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Introduction

Lipids are almost insoluble in aqueous media such as blood plasma and thus transported among the various tissues by water-soluble complexes called lipoproteins (LP). Elucidating the kinetic mechanisms involved in the formation, degradation and mutual interconversion of plasma lipoproteins is of high medical relevance as long-term perturbations of the lipoprotein distribution are considered the primary risk factor for atherosclerosis and cardiovascular diseases—the main cause of death in the western states [1]. Each lipoprotein complex contains a discrete number of apolipoproteins (e.g. apolipoprotein B-100) and lipid molecules (e.g. triglycerides, cholesterol), the number of which may vary between one and several hundreds or thousands, respectively. Basically, an enormous lipoprotein heterogeneity results from all possible stoichiometric combinations of lipid and apolipoprotein molecules. Despite this fact, for almost half a century, lipoproteins have usually been grouped into density classes named chylomicrons, VLDL, IDL, LDL, and HDL (very low-, intermediate-, low-, and high-density lipoproteins, respectively), separated by, for example, ultracentrifugation from blood plasma [2]. Accordingly, mathematical models of the systemic lipoprotein metabolism hitherto have considered lipoprotein density classes (compartments) to be dynamic variables of the system. The phenomenological transition rates between these compartments are usually determined by radioactive or stable isotope tracer experiments (for reviews see [3]–[5]). Except for the comprehensive compartment model proposed by Knoblauch et al. [6], compartment models have focused on specific parts of the lipoprotein metabolism based on kinetic measurements with, for example, labeled apoA-I [7]–[10], apoA-II [11], or apoB-100 [12]–[14]. Compartment models may provide a useful phenomenological description of the lipoprotein dynamics; however, they have some serious limitations. First, they neglect the possible heterogeneity of lipoproteins. Actually, a single density class comprises a huge number of lipoprotein complexes differing in their amount of lipids and proteins—an important fact that could be of relevance for the medical interpretation of lipoprotein density profiles. Second, the transition of a lipoprotein from one density class into another is not a single process but is accomplished in a series of successive elementary reactions in which, for example, triglycerides are removed, cholesterols are taken up from tissues, and apolipoproteins are exchanged. Therefore, phenomenological inter-compartment transition rates can hardly be related to the rate of the underlying molecular processes. To overcome these limitations of compartment models, we propose here a novel approach. It consists of the establishment of kinetic equations governing the temporal changes of individual lipoprotein complexes. Hence, in our modeling approach, the number of dynamic variables is in principle given by the number of different lipoprotein complexes that can be formed from a given number of apolipoproteins and lipids. From the calculated set of individual lipoprotein complexes in the system we can compute the distribution of lipoproteins over an arbitrary number of predefined density classes (the lipoprotein profile). In particular, we can compute such profiles over commonly defined density classes and compare them with experimental profiles obtained from normolipidemic subjects. Choosing a larger set of narrow density classes and subdividing the calculated lipoprotein distribution in, as we call it, high-resolution density sub-fractions (hrDS), we observe a remarkable heterogeneity of the lipoprotein distribution within commonly defined density classes. Finally, we calculate lipoprotein profiles associated with a disorder in one of the underlying molecular processes of the lipoprotein metabolism. In these simulations of pathological situations we altered the rate constants of the LDL receptor-mediated lipoprotein uptake, lipoprotein lipase, and ABCA1-mediated cholesterol transport.

Results

The Model

We avoid usage of predefined density classes but characterize lipoproteins regarding their protein and lipid composition. As described below, the model takes into account essential lipoprotein constituents and processes involved in the lipoprotein metabolism in human blood plasma. However, for the sake of numerical tractability we reduced the number of lipoprotein components to a manageable set and simplified the kinetic processes.

Lipoprotein components.

The lipoprotein complexes considered in the model are composed of three different types of apolipoproteins and lipids abbreviated as A, B, F, and C, T, P, respectively. The protein components A and B are thought to represent apoA-I (P02647, ENSG00000215756) as the primary protein constituent of HDL and apoB-100 (P04114, ENSG00000084674) as the characteristic apolipoprotein of VLDL, IDL, and LDL, respectively. We ignore apoB-48 and the related lipoprotein complexes (chylomicrons), which are rapidly formed and degraded within several hours after food intake. Each lipoprotein is either equipped with component A or component B. Thus, we will use the terms A-particles and B-particles in the following, respectively. All other apolipoproteins are lumped together into the protein component F. The lipid components C, T, and P represent total cholesterol (free cholesterol and cholesteryl esters), triglycerides, and phospholipids, respectively. The dynamics of phospholipids (P) is not explicitly modeled. Instead, the number of phospholipid molecules in an individual lipoprotein complex is calculated such that, together with the apolipoproteins, full occupancy of the lipoprotein surface is achieved (see Dataset S1 for details of calculation).

The component's densities vary between 1.35 and 0.886 g/ml for apolipoproteins and triglycerides, respectively. The possible number of lipid molecules may go up to several thousands. This results in a considerably diversity of lipoprotein complexes in the system. With the maximal number of molecules for each component given in Table 1, we would get 8.0×10⁸ lipoprotein complexes in total. Thus, to keep the model tractable we refrained from considering the actual number of molecules for total cholesterol (C) and triglycerides (T). Instead, their content was quantified in terms of lipid packages. The package size has to be chosen carefully to avoid sparsely occupied density ranges. In the calculations below, lipid packages of C and T comprise 2 molecules in A-particles and 20 molecules in B-particles, respectively.

Table 1. Composition properties of lipoprotein complexes.

doi:10.1371/journal.pcbi.1000079.t001

Kinetic processes.

From the reactions reported to affect the lipoprotein metabolism in human blood plasma we selected 20 elementary processes. As we lump together free cholesterol and cholesteryl ester into one component (total cholesterol), esterification of free cholesterol by the lecithin-cholesterol acyltransferase (LCAT, EC 2.3.1.43, P04180, ENSG00000213398) is not considered. A summary of the kinetic processes included in the model and their physiological meaning is given in Dataset S2. For a schematic representation see Figure 1. We grouped the processes into six categories: (1) Birth and death: the total amount of lipoprotein complexes is the result of de novo synthesis by the liver and the receptor-mediated uptake of whole particles from the blood by tissue cells (Figure 1A). Separate kinetic parameters are used for the generation and elimination of A- and B-particles. The initial composition of newly synthesized particles was set to fixed values given in Table 1. (2) Lipoprotein-tissue exchange: besides the synthesis and uptake of whole lipoprotein complexes (see Category 1), individual components are selectively altered by exchange processes with various tissue cells. The uptake of peripheral cholesterol by A-particles and the release of cholesteryl esters from both particle species are taken into account (Figure 1B). (3) Inter-lipoprotein exchange of neutral lipids among lipoproteins is mediated by the cholesteryl ester transfer protein (CETP; P11597, ENSG00000087237), which transfers preferentially cholesteryl esters from A- to B-particles and triglycerides and vice versa. To model this transfer, we introduce a non-lipid bound form of this carrier protein (called CETP(0) in the model) that can be loaded either with C (called CETP(C)) or T (called CETP(T)) which shuttles between A- and B-particles (Figure 1C). The transfer of triglycerides between B-particles is included as well and defined as a separate process. (4) Exchange of apolipoprotein A. The transfer of apolipoproteins that can be exchanged among lipoprotein complexes is modeled by decomposing it into (i) a release step from a lipoprotein complex into a common plasma pool of free apolipoprotein and (ii) an uptake process from this pool into a lipoprotein complex. The transfer process for the protein component A is restricted to A-particles and is thought to describe the re-modeling of apoA-containing HDL (Figure 1D). (5) Exchange of apolipoproteins F. The transfer of those apolipoproteins, mostly apoE (P02649, ENSG00000130203) and apoC, lumped together into the component F may take place between arbitrary lipoprotein complexes (Figure 1E). (6) Enzymatic conversion. One central enzymatic process effecting the re-modeling of lipoproteins is the hydrolysis of lipoproteins' triglycerides and phospholipids. This process is catalyzed by the lipoprotein lipase (LPL; EC 3.1.1.34, P06858, ENSG00000175445) or the hepatic lipase (HL; EC 3.1.1.3, P11150, ENSG00000166035), which are treated in the model as two separate processes (Figure 1F).

Figure 1. Schematic representation of the kinetic processes modeled.

(A) Synthesis of A- and B-particles and degradation via HDL and LDL receptors, respectively. (B) Influx of peripheral cholesterol (“C”) into A-particles via the ATP-binding cassette A1 (ABCA1) receptor and selective efflux of cholesteryl ester (“C”) by the scavenger receptor B1 (SR-B1). (C) Elementary processes of the cholesteryl ester transfer protein (CETP) mediating the exchange of triglycerides (“T”) and cholesteryl ester (“C”) between lipoprotein components. CETP(0), CETP(T), and CETP(C) represent non-lipid–bound, T- and C-loaded forms of CETP, respectively. (D, E) Exchange of apolipoproteins (“A,” “F”) among lipoprotein complexes via plasma pools (PoolA, PoolF). (F) Hydrolysis of triglycerides (“T”) from A- and B-particles by hepatic and/or lipoprotein lipase (HL, LPL).

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Stochastic Versus Deterministic Model Simulations

We modeled and simulated the system of lipoproteins by two different approaches, a stochastic and a deterministic one. For a detailed description the reader is referred to the Materials and Methods section. In brief, in the stochastic simulation the calculation of stationary lipoprotein distributions was performed by simulating the Master equation by means of the Gillespie algorithm until a stationary state was reached. The deterministic model (Equation 5) governing the lipoprotein concentrations comprises as many kinetic equations as there are different lipoprotein complexes. However, the concentrations of most complexes are practically zero because the elementary processes are highly specific. Moreover, the numerical solution of very large systems of equations poses serious numerical problems.

For the stochastic simulation of the Master equation this enormous complexity is not a problem as the stochastic trajectories generated by the Gillespie algorithm are located in a very small region within the space of all possible lipoprotein complexes. Hence, Gillespie's direct method does not suffer from the type of combinatorial explosion as the deterministic approach. Furthermore, using the stochastic simulation algorithm instead of the system of differential equations permits to deal with small package sizes for the lipid components, i.e. the number of lipid molecules per package. Small package sizes are needed to achieve a sufficiently high coverage of physiologically relevant density intervals containing a number of different lipoprotein complexes.

One problem, however, with the Gillespie algorithm is to conclude from the stochastic trajectories at which time point of the simulation the true stationary regime has been reached and a representative sampling of the state space has been accomplished. To test whether the criteria used to assess stationarity work well we have compared the concentrations calculated for one and the same set of kinetic parameters with both simulation variants, the Gillespie algorithm and the deterministic equation system (Figure 2).

Figure 2. Stochastic versus deterministic simulation.

Density distributions of the concentration of the sum of lipoprotein components (mg/dl) obtained by using the Gillespie algorithm with different numbers of simulation steps (events) and by the iterative solution of the deterministic equation system. The density space (0.93–1.35 g/ml) was subdivided into 30 equally sized intervals.

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To keep the deterministic model numerically tractable we fixed the package sizes for cholesterol (C) and triglyceride (T) to two molecules (A-particles) and 100 and 250 molecules (B-particles), respectively. We also restricted the maximal number of molecules of the components A, B, F, C and T to (4, 0, 5, 100, 40) in an A-particle and to (0, 1, 5, 5000, 10000) in a B-particle. With the package sizes given above, the possible C and T content of an A-particle complex decomposes into 50 and 20 packages; for B-particles they are 50 and 40 packages, respectively.

As the number of all components, with exception of A and B, can become zero, the total number of different lipoprotein complexes in this example is given by 4·(5+1)·(50+1)·(40+1) for A-particles plus 4·(5+1)·(50+1)·(40+1) for B-particles = 38,250, spanning a density range between 0.92 and 1.40 g/ml. Arbitrary values of the kinetic parameters (not shown) were chosen and the stationary concentration of lipoprotein complexes was computed using either Gillespie's algorithm or iterating the fix-point equation (Equation 6).

In order to compare the two solutions we subdivided the total density range covered by the 38,250 lipoprotein complexes into 30 intervals and calculated the occupancy of these density intervals by cumulating the calculated concentrations of the corresponding lipoprotein complexes. As shown in Figure 2, with increasing number of time steps used in the Gillespie simulation the stochastic solution of the Master equation converges toward the numerical solution of the deterministic model.

The striking advantage of performing stochastic simulations of the Master equation by means of Gillespie's algorithm is that increasing the number of lipoprotein components (e.g. by including cholesterol ester) or using smaller package sizes results only in a moderate increase of computing times because this algorithm per se only deals with such lipoprotein complexes that occur with significant concentrations. In contrast, the deterministic model has to deal with all possible complexes despite the fact that most of them never reach discernible concentrations. The following results were obtained with the stochastic simulation algorithm using the much smaller package size of 20 molecules for C and T in B-particles.

Calculation of Lipoprotein Density Profiles in Healthy Subjects

In the experiment, main density classes of lipoproteins (VLDL, IDL, LDL, HDL) and sub-fractions of LDL and HDL were isolated from blood plasma. The LDL class was separated into six density sub-fractions, which are grouped into the commonly named large buoyant (lb; LDL1/2), intermediate dense (id; LDL3/4), and small-dense LDL (sd; LDL5/6). The HDL class was subdivided into sub-fractions of HDL2b, HDL2a, and HDL3. In the simulation, stationary lipoprotein distributions were computed by using Gillespie's stochastic simulation algorithm as outlined in the Materials and Methods section. Simulation starts in a system state that is lipoprotein-free. Executing approximately five million reactions (one per time step), a steady state was reached, i.e. the average number of lipoproteins and their composition in the systems remained constant. An additional 10 million executions sampled the stationary distribution of individual lipoprotein complexes. We then calculated lipoprotein density profiles as they are typically determined in clinical investigations by assigning each of the lipoprotein complex according to its concentration to one of the experimentally defined density classes.

The set of model parameters that entail best agreement between the computed lipoprotein profile and the experimental data (see Table 2) was determined as follows. To keep the number of lipoprotein complexes in the simulation tractable, we scaled the system with an appropriate volume factor yielding a reaction volume of one tenth femto-liter. Parameter values for the synthesis of A- and B-particles were taken from [9] and [15], respectively, and fixed during parameter optimization. Numerical values of all other model parameters were obtained by minimizing the distance between simulated and clinically measured lipoprotein profiles (see Materials and Methods). They are listed in Table 3.

Table 2. Experimental lipoprotein composition data.

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Table 3. Model parameter values.

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The estimated parameter values are in most cases in a reasonable agreement with experimentally determined values taking into account the difficulties to extract rate constants of elementary processes from kinetic measurements settled on compartment analysis. The underlying reaction mechanism can either be monomolecular or bimolecular which is important to know while comparing the stochastic rate constants with rate constants obtained from tracer kinetic studies. A detailed comparison of calculated and measured rate constants is given in Dataset S3.

As shown in Figure 3, the calculated lipoprotein density profiles for each of the lipoprotein components by using the parameter values given in Table 3 are, to a large part, in a remarkable agreement with the clinical data. However, with respect to the distribution of apolipoprotein B (Figure 3B) and of triglycerides (Figure 3E) some discrepancies remain.

Figure 3. Simulated versus clinically measured distribution of lipoprotein components over main density classes including sub-fractions of LDL1-6, HDL2b, HDL2a and HDL3.

x-axis: number of lipoprotein density fraction; y-axis: simulated (circles) versus clinically measured (rectangles) concentration values in mg/dl of Apolipoprotein A–I (A), Apolipoprotein B-100 (B), Sum of further apolipoproteins (C), Total cholesterol (D), Triglycerides (E, logarithm) and Phospholipids (F). The error bars show the standard deviation of the experimental values.

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The total amount of 41.8±0.45 mg/dl of component B predicted by the model is lower than the mean value of 56.6 mg/dl determined experimentally for apoB-100, but within the expected interval (±21.4 mg/dl). Compared with experimental values the calculated concentration of apoB-100 is higher in the VLDL sub-fraction but lower in IDL and all LDL sub-fractions. This might be accounted by the simplifications made in our model for the kinetics of triglyceride removal from B-particles because regulatory influences of apolipoproteins C and E are ignored. Likewise, the simplified kinetics of triglyceride removal from B-particles might also explain the too low triglyceride content predicted for the IDL sub-fraction since the high rate of triglyceride hydrolysis obtained by the parameter optimization procedure yields a rapid delipidation of newly synthesized B-particles. The simplification to assume a definite initial composition of newly synthesized lipoproteins in the model might be another reason for the remaining discrepancy in the distribution of apoB-100.

The calculated distribution of model component F was compared with the clinically measured concentration values of the C apolipoproteins I–III and apoE. However, experimental data for component F are questionable, as only about one-half of the total plasma concentration of apoC and apoE (20.87±3.8 mg/dl) is associated with lipoprotein complexes. The other half reflects a free apolipoprotein pool in plasma, whose value is about 10-fold higher than 1.2 mg/dl reported by [16]. This might result from experimental difficulties, because it is well documented that apoE may dissociate from the surface of apoB-containing particles during prolonged ultracentrifugation [17],[18]. This may account for the fact that our model predicts higher levels of component F in almost all lipoprotein density classes as compared to the available experimental data (Figure 3C). In fact, the calculated distribution of F agrees much better with the experimental total plasma concentration (19.0 mg/dl) as well as with the concentration observed for the free plasma pool (1.2 mg/dl) by Batal et al.

Initial composition analysis.

The initial composition of A- and B-particles was kept constant during simulation (Table 1). However, depending on a variety of factors, including nutrition or cellular and regulatory processes, the liver generates a certain lipoprotein spectrum of different compositions. To analyze how the lipoprotein distribution in the blood plasma changes in response to different initial compositions, the molecule numbers of the lipoprotein components F, C, and T of B-particles were randomized from a normal distribution by taking the reference composition as the mean value. A total of 100 different initial compositions were analyzed in independent simulation runs. The compositions obtained by randomization provide ranges for F, C, and T between 0–18, 760–2,934, and 6,654–14,784 molecules, respectively. The variation of each of the random compositions relative to the reference composition was quantified using the euclidean distance.

Figure 4 illustrates the overall tendency, the more different the composition from the reference value (increasing euclidean distance) the larger the distance between the calculated and experimental lipoprotein distributions. However, a number of compositions considerably deviating from the default one fit comparably well or even improve the agreement with the experimental data (see LP composition #81 and #63, respectively). In our calculations, this predominantly pertains to compositions that are mainly increased in the amount of component F and C. The results suggest that certain variability in the initial composition can be partially compensated by the kinetic processes in the LP metabolism. Or, the other way around, specific inter-individual variations in the liver status are not necessarily reflected in altered distributions of the main lipoprotein classes in the blood.

Figure 4. Variation in the initial composition of B-particles.

The initial composition of B-particles, i.e. the molecule numbers of component F, C and T, was randomized. The filled black bars (x = 0) mark the default B-particle initial composition (initF = 10, initC = 2000, initT = 10000). A total of 100 different compositions (brown bars) were analyzed by independent simulation runs. The graphs are sorted by the euclidean distance (topmost sub-graph, black dashed line). The change in the error measure (distance between calculated and experimental lipoprotein distributions) relative to the value obtained for the default composition is shown in the topmost sub-graph (black continous line).

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hrDS

Characterizing the distribution of lipoproteins in the blood plasma by quantifying their abundance in a limited number of main density classes such as chylomicrons, VLDL, IDL, LDL, and HDL (the classical density profile) appears feasible as long as the distribution of lipoprotein components within these classes is sufficiently smooth. That means, any alteration in the kinetic properties of the underlying elementary processes ultimately gives rise to changes in the relative occupation of these density classes. On the other hand, alterations in the kinetic processes may not necessarily lead to visible changes in the average value of a density class, while the concentration and composition of individual lipoproteins within the class may vary significantly.

To reveal the heterogeneity of the lipoprotein distribution within the main density classes experimentally used, the width of each was decomposed into five equally sized sub-intervals. Subsequently, we quantified the calculated amount of lipoprotein components in these narrow density classes, for which we introduce the name hrDS. Since the main density classes exhibit differing density interval sizes, we normalized the distribution within each density class to its interval size.

As an example, Figure 5 shows the distribution of cholesterol across the hrDS, which allows to quantify the contribution of the hrDS to the average concentration of the main classes. Most significant intra-class variation appear in the ascending (LDL1, LDL2 and HDL2b) and descending (LDL5, LDL6 and HDL3) part of the overall distribution. For example, in LDL6 (density d = 1.044–1.063 g/ml), the calculated mean concentration of cholesterol amounts to 15.1±0.1 mg/dl (normalized value of 0.795 with LDL6 interval size of 19). The five hrDS named LDL6(I), (II), (III), (IV), and (V) relatively contribute with 53.1%, 25.7%, 11.7%, 5.5%, and 4.0% to the average cholesterol concentrations, respectively. According to this finding, more than one-half of the cholesterol content in LDL6 is contributed by lipoprotein complexes with densities in the narrow range of 1.044–1.0478 g/ml. Within the other classes the hrDS are nearly equally distributed.

Figure 5. High-resolution distribution of total cholesterol within main density classes including sub-fractions of LDL1-6, HDL2b, HDL2a, and HDL3.

Each density class was further decomposed into 5 equally sized sub-fractions, called hrDS (grey bars). x-axis: density in g/ml; y-axis: concentration of total cholesterol in mg/dl normalized to the density interval size.

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The specific intra-class variations of the main density classes show similar patterns for all the other lipoprotein components (Figure 6). However, some intra-class distributions vary for different components. For example, within VLDL triglycerides and phospholipids show monotonically decreasing instead of a nearly equal distribution (Figure 6E, F).

Figure 6. High-resolution distribution of lipoprotein components within main density classes including sub-fractions of LDL1-6, HDL2b, HDL2a, and HDL3.

Each density class was further decomposed into five equally sized sub-fractions, called hrDS (grey bars). x-axis: number of lipoprotein density subfraction; y-axis: concentration of apolipoprotein A–I (A), apolipoprotein B-100 (B), sum of further apolipoproteins (C), total cholesterol (D), triglycerides (E) and phospholipids (F) in mg/dl normalized to the density interval size.

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One may hypothesize that the amount and distribution of lipoprotein sub-populations differ between individual healthy subjects or even in pathological conditions due to inter-individual variations. To check whether the knowledge of the intra-class distribution provides additional and valuable information, we moderately varied each of the kinetic parameters by ±10% of the reference value.

The results indicate that, for example, marginal alterations in the delipidation process of B-particles (HydrolyzeB) shifts the high resolution distribution within a major density class either to lower or higher densities (Figure 7), while the concentration value of the major classes (e.g., LDL) remains nearly unchanged. Similar results were obtained for the selective cholesteryl ester uptake from B-particles (EffluxB) and the amount of CETP available (data not shown).

Figure 7. Variation in the distribution of hrDS cholesterol at moderately altered parameter values.

Alteration of the normal hrDS cholesterol distribution (grey bars) by (A) increasing and (B) decreasing the parameter value of the HydrolyzeB process by 10% of its normal value (black bars). x-axis: metrical density intervals in g/ml; y-axis: concentration of total cholesterol in mg/dl normalized to the density interval size.

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Simulated Pathological States

To check the predictive capacity of our model we simulated the impact of disorders in the kinetic properties of the LDL-receptor (LDLR; P01130, ENSG00000130164), the lipoprotein lipase (LPL) and ATP-binding cassette A1 (ABCA1; O95477, ENSG00000165029) on the stationary density distribution of lipoproteins (Figures 8–10).

Figure 8. Calculated pathological distribution compared with calculated normal data for LDL receptor deficiency.

Distributions of the lipoprotein components in the main density classes including sub-fractions of LDL1-6, HDL2b, HDL2a and HDL3. x-axis: number of lipoprotein density fractions; y-axis: calculated pathological (squares) and calculated normal (circles) concentration values of apolipoprotein A–I (A), apolipoprotein B-100 (B), sum of further apolipoproteins (C), total cholesterol (D), triglycerides ([E], logarithm) and phospholipids (F) in mg/dl.

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Figure 9. Calculated pathological distribution compared with calculated normal data for LPL deficiency.

Distributions of the lipoprotein components in the main density classes including sub-fractions of LDL1-6, HDL2b, HDL2a, and HDL3. x-axis: number of lipoprotein density fractions; y-axis: calculated pathological (squares) and calculated normal (circles) concentration values of apolipoprotein A-I (A), apolipoprotein B-100 (B), sum of further apolipoproteins (C), total cholesterol (D), triglycerides ([E], logarithm), and phospholipids (F, logarithm) in mg/dl.

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Figure 10. Calculated pathological distribution compared with calculated normal data for ABCA1 deficiency.

Distributions of the lipoprotein components in the main density classes, including sub-fractions of LDL1-6, HDL2b, HDL2a and HDL3. x-axis: number of lipoprotein density fractions; y-axis: calculated pathological (squares) and calculated normal (circles) concentration values of of apolipoprotein A–I (A), apolipoprotein B-100 (B), sum of further apolipoproteins (C), total cholesterol (D), triglycerides ([E], logarithm), and phospholipids (F) in mg/dl.

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Familial Hypercholesterolemia (FH) is an autosomal hereditary disease caused by a dysfunction of the LDLR. We simulated a reduced LDLR activity by decreasing the parameter for the process DestroyB to 75% of its normal value. The calculated lipoprotein distribution exhibits an increased concentration of LDL cholesterol at nearly unchanged cholesterol levels of VLDL and IDL (Figure 8D). It is suggesting that this arises from lowering the receptor-mediated uptake while maintaining a sufficient apoB-synthesis leading likewise to elevated LDL-apoB levels (Figure 8B).

Within LDL, the sub-fractions LDL1-6 behave differently. As compared to normolipidemic cholesterol values, we observe moderately increased levels of idLDL-C (25.2 vs. 31.8 mg/dl, +26%) and to a higher extent of sdLDL-C (28.7 vs. 49.2 mg/dl, +71%) whereas lbLDL-C (26.8 vs. 24.4 mg/dl) remains nearly unchanged. These results coincides with findings of low LDL-cholesterol to LDL-apoB ratios in carriers of the FH-Keuruu mutation (Asp235→Glu) suggesting that LDL particles are small and dense [19]. The LDL subfraction pattern is also similar to findings by März et al. in a patient with FDB (Familial defective apolipoproteinB-100) who has a mutation in codon 3500 of the apoB gene substituting glutamine for arginine [20].

Regarding the distribution of A-particles (equivalent to HDL), the model predicts a moderate decrease in HDL cholesterol as compared to the normolipidemic profile (Figure 8D). This is in good agreement with the reduced overall HDL cholesterol level observed in heterozygous FH patients [7].

However, an impaired interaction between B-particles, such as LDL, and the receptor may be due to several reasons. Either there is indeed a defect in the receptor itself (FH) due to mutations that cause a reduced expression or binding activity, or the ligand (potentially apolipoprotein B-100) carries a mutation (FDB). Since the uptake process is determined by solely one parameter in the present model, we cannot yet discriminate between different molecular determinants.

Hypertriglyceridemia is characterized by elevated levels of plasma triglycerides caused, among others, by deficiencies of the lipoprotein lipase (LPL), the key enzyme in the catabolism of triglyceride-rich lipoproteins by removing (hydrolyzing) triglycerides. We simulated the consequences of an impaired LPL activity by lowering the parameter value for the process HydrolyzeB to one-half of the original value.

The calculated lipoprotein distributions display markedly elevated levels of triglycerides as well as cholesterol, predominantly of VLDL, IDL and early LDL (Figure 9D and 9E). The total plasma concentration of triglycerides (159.5 vs. 95.2 mg/dl) is about 67.4% increased as compared to the simulated normolipidemic profile. According to our calculations, the total concentration of LDL cholesterol is only marginally affected (152 vs. 145 mg/dl), whereas substantial alterations in the cholesterol level of individual LDL subfractions are predicted. lbLDL (LDL1+2) is considerably increased. As compared to normolipidemic LDL cholesterol values, we obtain a strong reduction in idLDL (LDL3+4, 25.2 mg/dl vs. 1.8 mg/dl, −92%) and to a lower degree of sdLDL (LDL5+6, 28.7 mg/dl vs. 5.1 mg/dl, −82%). Elevated levels of sdLDL implicated with mild to moderate hypertriglyceridemia cannot be found in the model simulations [21]. Likewise, reduced HDL cholesterol are not observed as reported by Babirak et al. for the phenotype of heterozygous LPL deficiency [22]. In contrast, the calculated distribution shows increased HDL cholesterol levels of HDL2b while in HDL2a and HDL3 no discernible changes occur.

The calculated distributions might be mechanistically explained as follows. Due to the decreased hydrolysis parameter, during simulation, the HydrolyzeB process is executed about 1.2-fold less leading to the accumulation of B-particles enriched in triglycerides. Elevated triglyceride levels, in turn, promote the transfer of triglycerides to A-particles mediated by the CETP (1.4-fold higher frequency of the processes ExchangeT_B and ExchangeT_B). Likewise, the more CETP is loaded with triglycerides the less lipid-unloaded CETP is capable to transport cholesterol back from A-particles to B-particles. Subsequently, lipoprotein complexes in the density range of HDL become enriched in cholesterol and triglycerides. A 1.4-fold higher frequency is also observed for the HydrolyzeA process implicating a higher HL activity, the enzyme that catalyzes the hydrolysis of triglycerides from small B-particles and A-particles. That may explain the low triglyceride content in all LDL sub-fractions.

Hypoalphalipoproteinemia is a rare human metabolic disorder characterized by a severe decrease in HDL cholesterol and apoA-I levels. For example, Tangier Disease (TD) is assumed to be caused by defects in both alleles of the ABCA1 (ATP-binding cassette A1) transporter gene [23], the key mediator of the reverse cholesterol transport by transferring cholesterol and phospholipids from peripheral cells to acceptor lipoproteins in the plasma. A heterozygous ABCA1 defect correspond to the disorder known as familial hypoalphalipoproteinemia (FHA).

We simulated an impaired activity of ABCA1 by reducing the rate constant for the cholesterol uptake into A-particles (process Influx) to 50% of its normal value. Compared with simulated normolipidemic values, the model predicts low plasma cholesterol concentrations (144.5 mg/dl vs. 95.47 mg/dl, −34%). As reported for FHA, a remarkable reduction of cholesterol levels appears in all HDL fractions (Figure 10D). HDL cholesterol accounts for only ~10 mg/dl at nearly normal total plasma triglyceride levels (95.2 vs. 96.8 mg/dl). As a consequence, considerable levels of apoA-I occur in the density range d>1.21 g/ml which argues for the accumulation of pre-β–migrating lipoproteins (Figure 10A). Our model simulation predicts further a marginal reduction in apoA-I within the HDL3 fraction, while in HDL2b and HDL2a apoA-I is selectively depleted. As comparable results from clinical studies, a predominance of HDL particles being poor in cholesterol but enriched in apoA-I in patients with heterozygous TD [24], [25].

The distributions of apoB, apoF, cholesterol and triglycerides of B-particles display a shift to lipoproteins in the density range of LDL6 (Figure 10B–10E). Analyzing the stochastic trajectories of our model simulations shows that the Influx process occurred approximately 2-fold less frequent. The reduced uptake of cholesterol from the periphery to A-particles leads to a diminished cholesterol transfer to B-particles and to less CETP molecules loaded with cholesterol. This forces the rate for the back transfer of triglycerides from triglyceride-rich B-particles to A-particles or cholesterol-rich B-particles to increase. Accordingly, the model calculations show reduced concentrations of VLDL triglycerides and increased triglyceride levels in, e.g. HDL3 and LDL6, respectively.