Multi-compartmental modeling of APP processing in the presence or absence of SORLA
Probably more than any other major disease entity, AD is a pathological processes influenced by subtle quantitative changes in protein concentration and activity. Thus, common approaches in experimental AD research, using protein overexpression or gene-inactivation, are inadequate to study the effects of incremental changes in target protein levels on risk of neurodegeneration.
In our previous study[14], we have undertaken the first attempt to approach risk factors in AD through quantitative modeling. To do so, we have simulated the quantitative contribution of SORLA to proteolytic processing of APP, a central pathway in AD. We have chosen SORLA as a target for simulation because it represents one of the major genetic risk factors in AD. More importantly, solid experimental evidence had established the molecular mechanism of SORLA action, acting as an intracellular sorting receptor for APP that prevents proteolytic breakdown of the precursor protein into neurotoxic Aβ peptides. In[14], we have been able mathematically confirm hypotheses, derived from prior experimental work. In particular, we have confirmed the strict linear relationship between SORLA concentrations and efficiency of APP processing, and we have uncovered the ability of SORLA to prevent dimerization of APP, preventing the formation of high-affinity substrates for secretases.
While our initial study has been met with great enthusiasm in the field, it clearly falls short of addressing major aspects of SORLA activity in the cell biology of AD. Thus, for sake of simplification, our earlier study assumed a single-compartment model for simulation of the affects of SORLA levels on APP processing rates. Accordingly, it ignores the fact that APP follows a complex intracellular trafficking pathway whereby this protein moves between the TGN, cell surface, and endosomes where the various interacting proteins reside. In fact, it has the ability to show how SORLA affects APP transport between various cell compartments in neurons that initially sparked interest in this protein.
For the present work, a single-compartment model, describing the influence of SORLA in APP processing[14], was extended into a multi-compartmental model. The extended model addresses the important aspect of the cell biology of SORLA by assuming a three-compartment model that is based on experimental data. The biochemical network illustrating this multi-compartmental model can be found in Figure1. The notation that is used in this network is described in Additional file1: Table S1.
It is likely that there are many other proteins contribute to the processing of APP and the generation of neurotoxic Aβ peptides. However, unlike many proposed AD risk factors, the mechanism of action for SORLA has been established in numerous studies in cell cultures, in animal models, and even in patients providing a solid base for theoretical simulations. In particular, we have specifically addressed the caveat that this model focuses on pathways related to SORLA action, and that further studies will be required to sequentially add more risk factors to this model. Such approaches will require a profound understanding of the function of such risk factors; - an endeavor that clearly exceeds the scope of the present manuscript.
The choice of the compartments considered in Figure1 was based on the different locations where APP was shown to interact with SORLA, with α-, and with β-secretases. The corresponding three compartments are the TGN, the cell surface and the endosomes[9, 15, 20], respectively. Note that the transport of APP among these compartments indirectly interconnects these three compartments to one another. As SORLA affects the initial cleavage of APP by α- and β-secretases[11], the rate limiting steps that determine the extent of amyloidogenic processing, further processing steps involving γ-secretase were not included in this model.
In order to accommodate the monomeric and dimeric forms of APP, each compartment was further divided into two subcompartments (Figure1): a “red” subcompartment for APP monomer processing and a “green” one for APP dimer processing. Notice that the monomeric forms of APP, SORLA, α-secretase, and β-secretase, within the two subcompartments, were annotated differently: APP
G1
, SORLA
G1
, α
1
, and β
1
for monomer processing, and APP
G2
, SORLA
G2
, α
2
, and β
2
for dimer processing. Even so, the components from the two subcompartments were linked to each other via APP
init
, α
init
, and β
init
. Moreover, APP
G2
, α
2
, and β
2
undergo dimerization before the beginning of APP dimer processing. That is, two APP
G2
, α
2
, and β
2
monomers dimerize in order to give their corresponding dimeric forms. Conversely, these dimers can dissociate to generate their respective monomers. Note that subscript ‘1’ was assigned to the reactants and products in monomer processing while subscript ‘2’ for those in dimer processing. In addition, we used subscripts ‘G’, ‘CS’, and ‘E’ for APP in TGN, at the cell surface and in the endosomes, respectively.
Up to this point, we had described the different forms of APP, α-secretase, and β-secretase in the diverse compartments, prior to the beginning of APP processing. Because SORLA interacts with APP in a 1:1 stochiometric complex[9, 16], the model described how SORLA strictly interacts with APP-monomers (but not dimers) to form an APP-SORLA complex. Consequently, this interaction is responsible for the diminished amount of APP-monomers (APP
G1
) and APP-dimers (APP
G2d
) transported from the TGN to the cell surface. This interaction decreases the amount of APP-monomers (APP
CS1
) and APP-dimers (APP
CS2d
) ending up in the endosomes as APP
E1
and APP
E2d
. Moreover, in order to determine whether SORLA will have a similar influence on the monomer and dimer processing, the binding affinity assigned to APP
G1
SORLA
G1
in monomer processing is different to that of APP
G2
SORLA
G2
in dimer processing.
After the interaction of SORLA and APP in the TGN, the remaining APP
G1
and APP
G2d
are transported to the cell surface where APP processing begins within the non-amyloidogenic pathway. Then, a small part of APP
CS1
and APP
CS2d
, which is not cleaved by α-secretase, are further transported from the cell surface to the endosomes, where the amyloidogenic pathway takes over. Notably, the interaction of APP and α-secretase at the cell surface leads to the formation of non-amyloidogenic products like sAPPα and C83; whereas the interaction of APP and β-secretase in the endosomes yields to the amyloidogenic products such as sAPPβ and C99. Our model was established in such a way that the dimeric form of secretases act only on the dimeric form of APP and the monomeric form of secretases act only on the monomeric form of APP.
The biochemical network (Figure1) that we established, was translated into a system of ordinary differential equations (ODEs), describing temporal changes of molecular numbers for the network components as a function of interaction and cleavage processes. The model equations, their reduction, and a series of steps involved in model simulations are presented in the Materials and Methods section.
Decrease in total amounts of sAPP products is mainly due to the influence of SORLA in dimer processing
With the multi-compartmental model, we showed in Figure2 the corresponding model simulations for various APP products, namely, the products produced in monomer, in dimer, and in both processing pathways. The simulations of the parameterized mathematical model are in good agreement with recent experimental data by Schmidt and colleagues[14] (Figure2A-D).
In the absence of SORLA, the sigmoidal curve that is characteristic for products produced in dimer processing (green lines in Figure2A and B) has a strong impact on the sum of the products produced in monomer and in dimer processing pathways (black lines in Figure2A and B). As such, it very well describes the experimental data sets for sAPPα and sAPPβ (black dots in Figure2A and B, respectively).
Surprisingly, in the presence of SORLA, one observes from the simulations a significant decrease in the products produced in dimer processing (green lines in Figure2C and D) as compared to those in monomer processing (red lines in Figure2C and D). In particular, the analysis showed that at a high level of SORLA activity (i.e. 100% of SORLATot where SORLATot equals 2.43 x 105 fmol), there is obviously more APP bound to SORLA in dimer processing (Figure3B) than in monomer processing (Figure3A).
Taken together, our simulations shown in Figure2 and Figure3, strongly supported the hypothesis whereby SORLA prevents oligomerization of APP, thereby having a bigger impact on the products produced in dimer processing than in monomer processing.
Intermediate levels of SORLA
Up to this point, we only showed simulations of our model in the two most extreme scenarios: with no (Figure2A and B) or high levels of SORLA activity (Figure2C and D). However, subtle alterations of SORLA concentration are likely to be more relevant for the determination of its influence in APP processing pathways. Accordingly, we adapted our multi-compartment model to intermediate concentrations of SORLA. As shown in Figures4,5,6 and7 the simulations are all in dependence of three intermediate SORLA expression levels, namely, 3%, 12%, and 30% of SORLATot.
Remarkably, we observed in Figure4 that the simulations in dependence of the three intermediate SORLA expression levels are either “spread” (as in Figure4A and Figure4D) or “clustered” (as in Figure4B and Figure4C) into the two most extreme scenarios of SORLA concentration. This came as a surprised because the dose–response kinetics of total sAPPα production in dependence of the intermediate SORLA expression levels (Figure4A) is expected to be “clustered” like that of sAPPβ (Figure4B). Likewise in the case of the amount of APP bound to SORLA in monomer (Figure4C) and in dimer processing (Figure4D). We say that the simulations are “clustered” when
where Y = {3%, 12%, 30%}, and X denotes the amount of concentration at a given percentage value of SORLATot that is specified by its subscript. Otherwise, we say that the simulations are “spread”.
Next, we investigated what leads to the observation made in Figure4, in dependence of the intermediate SORLA expression levels.
SORLA indirectly affects the dynamical behavior of the β-secretase but not that of α-secretase
First, we analyzed the simulations of the influence of intermediate levels of SORLA on APP processing on the amount of α-secretase (Figure5A-F) and β-secretase (Figure5) concentration. In Figure5, the term “used” refers to the complex formation of the secretases and APP, while the term “free” refers to the secretases that are not bound in a complex.
The total amount of α-secretase and the total amount of β-secretase were assumed to be constant (depicted by the black lines in Figure5E-F and Figure5K-L, respectively). Due to the conservation law assumption, the total amount of each secretase in each subcompartment is conserved (i.e. αmonomer and βmonomer depicted by red lines in Figure5F and Figure5L; αdimer and βdimer depicted by green lines in Figure5F and Figure5L). Consequently, the total amount of each secretase in the whole system was thus also conserved (αTot and βTot shown by the black lines in Figure5E-F and Figure5K-L, respectively).
The simulations of the influence of intermediate levels of SORLA on APP processing on the amount of α-secretase (Figure5A-F) concentration showed that (i) there are more α-secretases that were used (Figure5C) than left free (Figure5A) in monomer processing, (ii) there are more α-secretases that are left free (Figure5B) than used (Figure5D) in the dimer processing, (iii) the total amount of α-secretase that is free and used (blue and orange lines in Figure5E, respectively) is dominated by the corresponding amount of α-secretase concentration in dimer (Figure5B) and in monomer processing (Figure5C), (iv) SORLA influences the amount of α-secretase concentration in dimer processing (Figure5B and5D), but not those in monomer processing (Figure5A and5C), and (v) its simulations in dependence of the three intermediate SORLA expression levels (Figure5D) is consistent to that of dose–response kinetics of total sAPPα production (Figure4A).
The significant difference in the free (Figure5B) and used (Figure5D) amounts of α-secretase in dimer processing is a consequence of the large amount of α-secretase used in monomer processing (shown in Figure5C). As the total amount of the APP concentration increases (from 0 nM to 400 nM), the amount of α-secretase, free in dimer processing, (Figure5B) decreases, while the amount of α-secretase used in monomer processing (Figure5C) increases. As the amount of SORLA concentration increases, the curves representing the secretases move from solid to dashed lines. SORLA does affect α-secretase in dimer processing (Figure5B and5D): those used in dimer processing decreases (Figure5D), while those that are free in dimer processing increases (Figure5B). In the later figure, the increase is not obvious because the amount of change is so small as compared to the concentration values of α-secretase.
As for the influence of intermediate levels of SORLA on APP processing on the amount of β-secretase (Figure5G-L) concentration, the simulations showed that (i) there are more β-secretases that are left free (Figure5G) than used (Figure5I) in monomer processing, (ii) SORLA has no influence on β-secretase in monomer processing (Figure5G and Figure5I), (iii) SORLA alters the dynamical behaviors of β-secretase in dimer processing (Figure5H and Figure5J), (iv) the total amount of β-secretase that is free and used (blue and orange lines in Figure5K, respectively) is dominated by the amount of β-secretase concentration in dimer processing (Figure5H and Figure5J, correspondingly), and (v) its simulations in dependence of the three intermediate SORLA expression levels (Figure5H and Figure5J) is consistent to that of dose–response kinetics of total sAPPβ production (Figure4B). The curves for beta-secretase with SORLA (dashed lines in Figure5H) are greater in values as compared to those without SORLA (solid line in Figure5H), as a consequence of SORLA’s influence on beta-secretase that is used in dimer processing (Figure5J).
When a comparison is made between the total amount of α- and β-secretase concentration that is free (blue lines in Figure5E and Figure5K) and used (orange lines in Figure5E and Figure5K) in dependence of the three intermediate SORLA expression levels, we observed that the total amount of β-secretase concentration for both free and used deviated (Figure5K), which was not the case for α-secretase (Figure5E). This observation suggested that SORLA is indirectly affecting the dynamics of β-secretase but not that of α-secretase. This result supports the hypothesis presented by Schmidt el al.[14]: “the global–local estimation of the parameter values in the model suggested a yet unidentified biological process whereby SORLA might indirectly affect the β-secretase, but not with the α-secretase”. The present result therefore clarifies what was unidentified in our previous study[14].
With SORLA concentration greater than the estimated total amount of SORLA concentration (i.e. SORLATot = 2.43 x 105 fmol), we arrived at Figure S1 shown in Additional file1: Figures S1D and S1J show that for a very large amount of SORLATot (greater than 1 x SORLATot for α-secretase and greater than 10 x SORLATot for β-secretase), the amount of α- and β-secretase are barely “used”. Consequently, the amount of α- (Figure B) and β-secretase (Figure H) are all “free” in dimer processing, and there will be no sAPP products produced in dimer processing.
SORLA is more influential in dimer processing than in monomer processing
We also investigated the amount of APP concentrations that is either free or used, in monomer or in dimer processing, and which is in the TGN, at the cell surface or in the endosomes (Figure6). The term “used” refers to the complex formation of (i) APP and SORLA in the TGN, (ii) APP and α-secretase at the cell surface, and (iii) APP and β-secretase in the endosomes. Wherein, the term “free” refers to the APP that is not bound in the respective compartments.
First, we showed the simulations of the amount of APP concentrations that is free or used in monomer and in dimer processing. The simulations under dimer processing showed that the amount of APP concentrations that is free or used in each compartment were significantly affected by the presence of SORLA (last two columns of Figure6: Figure6D-E, Figure6H-I, Figure6L-M, and Figure6P-Q), as compared to those under monomer processing (first three columns of Figure6: Figure6A-C, Figure6F-G, Figure6J-K, and Figure6N-O). In particular, one observes from the simulations that the amount of APP concentrations that is used to bind with SORLA in dimer processing of the TGN tremendously increases from 0 M to at most 300 nM (Figure6E), wherein those in monomer processing are so small that they can be neglected (Figure6C). Consequently, SORLA decreases the amount of APP concentrations that is free or used at the cell surface and in the endosomes (Figure6H-I and Figure6L-M, respectively). Also, the total amount of APP concentrations in dimer processing is dominated by the total amount of free APP in the absence of SORLA and by the total amount of used APP in the presence of SORLA (depicted by the two outermost lines in Figure6P and Figure6Q).
Next, in each compartment, the simulations for the total amount of APP concentrations that is free, used, in monomer processing, or in dimer processing, are shown in Figure7. Consistent to our previous observation (Figure6), the simulations for total amount of APP concentrations in monomer processing for the three different compartments (3rd column of Figure7) were not influenced by SORLA, while those in dimer processing were affected by the presence of SORLA (4th column of Figure7). Moreover, the simulations, in the first two columns of Figure7, also showed that the presence of SORLA in the TGN decreases the total amount of free APP (Figure7P), and increases the total amount of used APP (Figure7Q). In particular to the total amount of used APP under the influence of SORLA, it is (i) enormously increased in the TGN (Figure7B), (ii) not affected at the cell surface (Figure7G), and (iii) reduced by at most half in the endosomes (Figure7L). Taken together, the presence of SORLA increases the total amount of APP concentrations in the TGN (Figure7E), and subsequently decreases the total amount of APP concentrations at the cell surface (Figure7J) and in the endosomes (Figure7O).
The simulations for the total amount of APP concentrations in monomer processing (Figure7R), in dimer processing (Figure7S), and in both monomer and dimer processing (Figure7T) show that a conservation law was assumed for APP in monomer and in dimer processing. Above all, one observes that there are more APP concentrations in dimer processing (Figure7S) than in monomer processing (Figure7R).
The spread and clustering of SORLA expression levels
As noted in the subsection, Intermediate levels of SORLA, the simulations show that SORLA expression levels are either “spread” (Figure4A) or “clustered” (Figure4B). This is most likely due to the effect of SORLA on the processing of APP dimer. With respect to the total amount of APP, the amount of APP concentrations (Figure6I) and α-secretase concentrations (Figure5D) that are “used” at the cell surface in dimer processing “spread”. Considering the relevance of APP and α-secretase at the cell surface to the production of sAPPα, the observations thus suggest the “spread” observed in Figure4A for sAPPα. Similarly for the “clustering” observed in Figure4B for sAPPβ, it is a consequence of the “clustering” that is observed on APP (Figure6M) and β-secretase (Figure5J) that are “used” in the endosome in dimer processing, which are relevant in producing sAPPβ. Moreover, the change from “spread” at the cell surface (Figure4A) to “clustered” in the endosome (Figure4B) is probably due to the indirect influence of SORLA on the dynamical behavior of β-secretase that is observed in Figure5.
Effects of different SORLA concentrations in switching sAPPα and sAPPβ from preferred dimer-to-monomer processing
Lastly, in Figure8, are given simulations of the influence of SORLA on APP processing into sAPPα (Figure8A and Figure8C) and sAPPβ (Figure8B and Figure8D). The simulations show that the switch from preferred dimer-to-monomer processing is observed at 25% of SORLATot for α-secretase (Figure8A) and at 3% of SORLATot for β-secretase (Figure8B), where SORLATot equals 2.43 x 105 fmol. In agreement with the study performed by Schmidt and colleagues[14] previously, we therefore find that the switch from cooperative (dimer) to less efficient non-cooperative (monomer) processing occurs at small amount of SORLA concentration. Moreover, the end product obtained from monomer processing dominates the total amount of end product at 145% of SORLATot for α-secretase (Figure8C) and at 150% of SORLATot for β-secretase (Figure8D). In connection to what we observed in Figure4 for the simulations of the influence of intermediate levels of SORLA on APP processing into sAPPα (Figure4A) and sAPPβ (Figure4B), these two sets of results (Figure4 and Figure8) suggest that SORLA reduces the products produced in non-amyloidogenic and amyloidogenic pathways of APP processing at different rate.