Mathematical Model for Low Density Lipoprotein (LDL) Endocytosis by Hepatocytes
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Abstract
Individuals with elevated levels of plasma low density lipoprotein (LDL) cholesterol (LDLC) are considered to be at risk of developing coronary heart disease. LDL particles are removed from the blood by a process known as receptormediated endocytosis, which occurs mainly in the liver. A series of classical experiments delineated the major steps in the endocytotic process; apolipoprotein B100 present on LDL particles binds to a specific receptor (LDL receptor, LDLR) in specialized areas of the cell surface called clathrincoated pits. The pit comprising the LDL–LDLR complex is internalized forming a cytoplasmic endosome. Fusion of the endosome with a lysosome leads to degradation of the LDL into its constituent parts (that is, cholesterol, fatty acids, and amino acids), which are released for reuse by the cell, or are excreted.
In this paper, we formulate a mathematical model of LDL endocytosis, consisting of a system of ordinary differential equations. We validate our model against existing in vitro experimental data, and we use it to explore differences in system behavior when a single bolus of extracellular LDL is supplied to cells, compared to when a continuous supply of LDL particles is available. Whereas the former situation is common to in vitro experimental systems, the latter better reflects the in vivo situation. We use asymptotic analysis and numerical simulations to study the longtime behavior of model solutions. The implications of modelderived insights for experimental design are discussed.
Keywords
Endocytosis LDL uptake Regulation of LDLreceptor production PCSK91 1. Introduction
Complex biological mechanisms have evolved to transport cholesterol around the body, and to prevent accumulation of toxic levels of cholesterol within cells. A family of macromolecular complexes, known as lipoproteins, transport cholesterol through the bloodstream to the major tissues. In humans, low density lipoprotein (LDL) particles carry the majority of the plasma cholesterol, and elevated levels of LDL cholesterol (LDLC) is a widely accepted risk factor for the development of coronary heart disease (CHD). Advances in our understanding of LDL metabolism have led to the identification of lifestyle or pharmaceutical interventions which improve plasma LDLC levels; however, raised levels of plasma LDLC remain a concern for human health, and more effective interventions are required.
LDLC levels are in part determined by the rate at which LDL particles are taken up and removed from the circulation. The liver cells (hepatocytes) are responsible for the major part of whole body LDL uptake. The process by which LDL is taken up and processed by hepatocytes is known as receptor mediated endocytosis and involves a sequence of wellorchestrated mechanisms, which have been well defined in a series of classical experiments by Brown and Goldstein (1979). The first step in this process involves an LDL particle binding to hepatic LDL receptors (LDLR) in specialized regions of the cell surface known as clathrincoated pits. The interaction with the LDLR is mediated by LDLassociated apolipoprotein B (apo B100). Upon binding to the LDLR, LDL particles become internalized into the cell, forming intracellular vesicles known as endosomes. In this paper, we use the term internalized pits to refer to endosomes.
Fusion of endosomes with lysosomes results in degradation of the LDL particles, and the release of its constituent parts (e.g., cholesterol, fatty acids, and amino acids). The LDLRs may be either recycled prior to lysosomal fusion, or degraded. A negative feedback mechanism regulates the number of cell surface LDLR on the basis of the levels of intracellular free cholesterol, such that when they are elevated the number of LDLRs is reduced, whereas when they are low, the number of LDLRs is increased.
In vitro assays are widely used to study LDL cellular metabolism (Bradley et al., 1984; Brown and Goldstein, 1979; Cho et al., 2002; Jackson et al., 2005, 2006; Mamotte et al., 1999). These assays, which quantify the rate of LDL uptake by cultured cells, are used to investigate the steps of endocytosis, and to explore the mechanisms underlying the reduced rates of LDL uptake exhibited under specific experimental conditions. The assays typically involve adding an amount of lipoprotein spiked with radiolabeled LDL to the cell culture medium at a fixed timepoint, and tracking the movement of radiolabeled LDL into the cell over time. In vivo, rather than being exposed to a single bolus of lipoprotein, cells are typically exposed to a continuous source of lipoproteins, an effect which is technically challenging to reproduce in vitro. A major focus of this paper, is to assess the effect that such differences in the rate of lipoprotein delivery, has on the system behavior, and to identify methods to improve experimental design.
Over the past 20 years, a number of models of LDL metabolism have been formulated, which vary in both the mathematical approaches taken and biological scope, see August et al. (2007), Chun et al. (1985), Harwood and Pellarin (1997), Shankaran et al. (2007), for example. In several of these models, certain parameters were fitted in order to achieve agreement with experimental data from cultured human fibroblasts. Dynamic models have also been formulated. These tend to focus on specific areas of LDL metabolism, for example, fluid dynamics of lipid accumulation in the arterial wall, or the effect of antioxidants on the kinetics of LDL oxidation. August et al. (2007) developed a dynamic model of the lipoprotein delipidation cascade, consisting of a system of nonlinear differential equations which are more strongly linked to the underlying physiological processes. The model links lipoprotein metabolism at a physiological level, to the cellular level, by including a feedback mechanism whereby the cell regulates the amount of LDL receptors available for lipoprotein binding in response to the intracellular cholesterol concentration.
The model described in Section 2 of this paper is similar in form to that developed by August et al. (2007), that is, a system of ordinary differential equations (odes). However, rather than addressing the kinetics of the lipoprotein delipidation cascade, our model focuses in greater detail on the uptake of LDL by hepatocytes in culture, and incorporates greater detail on the endocytotic process than included in the model in August et al. (2007). In Section 2, we also estimate the model parameters and nondimensionalize the governing equations. In Section 3, we present typical numerical results and compare them to experimental data before performing a parameter sensitivity analysis. We confirm the numerical results and perform a steady state analysis to investigate the system's long term behavior in more detail in Section 4. Finally, in Section 5, we discuss the results. Our focus throughout the paper is on the endosome dynamics and the assumption that concentrations and fluxes are modulated by intracellular cholesterol levels.
2 2. Model development
In this section, we formulate the mathematical model that we use to study the dynamics of LDL endocytosis by hepatocytes or HepG2 cells, which is a cell line derived from a patient with hepatocellular carcinoma. We construct a model based on a system of ordinary differential equations which describe the evolution of spatiallyaveraged concentrations of LDL and cholesterol. Whilst the concentration of LDL in bulk extracellular medium may not be uniform, we will assume that there is a layer of fluid close to the cells in which the spatiallyaveraged concentration of LDL particles can be defined. The reason for this is our desire to focus on the dynamics of adhesion, internalization, and receptorregulation rather than the fluid dynamics of LDLdelivery to the cell surface.
2.1 2.1. Microscopic modeling of pit dynamics
2.2 2.2. Dynamics of LDL internalization and conversion to cholesterol
The quantity \(\bar k_L\) is the rate at which LDL particles are supplied to the system from an external store (e.g., replenished cell medium). Our model describes two different scenarios, depending on the value assigned to the rate \(\bar k_L\). Firstly, there is the “single bolus” model, in which the hepatocytes are deprived of LDL before a fixed dose of lipoprotein is delivered at time \(\bar t = 0\) (that is, at the start of the experiment). This is simulated by fixing \(\bar k_L = 0\) in Eq. (8) and prescribing \(\bar L_e (0)\) appropriately. The second scenario occurs when \(\bar k_L > 0\) so that there is a continuous source of LDL replenishing the extracellular pool of LDL. We note that \(\bar k_L > 0\) corresponds to the situation in which LDL particles are added continuously during the experiment. Therefore, we refer to it as the “continuouslyinfused” model. This case better reflects certain in vivo situations, in which, following a meal, lipoproteins are delivered to the liver continuously over a period of a few hours. We will investigate the single bolus and the continuouslyinfused models in Sections 3.1–3.2 and 3.3–4, respectively.
2.3 2.3. Reduction of pit dynamics to a macroscopic model
Characteristics and properties of a typical liver cell and a typical LDL particle. Values quoted are the postoptimization procedure, initial estimates for values for related systems available in the literature are quoted in brackets. Note that “mol” refers to the number of molecules or pits
Parameter  Description  Dimensional value 

Number of pits per cell  180  
Number of receptors per cell  32,000  
P _{ m }  Maximum number of receptors per pit  ∼200 
Radius of LDL particle  10 nm  
Number of receptors covered by LDL  ∼1  
Average radius of a pit  100 nm  
¯C _{ c }  Average cell volume  1 pl 
Average cell protein content  300 pg  
Cell volume of 1 mg cell protein  0.0033  
¯V _{ medium }  Volume of extracellular medium  10 ml 
¯V _{ cell }  Volume of cell culture  0.0066 ml 
¯L _{0}  Initial concentration of extracellular LDL  1.17 × 10^{13} mol/ml 
¯n _{0}  Concentration of pits  1.81 × 101^{11} mol/ml 
¯A  Rate of LDL binding to a receptor  6.64 × 10^{−17} mL/mol/sec 
¯g  Rate of release of pits from store  0.0108 per sec 
¯b  Rate of internalization of LDLbound pits  (0.0027) 0.0046 per sec 
¯b _{0}  Rate of internalization of empty pits  0.0061 per sec 
¯k _{ L }, ¯k _{ Ll },¯k _{ Lu }  Rate of addition of LDL to medium  0 mol/ml/sec 
¯C _{ e }  Intracellular Cholesterol Concentration  2.65 × 10^{19} mol/ml 
¯K  Cholesterol regulation of pit production  Varied around ¯C _{ e } 
¯k _{ s }  Rate of production of new pits  2.17 sx 10^{28} mol^{2}/ml^{2}/sec 
f  Fraction of internalized pits recycled  [0.7, 1] 
¯k _{ id }  Rate of degradation of LDL to cholesterol  (0.0033) 0.0002 sec^{−1} 
¯r  Number of cholesterol molecules per LDL  3400 
¯λ  Timescale of cholesterol regulation  0.0033 per sec 
¯λ{′}  Rate amino acid leaves the cell  (0.0027) 0.0015 per sec 
2.4 2.4. Parameter values
To calculate the model parameters, we use data from the available literature where possible and for other parameters consider different parameter ranges and their impact on model outputs. In deriving the model, we have implicitly assumed that concentration variables are measured in number of particles per volume. Therefore, before proceeding, we need to convert all the concentrations from mass per unit volume to numbers per unit volume.
2.4.1 2.4.1. Geometric parameters

\(\bar L_0\): Initial concentration of LDL (see Eq. (26)). In the experiments presented in Jackson et al. (2006), the initial concentration of LDL was 10 µg of ApoB100 per ml of cell medium. We convert this value to the number of particles by taking into account the molecular weight of apo B100 particles (MW apo B100 = 512723), that is, we write 10 µg per ml of cell medium, \({{\bar L}_0} = {{10 \times {{10}^{  6}}} \over {{\rm{MW apo b  100}}}}\) moles lipoprotein particles /ml × N _{ A } = 1.17×10^{13} particles/ml of cell medium.

\({{\bar L}_{b0}} = {{\bar L}_{bl}}(0)\): The initial value of bound radiolabeled LDL (see Eq. (26)); in the experiments of Brown and Goldstein (1979), this is 46 ng of LDL ApoB per mg of cell protein (Fig. 1 of Brown and Goldstein, 1979). In the units used in this paper, this is equivalent to 1.62 × 10^{13} particles per ml of cell volume. Note that this concentration is greater than L _{0} as the reference volumes used differ.
 \(\bar n_0\): Initial number of pits (see Eq. (26)). Taking a value of 92.4 ng of apo B100 per mg cell protein from Harwood and Pellarin (1997), we find \({{\bar R}_T} = {{92.4 \times {{10}^{  9}}} \over {{\rm{MW apo b  100}}}}{N_A} = 1.09 \times {10^{11}}\) receptors per mg of cell protein, which translates to \({{1.09 \times {{10}^{11}}} \over {{{\bar V}_c}}} = 3.26 \times {10^{13}}\) receptors per ml of cell volume, using estimates of cell volumes and protein content from Table 1. Assuming that 2% of cell area is covered by pits and using values for well protein content and well area to estimate the total number of cell receptors per well and the total number of pits per well,,$${R_{pit}} = {{{\rm{total number of receptors per well}}} \over {{\rm{total number of pits per well}}}}$$, and, therefore, \({{\bar n}_0} = 3.26 \times {10^{13}}/{R_{pit}} = 1.81 \times {10^{11}}\) pits per ml.$${R_{pit}} = {{{{\bar R}_T} \times {\rm{Well Protein}}} \over {0.02 \times {\rm{Well Area}}/\pi r_{pit}^2}} = {{1.09 \times {{10}^{11}}\pi {{(100)}^2}} \over {0.02}} = {{0.2} \over {0.00019}} = 180$$

p _{ m }: Maximum number of LDL particles in the pit; this occurs in Eqs. (13)–(15) and (17)–(18). Taking the pit radius as 100 nm, and noting that the radius of an LDL particle is 10 nm, we calculate that each pit can accommodate an upper limit of \(2\pi r_{pit}^2/\pi r_{LDL}^2 = 2{\left( {{{100} \over {10}}} \right)^2} = 200\) LDL particles.

\(\bar V_c\): Volume of a single cell (used in calculating \(\bar V_cell\), hence α, below). We assume that one cell has approximately 300 pg of cell protein and has a volume of 1 pl (Table 1). Hence, 1 mg of cell protein is equivalent to 0.0033 mL. This conversion factor is used to convert concentrations measured relative to cell protein to that relative to cell volume.

\(\bar V_cell\) : The total cell culture volume is estimated from the total mass of cell protein and \(\bar V_c\). Hence, for a total mass of cell protein of 0.2 mg, we estimate a cell culture volume of 0.00066 mL. This is required for the calculation of α, below.

\(\bar V_medium\): We assume the volume of medium is 10 ml.

α: \({\rm{Volume Ratio}} = {{\bar V}_{medium}}/{{\bar V}_{cell}} = 10{\rm{ ml}}/0.00066{\rm{ ml}} = 15000\) (nondimensional); this quantity appears in Eq. (17).
2.4.2 2.4.2. Kinetic rate parameters for LDL transport across cell membrane

\(\bar A\): Rate of LDL binding to free receptors, which influences \(\bar N_0\), \(\bar N\), \(\bar M\), \(\bar L_e\), \(\bar L_b\) in Eqs. (13)–(15) and (17)–(18). We equate this parameter to the rate of LDL binding to LDLR receptors in Harwood and Pellarin (1997) (denoted by k _{1} in their Table 2); and hence \(\bar A = {k_1} = 4 \times {10^4}\) per M per sec. Since molarity, M, is the number of moles per liter, we have \(\bar A = {{4 \times {{10}^4}} \over {0.001{N_A}}} = 6.64 \times {10^{  17}}\) per molecule per mL per sec.

\(\bar b_0\), \(\bar b\): these are the internalization rates of empty, and occupied pits (and appear in all the Eqs. (13)–(19). Basu et al. (1978) took the relative lifetime of LDL bound to empty pits to be 2.27. Hence, using \({{\bar b}_0} = 0.0027\) per second, from k _{3} in Harwood and Pellarin (1997), and since \({{\bar b}_0} = 2.27\bar b\) from Basu et al. (1978), \({{\bar b}_0} = 2.27 \times 0.0027 = 0.0061\) per second.
There is, however, a subtle difference between the definition of k _{3} in Harwood and Pellarin and our parameter, b. Let us write the reaction scheme of Harwood and Pellarin using our notation for free, bound, internalized LDL and pits,and compare it to ours$${L_e} + {R_f}L{R_b}\buildrel {{k_3}} \over \longrightarrow L{R_i}\buildrel {{k_5}} \over \longrightarrow {L_i} + {R_f}$$. Note that Harwood and Pellarin have a bound complex L _{ i } R _{ i }, which we do not have, and we have recycling to an internal store, whereas they have a direct flux of receptors from the internalized complex to free receptors on the cell's surface. In our scheme, the rate of conversion of L _{ b } R _{ b } to L _{ i } is simply b, whereas in Harwood and Pellarin (1997), it requires a combination of k _{3} and k _{5}. The conversion of L _{ b } R _{ b } to R _{ f } uses a similar combination of k _{3} and k _{5}, but in our scheme requires a combination of b and g. Thus, our parameters b and g are not directly equivalent to k _{3} and k _{5}; they are approximations.$${L_e} + {R_f}L{R_b}\buildrel b \over \longrightarrow {L_i} + {R_i}{\rm{ then }}{R_i}\buildrel g \over \longrightarrow {R_f}$$ 
\(\bar k_L\): Rate of LDL delivery, which only appears in Eq. (17) for the extracellular concentration \(\bar L_e\). In order to reproduce the experimental procedures of Brown and Goldstein (1979), this rate is set equal to zero since the bolus of LDL is modeled by prescribing \(\bar L_e\) at t = 0; to approximate the in vivo situation where LDL is transported to hepatocytes over periods of several hours so that we can investigate the impact on the model results of a continuous source of lipoprotein.
Nondimensional parameters for the model of LDL endocytosis (note that all rates are relative to the rate of internalization of occupied pits)
Parameter  Description  Value 

P _{ m }  Number of receptors in a pit  200 
α  Volume ratio of extracellular to cellular media  15,000 
g  Rate of return of internal pits to surface  2.35 
A  Rate of LDL binding  0.17 
b_{0}  Rate of internalization of empty pits  1.32 
k _{ L }k_{ Lu },k _{ Ll }  Rate of delivery of LDL to extracellular medium  0 
K  Determines cholesterol dependence of pit production  2.3 
k _{ r }  Rate of production of new pits  0.235 
f  Fraction of internalized pits which are recycled  0.7 
k _{ id }  Rate of conversion of LDL to cholesterol  0.0435 
r  Cholesterol content per LDL particle  0.0015 
λ  Rate of cholesterol regulation  0.717 
λ′  Rate at which labeled amino acids leave cell  0.326 
ψ  Ratio of pits to LDL particles at t = 0  0.0155 
γ _{ am }  Number of tagged amino acids per LDL  1.4 
ϑ  Initial internal cholesterol level as a fraction of equilibrium  0.7 
δ  Ratio of radiolabeled: unlabeled LDL  1.38 
2.4.3 2.4.3. Kinetic parameters for the cell's internal processes

\(\bar g\): Rate of pit release from the internal store back to the cell surface, and so only influences \(\bar N_0\) and \(\bar R_i\), which are governed by (13) and (16), respectively. We equate this parameter to the rate of receptor recycling which has been estimated in HepG2 cells as 0.01088 per second in Harwood and Pellarin (1997).

\(\bar C_e\): Ideal cholesterol level (see Eq. (20)). Aravindhan et al. (2006) find 56.8 µg of cholesterol per mg of cell protein and taking the molecular weight of cholesterol (MW chol = 386.65, with N _{ A } being Avogadro's number), we obtain \({{\bar C}_e} = {{56.8 \times {{10}^{  6}}} \over {{\rm{MW chol}} \times 0.0033}}{N_A}{\rm{ moles/ml}} = 2.65 \times {10^{19}}\) molecules per ml of cell volume.

\(\bar k_s\): Maximum rate of pit production by the cell, appearing in Eq. (16). This is of the order \(O(\bar b \bar n_0 \bar C_e)\) hence, \({\bar k_s} = 2.17 \times {10^{28}}\) mol^{2} ml^{−2} sec^{−1}. In the absence of an exact value from experiments, we have optimized the fit with experimental results by varying this parameter, in conjunction with K (below) by factors of up to 10.

\(\bar K\): Constant for the receptor production term in Eq. (16), this parameter should be of the order of \(O({{\bar C}_e}) = {10^{19}}\) molecules per ml of cell volume, which we use as the base value in our simulations. For the particular form of cholesterolregulated pit production, we use in this model and for fixed values of \(\bar k_s\), larger values of \(\bar K\)} reduce the rate at which receptor production decreases with cholesterol levels. Hence, smaller values of \(\bar K\) correspond to situations in which the cell increases pit (receptor) levels in response to deficiencies in cholesterol levels. Note that \(\bar K\) is the cholesterol level at which the rate of de novo pit production is half its maximum.

f: Fraction of internalized receptors returned to the cell surface, which influences \(\bar R_i\) through Eq. (16). Following Dunn et al. (1989), we assume f ∈ [0.7, 1].

\({{\bar k}_{id}}\): Rate at which internalized LDL particles are degraded to release cholesterol, and hence this parameter appears in both Eqs. (19) and (20). No data is currently available, but Brown and Goldstein (1979) quote a time of 10 minutes for marked particle ingestion to measurement of related cholesterol concentration. This time includes a number of mechanisms included in our model and we thus estimate 5 minutes to be more realistic. Therefore, we fix \({{\bar k}_{id}}\) per sec.

\(\bar r\): Average number of cholesterol molecules per LDL particle. This only appears in the cholesterol Eq. (20). Following Jackson (2005) and Panovska et al. (2006), we take \(\bar r = 3400\).

\({\bar \lambda }\): Rate of removal of cholesterol from free cholesterol pool (or conversely, the fractional rate of synthesis). This parameter also only appears in Eq. (20). We assume initially that \({\bar \lambda }\) is proportional to the rate at which LDL is digested (\({{\bar k}_{id}}\) ); we then vary \({\bar \lambda }\) in our simulations for a best fit to the experimental data.

\({{\bar \gamma }_{am}}\): The number of radiolabeled molecules per LDL particle, This parameter only appears in Eq. (23b) and is taken to be 1.4.

\({\bar \lambda '}\): The rate at which amino acid products from the breakdown of cholesterol leave the cell. We vary \({\bar \lambda '}\) in our simulations for a best fit to the experimental data. It influences both amino acid concentrations and and appears in Eqs. (23b)–(24b).
The parameter values listed above are summarized in Table 1.
2.5 2.5. Nondimensionalization of model
3 3. Numerical results
In this section, we present numerical solutions of Eqs. (30)—(41). Since the system of equations cannot be solved analytically, the numerical solver ode45 (in Matlab) is used to study the evolution in time of the model variables. In the first instance, using the parameters in Tables 1 and 2 as a starting point, we focus on Brown and Goldstein's classical experiments (Brown and Goldstein, 1979). In order to reproduce their results (and to calibrate our model), we need to vary some of the system parameters. We use fminsearch in Matlab to minimize the ℝ^{2} difference between experimental and simulation values. In Brown and Goldstein (1979), the dynamics of binding, internalization, and degradation of LDL by LDLR mediated uptake were studied using radio labeled LDL (^{125}ILDL). Fibroblasts were incubated at 4°C with ^{125}ILDL so that binding, but not internalization occurred. This resulted in labeled ^{125}ILDL particles being bound to the cell surface with concentration 46 ng/mg cell protein. The cells were then washed before being incubated in a medium containing 10 µg of unlabeled LDL per ml. The cells were then warmed to 37°C to allow internalization and degradation of LDL to commence. At a number of time points over a 2hour period, the concentrations of surfacebound, internalized, and culture medium ^{125}I were measured. The appearance of label in the medium was due to the degradation of amino acids of apo B100 within lysosomes in the cells (which we assume occurs at the same time that cholesterol esters are hydrolyzed) and their subsequent export across the cell membrane into the culture medium.
3.1 3.1. Evolution of the system
In our simulations (Fig. 3), we assume that the system reaches a steady state within the experimental timeframe of 2 hours. In order to achieve this, we set the fraction of internalized pits recycled to the store to f = 0.7 (with larger values of f our simulations show that the total number of pits N _{0} + N + R _{ i } does not reach a steadystate over the 2hour timecourse of the experiment).
There is a reduction in the total number of receptors (and pits) as cholesterol enters the cell within LDL, due to the imperfect recycling of receptors (pits); this is counterbalanced by the cholesterolregulated production term k _{ s }/(K + C). The parameters k _{ s } and K are chosen so that the total number of receptors (and thus pits) are maximal initially, to reflect the initial conditions of the experiment in which cells are incubated in a lipidpoor medium to upregulate LDL receptor levels. (See Section 4.2 for more details.)
Total extracellular LDL (L _{ e }) remains almost constant showing very little depletion from such a large external source. Labeled bound LDL falls from its initial concentration, at the same rate as it is internalized, ultimately this falls to zero since no more labeled LDL is introduced into the medium. Unlabeled bound LDL initially increases due to a high percentage of pits being recycled back to the surface, this allows further binding of unlabeled LDL until the maximum number of LDL particles per pit is reached and the faster rate of internalization takes over.
3.2 3.2. Sensitivity analysis
Sensitivity analysis examines how, near the chosen set of parameter values, a small variation in each model parameter affects the model outputs. For example, how the labeled internalized LDL concentration (\({{\bar L}_{il}}\)) varies with the rate of conversion of internal LDL to cholesterol (k _{ id }); we denote such a quantity by \(S({{\bar L}_{il}},{{\bar k}_{id}},t)\) since it is a timedependent quantity.
Since sensitivity analysis involves a calculation of how each initial condition and each parameter used in the model influences each model variable at each moment in time, vast amounts of data are generated. Much of this reinforces the fact that many of the processes are not ratelimiting, and so changing one rate or initial condition has little or no effect on the model variable. Here, we summarize those dependencies which sensitivity analysis has shown to be the most significant.
In Brown and Goldstein (1979), three quantities are measured: the concentration of surface bound labeled LDL (\({{\bar L}_{bl}}\)), internalized labeled LDL (\({{\bar L}_{il}}\)) and amino acid degradation products of the radiolabeled apo B100 in the external medium (\({{\bar A}_{ext}}\)).
Table of the measured variables (observed concentrations) and the parameters which they are sensitive to; parameters are listed in order of decreasing sensitivities
Variable  Parameters exerting the dominant influence 

\(\bar L_bl\)  \(\bar b\) 
\(\bar L_il\)  \(\bar b\),\(\bar k_id\) 
\(\bar A_ext\)  \( \bar b{\text{, }}\bar k_{id} {\text{, }}\bar \gamma _{am} ,{\text{ }}\bar \lambda \prime \) 
Hence, we find that the rate of internalization (\(\bar b\)) exclusively determines how the concentration of labeled bound LDL changes with time, but also has a significant in fluence on the other two measured variables, the labeled internalized LDL concentration (\({{\bar L}_{il}}\)) and the extracellular concentration of labeled amino acids (\({{\bar A}_{ext}}\)). The rate at which LDL is degraded (\({{\bar k}_{id}}\)) significantly influences both the concentration of internalized LDL and degradation products in the external medium (\({{\bar L}_{il}}\) and \({{\bar A}_{ext}}\), respectively). Sensitivity analysis of the remaining model variables to changes in the other parameters highlights the next most influential parameters. Of these, the most important is the fraction of pits returned to the cell surface (f), or inversely, the fraction degraded/lost from the cell. For both, the number of pits (internal, empty, and occupied) and the number of LDL particles (bound and internalized) an increase in this fraction causes a twofold increase in their concentration from half an hour onward. Sensitivity analysis aids the process of optimizing parameter values to reproduce experimental data since it highlights those parameters which have the greatest influence on each component of the system.
Table of the key parameters identified by sensitivity analysis which influence variables in the model. Determination of the “most influential” is based on the absolute maximum of S(t) over the 2hour
Variable  Parameters which influence variable 

Pit concentrations, \(\bar N\), \(\bar N_0\), \(\bar R\)  f, \(\bar g\), \(\bar A\), p _{ m }, \(\bar b_0\), \(\bar b\), \(\bar k_s\), \(\bar K\), \(\bar C_e\), (f is the most influential) 
LDL concentrations \(\bar L_b\), \(\bar L_i\)  f, \(\bar g\), \(\bar A\), p _{ m }, \(\bar b_0\), \(\bar b\), \(\bar k_s\), \(\bar K\), \(\bar k_id\), \(\bar CC_e\), (f is the most influential, followed by \(\bar b\) 
Cholesterol \(\bar C\)  \( \bar \lambda , \bar C_e , (\bar C_e being the more influential) \) 
Amino acid concentration, \(\bar A_int + \bar A_ext\)  \( \bar b{\text{, }}\bar k_{id} {\text{, }}\bar \gamma _{am} ,{\text{ }}\bar \lambda \prime \) 
3.3 3.3. Kinetics of convergence to quasisteadystates
Having identified the recycling fidelity parameter f as the dominant parameter in the fitting of experimental data, we now investigate its role in determining the rate at which the system attains its steadystate.
The upper panel of Fig. 5 shows the differences in the concentrations of internalized LDL in the two cases. The low degradation case (f ≈ 1) leads to significantly higher values of L _{ i } than the high degradation case. This suggests that the inflexible case is the more efficient in its uptake of LDL. Cells balance the need for efficient uptake of LDL, the cost of de novo manufacture of receptors, and their ability to respond to an environment of variable LDL concentration. There is a balance to be struck between timescale of equilibration on one hand and on the other, the speed of LDL uptake and cost of receptorsynthesis.
4 4. Steadystate behavior
4.1 4.1. Fixed number of pits (f = 1, k _{ r } = 0)
Here, we show that our model can reproduce the results of Harwood and Pellarin's (1997) model. Harwood and Pellarin quote values for the proportion of receptors which are free, bound, and internalized as a function of the extracellular concentration, assuming the system has reached steadystate. They considered a system in which the total number of pits or receptors was in steadystate, and hence constant. We reproduce this behavior by assuming perfect recycling of pits (f = 1) and neglecting the de novo synthesis of pits (that is, we impose k _{ r } = 0). In Harwood and Pellarin (1997) the internalization of empty pits was neglected and so we fix b _{0} = 0.
4.2 4.2. Variable number of pits (f < 1, k_{s} > 0)
In the above case, the number of receptors was fixed and determined by the initial conditions. Our more general model allows for a variable number of receptors since it includes the loss of receptors due to imperfect recycling of internalized pits (f < 1) and de novo production of pits via the term k _{ r }/(K +C). Since this introduces cholesteroldependence into the system, the resulting steadystate is more complex.
4.3 4.3. Steadystate for continuous delivery (0 < k _{ L } < k _{ Lc })
Including the input parameter k _{ L } > 0 adds to the complexity of the system (46)–(52) since now L _{ e } is unknown and must be determined as part of the solution
4.4 4.4. Pseudosteadystate for continuous delivery (k _{ L } > k _{ Lc })
As noted above, for k _{ L } > k _{ Lc }, there is no physicallyrelevant steadystate solution because the rate of delivery of LDL is too large for the cell to internalize all that arrives at its surface. Hence, there is a state in which the extracellular concentration grows linearly in time (and can become unboundedly large), whilst the cell variables approach a steadystate. We will refer to this situation as a pseudosteadystate.
Figure 7 shows how the behavior of the system changes as k _{ L } ranges from zero to 2.2k _{ Lc } in the case where k _{ Lc } = 1.6 × 10^{−4}. Nondimensional steadystates are plotted against k _{ L }. The proportion of empty pits drops from 0.25 to almost zero as k _{ L } increases from zero to 1/5k _{ Lc }, the steadystate is precisely zero for k _{ L } > k _{ Lc }. As k _{ L } rises from zero to 0.3 × 10^{−4} = 1/5k _{ Lc }, the number of occupied pits rapidly rises from zero to approximately 0.75. If we consider larger values of k _{ L }, from 1/5k _{ Lc } to 2.2k _{ Lc }, there is very little further increase in the number of occupied pits (N). There is no perceptible change in the number of any type of pits in the range 1/5k _{ Lc } < k _{ L } < 2.2k _{ Lc }; and almost no change in the number of internalized pits across the whole range of k _{ L }. Such quantities show no abrupt changes in the behavior of the cell in the cases k _{ L } < k _{ Lc } and k _{ L } > k _{ Lc }, rather, there is an abrupt change in the range 0 < k _{ L } < 1/5k _{ Lc }, then a plateau is reached which shows little variation for k _{ L } > 1/5k _{ Lc }. The cell's internal cholesterol level rises linearly from unity at k _{ L } = 0 to C = 1.003 at k _{ L } = k _{ Lc } and then stays at 1.003 for k _{ L } > k _{ Lc }. The quantities which do show abrupt changes are the pit occupancy, which rises linearly from zero to 100% in the range 0 < k _{ L } < k _{ Lc }, the external LDL concentration L _{ e } and the uptake (the last two of which are plotted in Fig. 7).
We might expect the steady increase in L _{ e } and the resulting decrease to zero of N _{0} to dramatically alter the cell's regulation of pit production. However, since intracellular cholesterol levels do not rise significantly, there is no significant increase or reduction in pit production or rate of internalization of LDL particles. The cell reaches a maximum rate of uptake of LDL and simply keeps operating at that level.
5 5. Discussion
We have proposed a model of LDL particle adhesion to clathrincoated pits on the surface of hepatocytes. Whilst this process has been modeled before, we believe that this is the first model to take account of the structure of receptors being grouped into pits of approximately 200 receptors which are all internalized simultaneously to form a lysosome. Furthermore, our model includes the breakdown of these internalized structures, the recycling of a proportion of the receptors, and the regulation of new receptorproduction by the cell's internal cholesterol concentration.
Following the construction of a detailed microscopic model, we reduce the system of over 200 ordinary differential equations (Sections 2.1–2.2) to just seven (Section 2.3). Parameters are found from the literature (2.4), themodel is nondimensionalized (Section 2.5) and verified against the experimental results of Brown and Goldstein (1979) (Section 3.1) and the model of Harwood and Pellarin (1997) (Section 4.1). Here, we show that the model correctly predicts the proportions of free, bound, and internalized receptors, as a function of external LDL concentration, assuming the system is at steadystate.
As well as presenting our bestfit to the results of Brown and Goldstein (1979) (Fig. 2), Section 3 contains details of the evolution of all the other (nonmeasurable) variables in the system. We also summarize the results of a detailed sensitivity analysis in which every parameter in the model has been varied, and its influence on all variables measured. Those parameters which have the dominant influence have been highlighted and discussed. The sensitivity coefficient of bound labeled LDL (denoted L _{ bl } in this paper) relative to the rate of endocytosis (\(\bar b\)), has been found to be an order of magnitude larger than all other sensitivity coefficients. This is an elegant quantitative illustration of the appropriateness of the Brown and Goldstein experimental setup (Brown and Goldstein, 1979) to the investigation of endocytosis processes, as the rate of disappearance of the LDL label is highly sensitive to the rate of receptor mediated endocytosis (\(\bar b\)) while at the same time robust to variation in all other parameters in the model. We have explored the effect of the recycling fidelity parameter f on the kinetics of the endocytosis process. This system provides a good example of the tradeoff between a robust, responsive system which rapidly equilibrates, but leads to less efficient use of pits at low values of f , and a more efficient process which takes longer to reach steadystate at higher values of f.
Elevated LDLC levels have been shown to play a role in the development and progression of CHD. In particular, individuals with familial hypercholesterolaemia (FH) have been shown to have circulating LDLC concentrations that are much higher than normal levels, which can be caused by defects in the LDLR, the binding protein apo B100, or more recently discovered protein involved in the degradation of the LDLreceptors, PCSK9 (Abifadel et al., 2003). The effects of these varying mechanisms can be explored using our model by varying the parameters associated with binding (\(\bar A\), p _{ m }), or intracellular recycling of receptors and pits (f) (Zhang et al., 2007; Lagace et al., 2006).
In Section 4, we also use the model to speculate on the form of solution in the case where LDL particles are added to the extracellular medium at a constant rate. The system will then approach a steadystate in which LDL particles are constantly being taken into the cell, broken down, most receptors being recycled, and new ones being manufactured, replacing the nonrecycled ones, so that the total number of receptors is dependent on the cell's internal cholesterol level. Such a situation can only be maintained if the rate of delivery of LDL is below a critical threshold value (k _{ L } < k _{ Lc }). For delivery rates above this value, the system approaches a pseudosteady state in which the extracellular LDL concentration grows linearly with time, the number of empty pits decreases to zero, the number of free receptors also approaches zero and the cell is at steadystate of maximum LDL processing capacity. This pseudosteady state is very similar to the exact steadystates for delivery rates below critical value (1/5k _{ Lc } < k _{ L } < k _{ Lc }). The main change in behavior of the cell occurring in the low LDL delivery rates of 0 < k _{ L } < 1/5k _{ Lc }.
Notes
Acknowledgement
We are grateful to Pieter de Groot for making helpful comments on the manuscript, also to Marcus Tindall for many useful discussions. JADW thanks Unilever for hospitality. We acknowledge financial support from the EPSRC for funding an Springboard fellowship for JADW [grant number EP/E032362/1], and for supporting the (2005) Mathematicsin Medicine study group where this problem was first analyzed (Panovska et al., 2006).
Open Access
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