A mathematical model to estimate cholesterylester transfer protein (CETP) triglycerides flux in human plasma

Subjects

Measurement imprecision

The data used for model validation are based on small changes due to the 1 h storage at 37 °C in lipid mass of lipoprotein fractions (Fig. 1). Therefore, measurement imprecision was considered. It is less than 4% for all lipoprotein fractions studied. In most cases the expected changes after one hour at 37 °C are relatively small compared to baseline. Comparing the relative mean changes in TG mass to the corresponding coefficients of variation, the change in VLDL is relatively low, while it is high in LDL and HDL.

We used one hour as incubation duration, as longer durations may lead to non-linear behavior of lipoprotein associated reactions happening in vitro and hence to data, which is less comparable to the in vivo situation.

Changes in CE and TG between plasma stored at 4 °C and 37 °C for one hour, [μmol/L]. Median (1st, 3rd quartile) of differences between ‘37 °C value’-‘4 °C value’. Experiment 1: n = 91 samples of various metabolic states. Experiment 2: n = 11 plasma samples with no inhibition (normal), with inhibition of LCAT (-LCAT) and with inhibition of CETP (-CETP). Wilcoxon signed rank test * < 0.05, ** < 0.001.

Considering measurement imprecision, it is obvious that the data of TG-redistribution from a single plasma sample at 37 °C may be too noisy to yield acceptable inferences of real TG fluxes mediated by CETP. However, sufficiently large collectives (n ≥ 10) may enable us to test our model.

CETP action

Figure 1 summarises changes in TG and CE concentrations in VLDL, IDL, LDL and HDL in experiments 1 and 2. The loss of TG in VLDL, as well as the increase in TG in LDL and HDL and the increase in CE in VLDL are significant in experiment 1. If CETP is inhibited (experiment 2), these changes are much smaller or even zero and lose their significance. If LCAT is inhibited the changes in TG in VLDL, LDL and HDL remain significant.

In experiment 1, there is a small but significant loss of 1% in VLDL ApoB (Wilcoxon: p = 0.001), but no significant corresponding changes in IDL or LDL. We assume that this ApoB is recovered in IDL or LDL but due to measurement imprecision we cannot statistically capture this fact.

Correlations

Figure 2 illustrates our model’s capability to estimate TG net flux via CETP in LDL, HDL and VLDL, as well as CE net flux in VLDL (assuming an equimolar exchange of TG and CE): Especially for HDL and LDL, the estimation correlates quite well to the measured changes. For VLDL, ΔCE correlates better than the corresponding ΔTG.

N = 91, subdivided into 4 metabolic states: n = 43 ‘normal’ (plus), n = 8 ‘high LDL’ (red triangle), n = 18 ‘high TG’ (blue diamond) and n = 22 ‘low HDL’ (green cross). Observed change: in TG (or CE) [μmol/L/h] in plasma stored at 37 °C. A: TG change HDL, B: TG change LDL, C: TG change VLDL, D: CE change VLDL (assuming equimolar exchange). Pearson’s correlation coefficient R².

Note that in experiment 1 considering ΔTG, there is no significant correlation between HDL and LDL (R = 0.174, p = 0.098), but rather between VLDL and LDL (R = − 0.248, p = 0.018) as well as between VLDL and HDL (R = − 0.478, p < 0.0001), respectively.

In the VLDL fraction, there is no significant difference between LCAT- and non-LCAT-inhibited plasma in ΔCE. This suggests that LCAT activity in VLDL in experiment 2 is negligible.

Assuming an equimolar exchange between TG and CE in VLDL and no significant LCAT action on VLDL, the correlation between ΔTG and ΔCE in VLDL appears to be relatively weak (R = − 0.160, p = 0.130).

In contrast to the three LDL subfractions, the TG flux in the three measured HDL subfractions resulted in a much poorer fit compared to total HDL. This is not surprising, as the observed data are corrupted by the significant redistribution of ApoA1 mass among those fractions during the 1 h storage at 37 °C [11].

CETP_TG and r_cetp

Not surprisingly, correlations between the total calculated surface (as described in the method section) and corresponding PL concentration in HDL, IDL, LDL and VLDL are strong (R = 0.937, 0.994, 0.989 and 0.996, respectively). However, using calculated surfaces leads to slightly better results than using PL mass. Distribution of surfaces of lipoprotein (sub-)fractions is displayed in Fig. 3.

Calculated surfaces median (1st, 3rd quartile) of lipoprotein fractions in the different subgroups (‘normal’: white; ‘high LDL’: light grey; ‘high TG’: dark grey; ‘low HDL’: black).

Figure 4 illustrates quartiles of the predicted and observed TG net fluxes for VLDL, IDL, LDL and HDL. Note that there is not only good correlation between the model and the observed data in each lipoprotein fraction, but the model is also capable of predicting the proportion of TG mass distributed among the lipoprotein fractions by CETP. The corresponding broken-down fluxes of TG into- and out of the fractions are displayed in Fig. 5.

Predicted TG net flux (black) and observed flux (white) of TG; Median (1st, 3rd quartile), 4 groups, n = 43/8/18/22, respectively.

Predicted median (1st, 3rd quartile) flux of TG entering via heteroexchange (white), leaving via heteroexchange (grey) and the resulting net flux (black) in ‘normal’ (n = 43).

Figure 6 displays the v/v TG/(TG + CE) threshold at which –according to the model- there is no CETP mediated TG-net flux, a ratio that is here defined as “isotransfer point” (ITP). The here called ITP equals the parameter CETP_TG. Considering all four metabolic situations investigated, the ITPs are always located between the corresponding means of the IDL and LDL fraction.

In experiment 1 the ITP shifts among the four subgroups: ‘high LDL’ 0.17 (0.16, 0.19), ‘normal’ 0.18 (0.15, 0.21), ‘low HDL’ 0.26 (0.21, 0.29), and ‘high TG’ 0.30 (0.26, 0.34). The correlation between CETP_TG and the ratio TG/(TG + CE) in plasma appears to be remarkably strong (R = 0.968, p = 3.7*E^− 55).

The ITP equals a functional characteristic of lipoproteins in a defined metabolic state and shifts from LDL in normolipidemia to IDL in hyperlipidemia.

As described in the model the rate for CETP interacting with a lipoprotein in plasma per hour per lipoprotein surface in\( \raisebox{1ex}{${m}^2$}\!\left/ \!\raisebox{-1ex}{$l$}\right. \) is estimated as r_cetp = 10^− 5*4.116.

The model’s structure

Potter’s model describes an equimolar shuttle exchange of TG and CE by CETP between lipoproteins on the molecular level. Hence, on a formal basis, CETP is discriminated into a free and a lipoprotein-bound fraction. Correspondingly, lipoproteins are discriminated into free and CETP-bound. The model presented here simplifies and generalises this model. Our model is independent of the exact biochemical mode of CETP action (shuttle or ternary complex) - an obvious advantage, since the mode of action has still not been elucidated. Our generalised model is compatible with both the shuttle and the ternary complex models. Although differing on the molecular level, both approaches result in the same dynamics on the macroscopic level (by the law of large numbers).

As we point out later, our data do not support equimolar change in all fractions, thus we model the TG change only. However, at least in VLDL, our model seems to predict the CE flux to a strong degree. Furthermore, we are not modelling free and bound lipoproteins and CETP-molecules explicitly. The ‘CETP binding to lipoprotein’-reaction is disregarded, as the dynamics of those reactions are only relevant on a mechanistic level.

Note that there are several mathematical model approaches in the field of human lipid metabolism dealing with for example topics like oxidized lipoproteins [14], LDL endocytosis [15] or the cholesterol biosynthesis pathway [16].

Comparison to other models

There are other models besides Potter’s describing in vivo CETP dynamics. Figure 7 summarises two thereof. The model presented by Lu et al. [17] describes the flux of CE from HDL to VLDL as well as CE’s flux from LDL to HDL and vice-versa. In Lu’s model the fluxes are simple first-grade reactions\( {R}_i:C{E}_{donor}\overset{k_i}{\to }C{E}_{acceptor} \) (i = 1, 2, 3) with three different corresponding reaction rates. Applying our own data to both models, the observed ΔCE correlates much weaker to Lu’s predicted ΔCE in VLDL than to our predicted ΔCE: R = 0.315, p = 0.002 vs. R = 0.705, p = 6.4E^− 15. As we are not assuming equimolar exchange and just model TG, we cannot compare Lu’s model directly to ours. However, considering the LCAT-inhibited data from experiment 2, we detected no correlation between the CE enrichment in HDL and CE mass in LDL (or VLDL).

Hübner’s model [18] considers TG and cholesterol (CH), which corresponds to FC + CE. A shuttle transport is modeled in which CETP is either free or TG- or CH-loaded. Let us replace CH by CE to better compare their model to ours. Here, five rates are used for five reactions (Fig. 7). Their model neglects CETP-mediated transport of TG out of HDL, and CE out of ApoB-containing particles. However, their model somehow makes use of the ratio of TG to (CE + TG) associated to CETP. Comparing the strength of correlation between Hübner’s model and our model, ours appears to be superior with regard to HDL-ΔTG, LDL-ΔTG, and slightly better for VLDL-ΔTG (R = 0.515, R = 0.563, R = 0.544 vs our model’s: R = 0.683, R = 0.753, R = 0.562, respectively).

Comparison between the CETP model of Lu et al. [17], Hübner et al. [18] and ours presented. Lu’s model uses first-order reactions; Hübner’s model uses 2 second-order and 3 first-order reactions. The model presented here assumes a steady state and uses one reaction rate for CETP. Due to CETP’s diffusion-like property, the amount of TG entering and leaving a lipoprotein depends on the surface and the TG and CE load of the lipoprotein and all other lipoproteins.

Comparing the Hübner and Lu models to ours presented here, note that ours is less complex, as it only requires one reaction rate instead of 5 or 3, respectively. Hence, unlike their models, which describe each flux from one fraction to another with their own reaction, our newly derived model represents a rather holistic model.

Concept of isotransfer point (ITP)

Figure 6a displays the medians of the ITP in the ‘normal’ and the ‘high TG’ group. Intuitively speaking, the point is lower in the ‘normal’-group. Based on our model, the ITP ‘CETP_TG’ is the mean of all fraction’s TG/(TG + CE) ratios weighted by the corresponding surface. As clarified in Fig. 6 the greater the distance between the TG/(TG + CE) ratio of a lipoprotein to the ITP, the higher is the net flux of TG into/out of this fraction (if the ratio is smaller/greater than the ITP). Thus we expect LDL and HDL to have a higher TG turnover in the ‘high TG’ situation.

The ITP may be changed due to lipid-lowering medication or during the postprandial state. This change may also exert a significant influence on HDL metabolism. The TG/(CE + TG) ratio in IDL in the ‘high TG’ group is surprisingly low. This might be caused by a prolonged retention time in plasma, which may be characteristic in the hypertriglyceridemic state. The clinical significance of the shift of the ITP from LDL to IDL in hyperlipidemia remains unclear. However, the atherogenity of certain lipoproteins (e.g. IDL) may be altered if its metabolic function (CETP-mediated TG transfer) changes.

Data

Our in vitro data mirror the corresponding in vivo redistribution of lipids to a fairly good degree, as data from Liu et al. [19] as well as our group’s experiments have revealed evidence that the net change in TG in lipoprotein fractions by CETP action during plasma storage at 37 °C is linear for at least four hours.

Further data from experiment 2 suggest that CETP is the only factor responsible for TG redistribution among lipoprotein subfractions. CETP inhibition suggests that the esterification rate on non-HDL is very low or even non-existent. However, the data may be too noisy to make precise claims.

In contrast to other methods measuring CETP-activity, the data we used to estimate the CETP-mediated TG-change enables the estimation of TG in all fractions, namely VLDL, IDL, LDL and HDL, simultaneously. The data cover a broad physiological range of TG changes mediated by CETP, as individuals with healthy and various pathologic lipid profiles have been considered.

Very probably CETP activity differs among individuals. However, as the first step, our model estimates the CETP-mediated relative redistribution of TG among the lipoproteins, which is independent of the CETP reaction rate, given only the TG, CE and ApoB (or ApoA1) concentration of VLDL, IDL, LDL and HDL. Based on this estimation, absolute fluxes may be inferred assuming a fixed reaction rate in a second step, if no additional information is given.

Model fit

Despite the noise of TG change data in lipoprotein fractions, the predicted TG fluxes correlate well with the observed corresponding changes (Fig. 2). While HDL and LDL fit well, VLDL and especially IDL appear more problematic. This could be mainly caused by relatively greater measurement imprecision, but also by a shift of ApoB from IDL to VLDL or vice-versa during the one hour storage at 37 °C – as our group already observed [11]. Considering the R² values between modelled and measured TG changes in VLDL, LDL and HDL (Fig. 2), mind that on the one hand our rather imprecise method of measurement may strongly lower the ‘true’ R² value and on the other hand that our prediction is superior compared to the other models mentioned above, as well as to possible linear correlations mentioned in the results part (Table 2).

Equimolarity

Assuming an equimolar lipid-exchange it is expected that the amount of TG entering/leaving a lipoprotein fraction equals the amount of CE leaving/entering it if LCAT inhibition is given. Considering ΔCE and ΔTG in experiment 2 (Fig. 1) we noted a significant increase in the TG-median of 11.3 μmol/L and 19.2 μmol/L in LDL and HDL under LCAT inhibition. Surprisingly no corresponding loss of CE (0 and − 5.2 μmol/L, respectively) was observed. However, due to the small sample size and measurement imprecision, we cannot make a solid statement on this point.

Taken together, we identified some indicators suggesting that the CETP exchange may not necessarily be equimolar. However, due to experiment 2’s low case numbers, our data do not suffice to clarify this issue.

Lipid transfer inhibitor protein (LTIP)

There is strong evidence of a protein inhibiting CETP, namely lipid transfer inhibitor protein (LTIP) or Apolipoprotein F, which mainly interacts with LDL particles [20]. We investigated whether the resulting lower availability of LDL would have a positive effect on our model’s fit by reducing s(P_LDL), the surface of LDL. Potter uses a factor of 0.73 for LDL inhibition by LTIP. Such a reduction in our case leads in fact to a better correlation between predicted and observed changes in LDL and especially HDL: If s(P_LDL)*0.4 is used instead of s(P_LDL), the correlation is optimised. If s(P_LDL) is reduced, the correlation coefficients R in HDL and LDL increase slightly, while the corresponding coefficient in VLDL decreases simultaneously. Further the estimated and observed proportion of TG enrichment between LDL and HDL tends to impair as s(P_LDL) is reduced. A possible inhibitory effect of ApoF is thus not implemented in our model.

CETP mass

Estimating TG fluxes in experiment 1, we assumed all r_cetp to be equal. This of course did not hold true. The parameter r_cetp rather depends on CETP’s mass concentration and the ability of the particular CETP-phenotype to mediate lipid exchange.

In the literature, CETP’s plasma concentration does not differ that much depending on gender or pathologies [8, 21–25]. We are thus assuming that the mass is equal in experiment 1’s different subgroups. However, our model may not account for changes in r_cetp due to genetic variants of CETP.

Properties of CETP exchange

Considering our model and applying it to several lipid-profiles, two issues deserve attention: First, the driving force for CETP exchange is the mass and composition of triglyceride-rich lipoproteins such as VLDL and chylomicrons, as they contain the most TG in plasma. Figure 4 illustrates this fact, as in the ‘high TG’ case, LDL and HDL both revealed the highest increase in TG mass. This issue might be an important factor, when assessing TG influence on lipid metabolism (for example the postprandial state).

Second, TG from VLDL is distributed to LDL and HDL to a similar degree in all four metabolic states investigated (Fig. 4). However, depending on their surface and lipid composition, the proportion of TG entering HDL compared to TG entering LDL may differ strongly, as both lipoprotein-species compete against each other (consider for example the ‘high LDL’ case in Fig. 4).

Let us assume that TG’s high flux to LDL or HDL is atherogenic, as (in combination with hepatic lipase) it can lead to small, dense LDL and small HDL. Following the model of Lu et al. [17], lipolysis by hepatic lipase is the main pathway causing HDL particles to shrink. Small HDL are more likely to exit the plasma. Hence, via CETP hypertriglyceridemic plasma may lead to an atherogenic lipoprotein phenotype characterized by low HDL particle concentrations and small HDL and LDL particles. Our model might thus help to quantify the causal relationship between hypertriglyceridemia and those atherogenic characteristics. Furthermore, following our model’s dynamics, a high HDL particle concentration may prevent small dense LDL particles from forming by reducing the net flux of TG to LDL in such a hypertriglyceridemic state. However, the focus of this paper making is not claims about HDL functionality by relying on its lipid-composition.

Note that only 43 normolipidemic persons out of experiment 1 were used for model calibration. Hence, the results for ‘high TG’, high LDL’ and ‘low HDL’ presented in Fig. 4 are extrapolated and demonstrate our model’s ability to predict CETP redistribution of TG based on lipid and apolipoprotein concentrations only.

When studying lipoprotein metabolism, the concepts considered are usually delipidation of ApoB containing lipoproteins and the reversed cholesterol transport mediated by HDL. However, both metabolic pathways are intimately connected by the CETP-mediated flux of TG and CE. This paper is intended to give a new perspective on lipoprotein metabolism concerning CETP-mediated fluxes of TG and CE.

Subjects & experimental procedure

For experiment 2 we used plasma samples of n = 11 volunteers (n = 9 normal, n = 2 non-normal). The limits of subgroup ‘normal’ are based on guidelines from the German medical association ‘Deutsche Gesellschaft zur Bekämpfung von Fettstoffwechselstörungen und ihren Folgeerkrankungen DGFF (Lipid-Liga) e. V.’ (https://www.lipid-liga.de/fuer-aertze/empfehlungen). We followed a method described by Jansen and colleagues [11]. In short, EDTA-plasma was separated into two samples for both experiments. While the first was stored at 4 °C immediately, the second was exposed to 37 °C for one hour (water bath) and then stored at 4 °C as well. Subsequently, lipoprotein fractions were isolated via ultracentrifugation as described before [26]. Lipoproteins were separated into VLDL, IDL, LDL, HDL and lipid deficient serum (LDS) and further into three LDL subfractions LDL-I, LDL-II and LDL-III and three HDL subfractions HDL2b, HDL2a and HDL3. TG, CE, free cholesterol (FC), PL as well as ApoB and Apolipoprotein A1 (ApoA1) were measured in plasma and each separated lipoprotein fraction. Further the ratios ‘lipid to ApoB’ and ‘lipid to ApoA1’ were calculated to estimate the lipoprotein composition. These values were measured with storage at 37 °C or 4 °C, the corresponding differences in lipid and apolipoprotein mass and lipoprotein composition were calculated as the differences of baseline (4 °C) minus the 1 h 37 °C value and denoted with Δ.In experiment 2, each plasma sample was additionally divided into 4 groups in which only CETP, only LCAT, or none of both enzymes were inhibited. CETP was inhibited via Torcetrapib (1 μM, resolved in dimethyl sulfoxide), LCAT was inhibited by sodium iodacetate (5 mM resolved in Tris buffer [27]).

The coefficient of variation (CV: SD/mean*100) was determined by taking 20 repeated measurements of lipoprotein fractions.

The model

This work is based on the model by Potter et al. [1], which was modified by our group recently [11]. Briefly, metabolism of FC, CE and TG in LDL was modelled on the molecular level. In contrast to the recently published model, which assumed a one-to-one exchange of TG and CE, this restriction for equimolarity is now relaxed, as only TG are considered for flux calculations. If CETP mediates an exchange between two lipoproteins (ternary model) or between a CETP loaded with CE or TG (shuttle model), neither CE nor TG is preferred. Further, in line with Morton [12, 13] we assume that the probability of the association of CETP to a lipoprotein is proportional to its surface. To simplify our model, the lipoprotein core consists of only TG and CE.

Considering event 1, the probability that a TG molecule leaves the particle equals the particle’s \( \frac{TG}{CE+ TG} \) v/v ratio and the complementary probability (no TG molecule leaves the particle) equals the particle’s \( \frac{CE}{CE+ TG} \) v/v ratio.

Considering event 2, the probability that a TG molecule enters the particle equals a parameter called CETP_TG while (1- CETP_TG) is the complementary probability (no TG molecule enters the particle).

The methods applied for calculating a particle’s surface and the concentration of particles in the HDL subfractions are described previously [11].

Taken together, given an interaction of CETP with a lipoprotein of fraction P_i, the lipoprotein acquires a TG molecule in total, if no TG molecule leaves the particle and one TG molecule enters the particle. The corresponding joint probability is q^′(P_i) ∙ CETP_TG. The probability of a TG molecule leaving the lipoprotein in total can be derived analogously as q(P_i) ∙ (1 − CETP_TG). These exchanges are so-called heteroexchanges, as a TG molecule is replaced by a non-defined number (due to the assumption of non-equimolary exchange) of CE molecules. The corresponding probability of a TG-homoexchange (a TG leaves and a TG enters the lipoprotein) is q(P_i) ∙ CETP_TG.

With k, s and q calculated by measured data, only the reaction rate r_cetp needs to be inferred to calibrate our model. The proportion of TG mass entering and leaving the single fractions is already determined by the model’s structure (it depends on TG, CE and particle concentrations in the single fractions). We used the subgroup ‘normal’ (n = 43) to infer r_cetp by minimising the difference between measured and predicted TG net flux in VLDL, IDL, LDL and HDL (weighted by the corresponding surface). All other non-‘normal’ cases were used as a validation group.

The Wilcoxon signed rank test was performed to test for differences between the change in CE and TG due to plasma storage at 37 °C for one hour. The Pearson’s correlation coefficient R was used to describe relationships between variables.

Statistics were analysed with IBM SPSS version 21.0 (IBM SPSS Statistics, IBM Corporation, Chicago, IL), all other computations were done using Scilab 5.4.0 (Scilab enterprises, Le Chesnay). Graphs were depicted by GraphPad Prism Software V7.00.