A Mathematical Model of the Human Metabolic System and Metabolic Flexibility

Abstract

In healthy subjects some tissues in the human body display metabolic flexibility, by this we mean the ability for the tissue to switch its fuel source between predominantly carbohydrates in the postprandial state and predominantly fats in the fasted state. Many of the pathways involved with human metabolism are controlled by insulin and insulin-resistant states such as obesity and type-2 diabetes are characterised by a loss or impairment of metabolic flexibility. In this paper we derive a system of 12 first-order coupled differential equations that describe the transport between and storage in different tissues of the human body. We find steady state solutions to these equations and use these results to nondimensionalise the model. We then solve the model numerically to simulate a healthy balanced meal and a high fat meal and we discuss and compare these results. Our numerical results show good agreement with experimental data where we have data available to us and the results show behaviour that agrees with intuition where we currently have no data with which to compare.

Appendices

Appendix 1: Properties of Solutions of the Governing Equations

1.1 Uniqueness of Steady State

In this appendix we will prove the uniqueness of our steady states. We begin by writing down which variables each steady state is a function of (except $I^*$),

$$\begin{aligned} G_b^*&= \frac{\beta _G }{(1 + k_{GL}{I^*}^2)(S_G + k_G(1 + k_{GI}I^*) + k_LI^* + k_{AL}I^*)} = f_1(I^*), \nonumber \\ T_b^*&= \frac{\beta _T}{(1+k_{TL}I^*)(k_{TA}(1+k_{AI}I^*)+k_T)} = f_2(I^*), \nonumber \\ A_l^*&= \frac{k_{AL}I^*G^*_b+k_{BL}A^*_b}{\frac{S_L}{1+k_{AS}I^*} + k_{LB} + k_{TH}I^*} = f_3(I^*, G_b^*, A_b^*), \nonumber \\ A_b^*&= \frac{\beta _A}{(1+k_{AA}{I^*}^2)(k_A + k_{BL})} +\frac{k_{LB}A^*_L}{k_A + k_{BL}} = f_4(I^*, A_l^*), \nonumber \\ G_m^*&= \frac{\beta _G k_G(1 + k_{GI}I^*) }{M_GP^*I^*(S_G + k_G(1 + k_{GI}I^*) + k_LI^* + k_{AL}I^*)} = f_5(I^*, P^*), \nonumber \\ A_m^*&= \frac{1}{M_AP^*} \left( \frac{\beta _Tk_T}{(1+k_{TL}I^*) (k_{TA}(1+k_{AI}I^*)+k_T)} \right) = f_6(I^*, P^*), \nonumber \\ Y_m^*&= k_Y G^*_m \left( \frac{1+k_{YI}I^*}{1+k_{YP}P^*} \right) \left( \frac{1+k_{CI}I^*}{k_{CP}P^*} \right) = f_7(I^*, P^*, G_m^*), \nonumber \\ T_m^*&= k_X A^*_m \left( \frac{1+k_{XI}I^*}{1+k_{XP}P^*} \right) \left( \frac{1+k_{DI}I^*}{k_{DP}P^*} \right) = f_8(I^*, P^*, A_m^*), \nonumber \\ P^*&= \frac{\mu - \mu _c}{\lambda _P} = f_9(I^*). \end{aligned}$$

(32)

The expressions for $A_b^ =f_4$ and $A_l^*=f_3$ form a linear system for $A_b^*$ and $A_l^*$ which has a unique solution for each quantity in terms of $I^*$ and $G_b^*$, and hence the solutions can be written explicitly in terms of $I^*$ using $G_b^*=f_1(I^*)$. Now we substitute the expression for $P^*=f_9$ into the equations for $G_m^*=f_5$, $A_m^*=f_6$, $Y_m^*=f_7$, etc., to obtain a simpler system given by

$$\begin{aligned}&G_b^* = f_1(I^*), \quad T_b^* = f_2(I^*), \quad A_l^* = \widetilde{f}_3(I^*), \quad A_b^* = \widetilde{f}_4(I^*), \quad G_m^* = \widetilde{f}_5(I^*), \nonumber \\&A_m^* = \widetilde{f}_6(I^*), \quad Y_m^* = \widetilde{f}_7(I^*, G_m^*), \quad T_m^* = \widetilde{f}_8(I^*, A_m^*), \quad P^* = f_9(I^*). \end{aligned}$$

(33)

Thus, all quantities can now be written as a function of $I^*$, only $T_m^*$ still depends on other quantities, and these can be eliminated using other expressions in (33), for example, $T_m^*=\tilde{f}_8(I^*,\tilde{f}_6(I^*))$.

Considering the steady-state equation for insulin, we write down an expression for $I^*$,

$$\begin{aligned} I^* = \frac{ k_1 + k_2 \mathrm{erf}((G_b-v)/c) + k_{IA}A_b^*}{\lambda _I}. \end{aligned}$$

(34)

We now examine the steady-state values for $G_b^*$ and $A_b^*$ given in Table 2 and note that $G_b$ terms contribute more significantly to the steady state than the $A_b^*$ terms. Hence we temporarily neglect the $A_b^*$ term. Under this assumption we substitute in the expression for $G_b^*$ to find

$$\begin{aligned} \lambda _I I^*&= k_1 + k_2 \mathrm{erf}\left( \frac{ f_1(I^*) -v}{c} \right) . \end{aligned}$$

The left-hand side of Eq. 35 is an increasing function of $I^*$ for $I^* > 0$, satisfying rhs $=0$ when $I^*=0$, and the right-hand side is a decreasing function of $I^*$ for $I^* > 0$, with rhs $>0$ at $I^*=0$; therefore, there is a unique solution for $I^*$.

Given that $I^*$ has a unique solution, we can now follow the chain of reasoning

$$\begin{aligned} I^* \hbox { has unique steady-state}&\Rightarrow G_b^*, T_b^*, P^* \hbox { have unique steady-states}, \\ I^* \hbox { and } P^* \hbox { have unique steady-states }&\Rightarrow G_m^* \hbox { and } A_m^* \hbox { have unique steady-states}, \\ I^* \hbox { and } G_b^* \hbox { have unique steady-states }&\Rightarrow A_l* \hbox { and } A_b^* \hbox { have unique steady-state}, \\ I^* , P^* \hbox { and } G_m^* \hbox { have unique steady-states }&\Rightarrow Y_m^* \hbox { has unique steady-state}, \\ I^* , P^* \hbox { and } A_m^* \hbox { have unique steady-states }&\Rightarrow T_m^* \hbox { has unique}. \end{aligned}$$

Hence the system as whole has a unique steady-state solution.

Reinstating the $A_b^*$ term theoretically could make the rhs of (35) nonmonotone, and so there could be multiple steady-states, however, for the physically realistic parameter values of interest to us, this does not occur.

1.2 Positivity of Solutions

Since all of the governing Eqs. (1)–(4), (7)–(14) have the form $\mathrm{d}X/\mathrm{d}t = A - B X$ with $A>0$, we can be sure that if $X=0$ ever occurs, then $X$ would increase, and so, provided we start with positive initial data, the concentrations will remain positive for all time.

Since the nonlinearities in the model are all analytic and have at most linear growth, the standard theory of ordinary differential equations implies uniqueness for the initial value problem.

Appendix 2: NonDimensionalisation

Before we attempt to solve the system numerically, we nondimensionalise it so that we may reduce the number of parameters in the model. This process also allows for simpler numerical simulations. We rescale each variable by its steady-state value, except for $Y_L$ which we rescale by $Y_{max}$, and $T_L$ which we rescale by a typical healthy liver fat concentration, denoted by $T_L^H$. This means that we are now concerned with the following nondimensional variables

$$\begin{aligned}&\widehat{Y}_L = \frac{Y_L}{Y_{max}} , \quad \widehat{A}_L = \frac{A_L}{A_L^*} , \quad \widehat{T}_L = \frac{T_L}{T_L^H} , \quad \widehat{G}_b = \frac{G_b}{G_b^*} ,\\&\widehat{A}_b = \frac{A_b}{A_b^*} , \quad \widehat{T}_b = \frac{T_b}{T_b^*} , \quad \widehat{I} = \frac{I}{I^*} , \quad \widehat{G}_m = \frac{G_m}{G_m^*} , \quad \widehat{Y}_m = \frac{Y_m}{Y_m^*} ,\\&\widehat{Y}_m = \frac{Y_m}{Y_m^*} , \quad \widehat{A}_m = \frac{A_m}{A_m^*} , \quad \widehat{T}_m = \frac{T_m}{T^*_m} , \quad \widehat{P} = \frac{P}{P_b^*}, \quad t = \frac{V_b G_B^*}{\beta _G} \,\widehat{t} . \end{aligned}$$

The forcing functions are nondimensionalised by $F_G(t) = \beta _G \widehat{F}_G(\widehat{t})$ and $F_T(t)=\beta _T \widehat{F}_T(\widehat{t})$, where the nondimensional forcing functions are given by

$$\begin{aligned} \widehat{F}_G(\widehat{t}) = \frac{\widehat{k}_{FG}\widehat{t}}{\widehat{\tau }_G} e^{-\widehat{t}^{\,2} / 2\widehat{\tau }^2_G},&\quad&\widehat{k}_{FG} = \frac{\theta _G}{B_G\beta _G}, \quad \widehat{\tau }_G = \frac{B_G\beta _G}{V_b G_B^*},\end{aligned}$$

(35)

$$\begin{aligned} \widehat{F}_T(\widehat{t}) = \frac{\widehat{k}_{FT}\widehat{t}}{\widehat{\tau }_T} e^{-\widehat{t}^{\,2} / 2\widehat{\tau }^2_T},&\quad&\widehat{k}_{FT} = \frac{\theta _T}{B_T\beta _T}, \quad \widehat{\tau }_T = \frac{B_T\beta _G}{V_b G_B^*}, \end{aligned}$$

(36)

In terms of their nondimensional variables, the functions $f_1$, $f_2$ and $f_3$ are given by

$$\begin{aligned} f_1(\widehat{Y}_L)&= \frac{\widehat{Y}_L}{Y_c+\widehat{Y}_L}, \qquad Y_c = \frac{Y_0}{Y_{max}}, \nonumber \\ f_2(\widehat{Y}_L)&= \frac{1- \widehat{Y}_L}{Y_c+ 1 - \widehat{Y}_L}, \quad Y_c = \frac{Y_0}{Y_{max}},\nonumber \\ f_3(\widehat{T}_L)&= \frac{\widehat{T}_L}{T_c+\widehat{T}_L}, \quad T_c = \frac{T_0}{T_L^H}. \end{aligned}$$

(37)

Using the above rescalings we obtain the following system of nondimensional equations where we have dropped the hats for convenience

$$\begin{aligned} \eta \frac{dY_L}{dt}&= \psi _L \left( \beta _1IG_b f_2(Y_L) + \frac{f_1(Y_L)}{1+\delta _GI^2} \right) , \end{aligned}$$

(38)

$$\begin{aligned} \eta \frac{dA_L}{dt}&= \epsilon _{ga} \theta _1 IG_b -\left( \frac{\theta _3}{1+\delta _HI}+\theta _2+\theta _4I \right) A_L +\frac{\theta _5}{\epsilon _{lb}} A_b, \end{aligned}$$

(39)

$$\begin{aligned} \eta \frac{dT_L}{dt}&= \epsilon _{at} \theta _4 IA_L -\frac{\psi _V f_3(T_L)}{1+\delta _TI}, \end{aligned}$$

(40)

$$\begin{aligned} \frac{dG_b}{dt}&= F_G(t) + \frac{f_1(Y_L)}{1+\delta _GI^2} - \frac{G_b}{1+\delta _G}+ \beta _1 (1-If_2(Y_L))G_b \nonumber \\&+(\beta _0+\theta _1)(1-I)G_b , \end{aligned}$$

(41)

$$\begin{aligned} \frac{dA_b}{dt}&= \psi _a \left( \frac{1}{1+\delta _AI^2} - \frac{A_b}{1+\delta _A} \right) + \epsilon _{lb} \theta _2 (A_L-A_b) , \end{aligned}$$

(42)

$$\begin{aligned} \frac{dT_b}{dt}&= \psi _t \left( F_T(t) + \frac{f_3(T_L)}{1+\delta _TI} - \frac{T_b}{1+\delta _T} \right) + (\beta _4 + \beta _5)(1-I)T_b , \end{aligned}$$

(43)

$$\begin{aligned} \frac{dI}{dt}&= \beta _6(1-I)+\beta _7(A_b-I) +\beta _8( \mathrm{erf}(w G_b-\rho ) -I \mathrm{erf}(w-\rho ) ) , \end{aligned}$$

(44)

$$\begin{aligned} \alpha \frac{\mathrm{d}G_m}{\mathrm{d}t}&= \mu _g (G_b-PIG_m) -\epsilon _{gg}\beta _0 G_m(1-I) \nonumber \\&- \epsilon _{yg}\beta _2 \left( \frac{(1+\gamma _p)}{(1+\gamma _p P)} \frac{(1+\gamma _y I)}{(1+\gamma _y)} G_m \right) -\epsilon _{yg}\beta _2\left( \frac{(1+\gamma _I)}{(1+\gamma _I I)} P Y_m\right) ,\qquad \end{aligned}$$

(45)

$$\begin{aligned} \alpha \frac{\mathrm{d}A_m}{\mathrm{d}t}&= \mu _a (A_b-A_m P) - \epsilon _{ta}\beta _4 (A_b-T_b) +\epsilon _{ta}\beta _4\gamma _t(A_b-IT_b) \nonumber \\&- \epsilon _{tg} \beta _3 \left( \frac{(1+\gamma _q)}{(1+\gamma _q P)} \frac{(1+\gamma _x I)}{(1+\gamma _x)} A_m \right) -\epsilon _{tg} \beta 3 \left( \frac{(1+\gamma _j)}{(1+\gamma _j I)} P T_m\right) , \end{aligned}$$

(46)

$$\begin{aligned} \alpha \frac{\mathrm{d}Y_m}{\mathrm{d}t}&= \beta _2 \left( \frac{(1+\gamma _p)}{(1+\gamma _p P)} \frac{(1+\gamma _y I)}{(1+\gamma _y)} G_m \right) - \beta _2 \left( \frac{(1+\gamma _I)}{(1+\gamma _I I)} P Y_m\right) , \end{aligned}$$

(47)

$$\begin{aligned} \alpha \frac{\mathrm{d}T_m}{\mathrm{d}t}&= \beta _3 \left( \frac{(1+\gamma _q)}{(1+\gamma _q P)} \frac{(1+\gamma _x I)}{(1+\gamma _x)} A_m \right) -\beta _3 \left( \frac{(1+\gamma _j)}{(1+\gamma _j I)} P T_m\right) , \end{aligned}$$

(48)

$$\begin{aligned} \alpha \frac{\mathrm{d}P}{\mathrm{d}t}&= \mu _p (1-P) + \epsilon _{gp} \gamma _g \mu _g P (1-I G_m) + \epsilon _{ap} \gamma _a \mu _a P (1-A_m) . \end{aligned}$$

(49)

where the new dimensionless parameters are given by

$$\begin{aligned}&\beta _0 \!=\! \frac{k_Gk_{GI}I^*G_b^*}{\beta _G} , \quad \beta _1 \!=\! \frac{k_LI^*G_b^*}{\beta _G} , \quad \beta _2 \!=\! \frac{k_{CP}P^*G_b^*}{\beta _GG_m^*(1\!+\!k_{CI}I^*)} , \quad \rho \!=\! \frac{v}{c} , \quad w \!=\! \frac{G^*_b}{c} ,\nonumber \\&\beta _3 \!=\! \frac{k_{DP}P^*G_b^*}{\beta _GA_m^*(1+k_{DI}I^*)} , \quad \beta _5 \!=\! \frac{k_{TA}I^*G_b^*}{\beta _G} , \quad \beta _4 \!=\! \frac{k_TI^*G_b^*}{\beta _G} , \quad \beta _6 \!=\! \frac{k_{1}{G^*_b}}{\beta _GI^*} ,\nonumber \\&\beta _7 \!=\! \frac{k_{IA}A^*_bG_b^*}{\beta _GI^*} , \quad \beta _8 \!=\! \frac{k_{2}{G^*_b}}{\beta _GI^*} , \quad \theta _1 \!=\! \frac{k_{AL}I^*G_b^*}{\beta _G} , \quad \theta _2 \!=\! \frac{k_{LB}G_b^*}{\beta _G} , \quad \theta _3 \!=\! \frac{S_LG_b^*}{\beta _G} ,\nonumber \\&\theta _4 \!=\! \frac{k_{TH}I^*G_b^*}{\beta _G} , \quad \theta _5 \!=\! \frac{k_{BL}G_b^*}{\beta _G} , \quad \gamma _Y \!=\! k_{YI}I^* , \quad \gamma _P \!=\! k_{YP}I^* , \quad \gamma _Q \!=\! k_{XP}I^* ,\nonumber \\&\gamma _I \!=\! k_{CI}I^* , \quad \gamma _J \!=\! k_{DI}I^* , \quad \gamma _X \!=\! k_{XI}I^* , \quad \mu _p \!=\! \frac{ \mu G_b^*}{ \beta _GP^*}, \quad \mu _g \!=\! \frac{ M_G P^* I^* G_b^* }{ \beta _G} ,\nonumber \\&\mu _a \!=\! \frac{ M_A P^* G_b^*}{ \beta _G}, \quad \psi _L \!=\! \frac{G_b^*}{Y_{max}}, \quad \psi _a \!=\! \frac{ \beta _A G_b^*}{ \beta _GA_b^*} , \quad \psi _T \!=\! \frac{ \beta _T G_b^*}{ \beta _GT_b^*} , \quad \psi _V \!=\! \frac{ \beta _T G_b^*}{ \beta _GT_L^H} ,\nonumber \\&\delta _G \!=\! k_{GL}{I^*}^2 , \quad \delta _T \!=\! k_{TL}I^* , \quad \delta _A \!=\! k_{AA}{I^*}^2 , \quad \delta _H \!=\! k_{AS}I^* , \quad \epsilon _{ga} \!=\! \frac{G^*_b}{A^*_L},\nonumber \\&\epsilon _{lb} \!=\! \frac{A^*_L}{A^*_b}, \quad \epsilon _{gg} \!=\! \frac{G^*_b}{G^*_m}, \quad \epsilon _{yg} \!=\! \frac{Y_m^*}{G_m^*} , \quad \epsilon _{ta} \!=\! \frac{T_b^*}{A_m^*}, \quad \epsilon _{at} \!=\! \frac{A_L^*}{T_L^H} , \quad \epsilon _{gp} \!=\! \frac{G_m^*}{P^*} ,\nonumber \\&\epsilon _{ap} \!=\! \frac{A_m^*}{P^*} , \quad \epsilon _{tg} \!=\! \frac{T^*_m}{G^*_m}, \quad \alpha \!=\! \frac{V_s}{V_b}, \quad \eta \!=\! \frac{V_l}{V_b}, \quad Y_c \!=\! \frac{Y_0}{Y_{max}}, \quad T_c \!=\! \frac{T_0}{T_L^H}. \end{aligned}$$

(50)

The values for the nondimensional parameters are given in Table 4.

Table 4 List of nondimensional parameters, their interpretation, and values used to produce the example results

Appendix 3: Graphs of Fluxes

In this section we plot various combinations of the governing concentration variables to illustrate the evolution of the fluxes in the model. Noting that the three functions $f_1,f_2,f_3$ are defined to be unity over the vast majority of their ranges, we only need to consider 10 combinations of concentrations of functions, which are plotted in Figs. 10 and 11.

Fig. 10

Top dimensionless fluxes involved in the model, namely the products of insulin concentration with plasma glucose (left), plasma TAG (centre) and liver FFA (right). Bottom, fluxes which involve the concentration $P$, muscle glucose with insulin (left), muscle FFA (centre) and muscle TAG (right). All graphs plotted against time in hours. Dotted lines indicate the results for the high fat meal, whilst the solid lines represent the balanced meal

Fig. 11

Dimensionless fluxes which are suppressed by insulin, plotted again time (hours). Top left flux from liver glucagon to plasma glucose; top right flux of FFA into plasma from adipose tissue; lower left flux of TAG from liver to plasma; lower right oxidation of liver FFA. Dotted lines indicate the results for the high fat meal, whilst the solid lines represent the balanced meal

The flux of glucose from plasma to muscle has a component which depends on insulin according to the product $IG_b$, which we plot in the top left panel of Fig. 10. This quantity also influences the flux from plasma glucose to liver glycogen and liver FFA, see Eqs. (2), (7), (8) and (11). The flux from plasma TAG to adipose tissue as modelled in (3) depends on the product $IT_b$, which is plotted in the top central panel. This is the only case where the high fat meal induces a higher response than the balanced meal. The product $IA_L$ describes the rate of conversion of FFA into TAG in the liver (8)–(9).

In the muscle, both glucose and FFA are used to convert $P$ into ATP, with rates that depend on $PIG_m$ and $PA_m$, respectively, as described by Eqs. (10), (11) and (13). These two fluxes are plotted in the lower left and lower centre plots in Fig. 10. The difference between balanced and high fat meals is more pronounced in the glucose term than the FFA term. The product $P T_m$ determines the rate of conversion of TAG to FFA in muscle as described in Eqs. (13) and (14), (note that in our current parameterisation, $k_{DI}=0$).

In the top left panel of Fig. 11, we plot the rate of release of glucose into the plasma from liver glycogen, this is inhibited by insulin, which we have modelled by $\beta _G/(1+k_{GL}I^2)$ in Eqs. (2) and (7). The top right panel shows the flow of FFA from adipose tissue to plasma, which has a similar form, but with $k_{AA}$ replacing $k_{GL}$, see Eq. (4). The flux from liver TAG to plasma TAG is shown in the lower left panel. This has weaker insulin dependence, being of the form $\beta _T/(1+k_{TL}I)$ in Eqs. (3) and (9). The oxidation of hepatic FFA is shown in the lower right panel of Fig. 11, this has the form $S_L A_L / (1+k_{AS}I)$, see Eq. (8). All four fluxes show significant reductions for the time that insulin is elevated, and not a great difference between the high fat mean and the balanced meal.