposition of serum holesteryl esters (%) from healthy subjets supplemented with hempseed and flaxseed oils for 4 weeks: Data obtained from Shool of Siene Fatty aids Hempseed oil Flaxseed oil 0 week 4 week 0 week 4 week n-6 18:2n-6- LA :3n-6- GLA :3n-6 -DGLA :4n-6-AA n-3 18:3n-3-ALA :5n-3-EPA :6n-3-DHA

10 Fatty aids omposition of serum triglyerides (%) from healthy subjets supplemented with hempseed and flaxseed oils for 4 weeks: Data obtained from Shool of Siene Fatty aids Hempseed oil Flaxseed oil n-6 0 week 4 week 0 week 4 week 18:2n :3n :3n :4n n-3 18:3n :5n :6n We need at least three olumns of data for Hempseed oil and three olumns of data for Flaxseed oil not enough data at different time points in alulating the derivatives. The number of unknowns in one equation is more than the number of available data Need to use adjoint method [5] or regularisation ( urrent work).

11 Time-ourse of fatty aids onentration (µg/ml) of plasma triaylglyerol and holesterol ester from healthy subjets Time (h) Triaylglyerol Cholesterol ester LA LNA LA LNA The model that we are onsidering is a very simple metabolism model of 18:3n-3 (LNA) to 20:5n-3 (EPA) and finally to 22:6n-3 (DHA). The equations are inter-related. The data provided above, say the seond olumn, refleting the LNA onentration at various times for 3 g of deuterium-labelled LNA intake. A graph an be easily drawn for this LNA onentration. However it is possible to use the data in an absorption model [3,4]. ( urrent work)

12 Absorption & Elimination model C 0 =3g (1) 18:3n:3 (stomah) K 2,1 (2) 18:3n:3 ( small intestine) K 3,2 (3) 18:3n:3 (Plasma) Elimination Proess (Mihaelis Menten Equation) V m K m Where V m and K m M.M oeffiient The struture of the open model onsists of Three ompartments. The retangle represents 18:3n:3 in the absorption sheme The system differential equations are: d 1 dd = k 21 1 d 2 dd = k 21 1 k 32 2 d 3 dd = k 32 2 V m 3 K m + 3

13 Referenes [1] Pawlosky, R. J., et al. Physiologial ompartmental analysis of alpha-linoleni aid metabolism in adult humans. J. Lipid Res. 42(8): , [2] G. Golub Numerial Methods for solving Linear Least Squares Problems. [3] J. E. PIETERS, M. WEDEL and G. SCHAAFSMA Parameter estimation in a three-ompartment model for blood alohol urves [4] Bus, J. C. P., Domselaar, B. V. and Kok, J. (1975) Nonlinear least squares estimation. Report NW 17/75, Mathematial Centre, Amsterdam. [5] S.Muller,J.Lu,P.Kugler,H.W.Engl Parameter Identifiation in Systems Biology: Solving Ill-posed Inverse Problems using Regularization.