Aggregation, impaired degradation and immunization targeting of amyloid-beta dimers in Alzheimer’s disease: a stochastic modelling approach
© Proctor et al.; licensee BioMed Central Ltd. 2012
Received: 2 March 2012
Accepted: 11 June 2012
Published: 2 July 2012
Alzheimer’s disease (AD) is the most frequently diagnosed neurodegenerative disorder affecting humans, with advanced age being the most prominent risk factor for developing AD. Despite intense research efforts aimed at elucidating the precise molecular underpinnings of AD, a definitive answer is still lacking. In recent years, consensus has grown that dimerisation of the polypeptide amyloid-beta (Aß), particularly Aß42, plays a crucial role in the neuropathology that characterise AD-affected post-mortem brains, including the large-scale accumulation of fibrils, also referred to as senile plaques. This has led to the realistic hope that targeting Aß42 immunotherapeutically could drastically reduce plaque burden in the ageing brain, thus delaying AD onset or symptom progression. Stochastic modelling is a useful tool for increasing understanding of the processes underlying complex systems-affecting disorders such as AD, providing a rapid and inexpensive strategy for testing putative new therapies. In light of the tool’s utility, we developed computer simulation models to examine Aß42 turnover and its aggregation in detail and to test the effect of immunization against Aß dimers.
Our model demonstrates for the first time that even a slight decrease in the clearance rate of Aß42 monomers is sufficient to increase the chance of dimers forming, which could act as instigators of protofibril and fibril formation, resulting in increased plaque levels. As the process is slow and levels of Aβ are normally low, stochastic effects are important. Our model predicts that reducing the rate of dimerisation leads to a significant reduction in plaque levels and delays onset of plaque formation. The model was used to test the effect of an antibody mediated immunological response. Our results showed that plaque levels were reduced compared to conditions where antibodies are not present.
Our model supports the current thinking that levels of dimers are important in initiating the aggregation process. Although substantial knowledge exists regarding the process, no therapeutic intervention is on offer that reliably decreases disease burden in AD patients. Computer modelling could serve as one of a number of tools to examine both the validity of reliable biomarkers and aid the discovery of successful intervention strategies.
KeywordsAlzheimer’s disease Amyloid-beta Dimers Down’s syndrome Intervention Immunotherapy Mathematical model Protein aggregation Stochastic simulation
Alzheimer’s disease (AD) is characterised by cholinergic neuron loss, synaptic dysfunction and the accumulation of protein aggregates in specific regions of the brain . The main brain regions affected are the cortex and hippocampus leading to clinical symptoms including the inability to form new memories and altered personality [2, 3]. Although the underlying molecular causes remain unresolved, two functionally different proteins are involved in the aggregation process: amyloid-beta (Aβ), which forms extracellular plaques, and the microtubule binding protein tau, the main component of neurofibrillary tangles, with the relative contribution of each to the manifestation of disease symptoms remaining a matter of considerable controversy [4–6]. The order of events in the disease process is also unclear. Aβ is produced by the sequential processing of the amyloid precursor protein (APP), by means of a catalytic process involving the β- and γ-secretase enzymes. The Aβ peptide contains 36–43 amino acids with the actual length affecting its properties. For example, the isoform Aβ40 is normally secreted and less cytotoxic than Aβ42, which has a strong tendency to aggregate in the brain and associates with AD . Levels of Aβ40 and Aβ42 also increase in normal ageing  and have been shown to be elevated in the brains of Down’s syndrome patients . An association has been made between the underlying genetic-molecular profile of AD and that of Down’s, with the detection of overexpressed APP, located on chromosome 21 . Henceforth, elevated plasma concentrations of Aβ40 and Aβ42 were regarded as prominent risk factors for the onset of dementia in persons suffering from Down’s syndrome .
The molecular mechanisms involved in AD are therefore complex and as yet we lack clear understanding of which part of the aggregation process would provide the best target for stopping or slowing down AD-related disease processes. In such complex scenarios where it has proved difficult to disentangle the contributing disease processes, mathematical modelling is increasingly being used to complement experimental approaches [29–32]. For instance, modelling can be used to integrate multiple mechanisms, refine complex hypotheses, make testable predictions and explore in silico the possible effectiveness of interventions. We have used dynamic modelling which means that the model can be used to examine mechanisms leading to the disease state. This has an advantage over experimental approaches which are often limited to the end time-point of the disease. Many biological processes are inherently stochastic. For example, protein aggregation involves random molecular interactions to initiate the formation of pathogenic structures . Therefore stochastic computer models can be essential for examining the plausibility of alternative mechanisms. To this end, we built a stochastic model of turnover of Aβ42 (hereafter referred to as Aβ) to investigate the effects of the degradation rate on Aβ levels and specifically to discover whether a slower degradation rate is itself sufficient to explain the increased level of soluble Aβ and senile plaques seen in AD-affected brains, a molecular phenomenon which is strongly believed to underlie disruption of memory function [16, 18, 34]. The potential clinical importance associated with an intervention strategy to lower soluble Aβ levels was demonstrated by Busche and others , who recently provided evidence in support of the concept that soluble Aβ induces neuronal dysfunction. This was experimentally revealed by direct application of soluble Aβ in wild-type mice, who then displayed pathological neuronal hyperactivity within the hippocampus region. This deficit was rescued through acute treatment with a γ-secretase inhibitor, which reduced soluble Aβ levels. Importantly, the authors observed that the selective increase in neuronal hyperactivity preceded the formation of plaques, to suggest that soluble species of Aβ may underlie this functional neuronal impairment. We then extended this model to include details of Aβ aggregation, running simulations over long periods of simulated time (up to 100 years) in order to examine the aggregation process in human ageing. We used the model to compare the plausibility of different candidate mechanisms and, by varying key parameters in the model, to identify which processes are most sensitive to modification and might therefore be best to target in order to reduce the burden of plaques.
Model 1: Aβ turnover
A lower degradation rate of Aβ has only a small effect on soluble levels of the protein but increases the probability that monomers will exceed a critical threshold
Model 2: Aβ turnover and aggregation
Model 1 was extended by adding reactions for the formation of dimers and aggregates, assuming that these processes were reversible. We defined a plaque to be an aggregate of size greater than 30 monomers as this was the size at which according to the model output, an aggregate had a very low probability of disaggregating completely. Full details of our terminology are given in the Methods section.
Decreased clearance of soluble Aβ leads to plaque formation
We assumed that the rate of Aβ clearance declined with age in AD due to the decline in neprilysin protein levels over time. To see whether the model also predicted a change in neprilysin activity with age, we added a dummy variable to the model which counted each neprilysin-dependent Aβ degradation reaction. A plot of this dummy variable versus age is linear indicating that the reaction rate does not change with time (not shown). The reaction contains both neprilysin and Aβ as reactants and so a decline in neprilysin with time means that there must be an increase in the number of Aβ monomers in order that the reaction rate remains constant. Therefore our model predicts that the actual activity per neprilysin molecule gradually increases with time due to higher levels of monomers. However, despite the increase in neprilysin activity, the levels of monomers are higher so that there is a greater chance of dimers forming which may then initiate the aggregation process.
Parameter scans for model 2
Dimerisation rate has the largest effect on the number of plaques formed and the age of onset of plaque formation
Effect of increasing parameter values on model outputs for normal Aβ degradation rate (Model 2)
Number of plaques (%)
Increases to max of 100
Increases to max of ~50
Slight decrease for large values
Increase at high values
Varying k dimer
Varying k pf
Increasing the value of k pf leads to an increase in the percentage of cells associated with plaques, up to a maximum of about 50% (Figure 6B). Even increasing k pf over four orders of magnitudes did not result in all cells being associated with plaques by age 100 years. This is because the rate of plaque formation is dependent on the pool of dimers and so can only increase if there are dimers available. Changing k pf has no effect on the size of plaques or the lag time.
Varying parameters for plaque growth
The parameters which affect plaque growth are: k pg (which affects the maximum rate) and k pghalf (which is the plaque size at which the growth rate is 50% of the maximum rate). The rate of growth also depends on the plaque size and the availability of Aβ monomers. The model predicts that increasing k pg initially leads to an increase in the percentage of cells associated with plaques but reaches a maximum of just over 12% and then there is a sharp decline to about 10% which remains fairly constant with further increases of the parameter (Figure 6C). However, this behaviour is due to stochastic effects as only small numbers of cells developed an association with plaques, as can be seen by the 95% confidence interval (CI) error bars. Overall, the effect of increasing k pg on the number of affected cells is small since if plaques are not formed in the first place, they cannot grow. As expected increasing the value of k pg leads to larger plaques and has no effect on the lag time. Increasing the value of k pghalf reduces the aggregation rate and so the number of plaques decreases, but there is no effect on plaque size.
Varying k disagg
Increasing the rate of disaggregation k disagg has no effect on the percentage of cells associated with plaques or the lag time. However, increasing this parameter has a large negative effect on plaque size. This is not surprising as disaggregation removes monomers from the plaques but is not involved in the actual formation of plaques.
Varying rate of decline of neprilysin
List of parameter values for Model 2
Aβ production rate
1.86e-5 molecules s-1
Value from Mawuenyega at al. 2010 
Aβ degradation rate
Values from Mawuenyega et al. 2010 
Aβ dimerisation rate
1.2e-7 molecules-1 s-1
Chosen from global parameter scan
Dissociation rate of dimers
Chosen from global parameter scan
Formation rate of plaques
2.8e-6 molecules-1 s-1
Value in range of 1 new aggregate per 5 days (Lomakin et al., 1997) 
Maximum rate of plaque growth
Approx 5 monomers per min (Lomakin et al. 1997) 
Size of plaque at which growth rate is half of k pg
Chosen from global parameter scan
Rate of plaque disaggregation
Chosen from global parameter scan
Neprilysin degradation rate
Normal rate (neprilysin remains constant with age)
Set so that neprilysin declines to about 700 by age 60 years.
Addition of antibodies against Aβ dimers
Model predicts that early interventions have largest benefits, although late interventions are also beneficial
Addition of antibodies against Aβ monomers
Model predictions suggest that this intervention may be less effective than antibodies against Aβ dimers
Another therapeutic approach which is currently under consideration is the use of antibodies against Aβ monomers [, reviewed by]. We do not believe that this approach will be as effective as antibodies against dimers for two important reasons. Firstly, it has been established that dimers are toxic species but there is no clear evidence that monomers have in themselves a pathologic role. Secondly, monomers may well have a functional role and so the use of antibodies against them may have detrimental effects. However, this approach could prevent the formation of dimers and hence reduce plaque formation and so it is of interest to use the model to make predictions about the outcomes of such a procedure and to compare the efficiency of preventing plaque formation using antibodies against monomers and dimers. We used the same method as for antibodies against dimers except we replaced antiAbDim with a molecular species named antiAbMon which we assumed could bind to monomers and so prevent the monomers from forming dimers or being sequestered into plaques. Interestingly our model predicted that using antibodies against monomers requires that the addition of antibodies is two orders of magnitude higher than for dimers in order to reduce the level of plaques to those shown in Figure 8. This is because Aβ monomers are continually being produced and so a large number of antibodies are required to prevent any dimerisation taking place and to prevent the sequestering of monomers into aggregates. Our model also predicted that if the addition of antibodies against monomers was carried out at age 70 or 80, there was an initial significant decrease in the number of cells associated with plaques which was due to increased disaggregation. However, at any age of administration the maximal effect was only temporary as antibodies became bound to monomers depleting the available pool (Figure 8B).
Inhibition of Aβ production
Model predicts that inhibition of Aβ production is less effective than targeting dimers especially for interventions at early ages
Interventions to decrease Aβ production, such as secretase inhibitors, are also being tested . Therefore it is also of interest to use our model to mimic the action of such an intervention by decreasing the rate of Aβ production. As in the antibody simulations, we simulated the intervention starting at different ages. The model predicts that a 25% reduction in the production rate slows down the rate at which plaques form but does not completely prevent the increase in plaque formation (Figure 8C). In particular, an early intervention at age one year greatly reduces the plaque burden but does not completely prevent it as was seen in the immunization simulations (compare pink curves in Figures 8A and 8C). Therefore our model predicts that interventions to reduce Aβ production may not be sufficient to prevent plaque formation unless administered at high doses and for prolonged time periods.
Model 3 plaques grow by addition of dimers
The hypothesis that plaques grow by addition of dimers is not supported by our model
We have shown how a modelling approach can be used to address questions concerning the aggregation of amyloid-beta in Alzheimer’s disease. For many readers, this approach may be unfamiliar and some may consider it premature and incomplete. However, we would like to point out that this is exactly why this paper is both timely and valuable. We developed a stochastic model which could be simulated over long time periods to investigate the effects of Aβ turnover on the accumulation of plaques with age. The model showed that dimers play an essential role in seeding (or starting) the aggregation process and that targeting Aβ dimers rather than monomers is likely to be the best approach for future interventions. The motivation for the model was to examine in detail the basic steps in the process of amyloid aggregation and to use the model to make predictions about the effects of proposed interventions. There are many different hypotheses concerning the aggregation process in AD and great efforts are being made to find effective interventions. Modelling is a useful tool for testing ideas and examining the effects of proposed interventions being able to make predictions which can be tested experimentally. For example, our model predicted that very early interventions could prevent the formation of aggregates at later ages. This could be tested experimentally in an animal model of AD and/or an animal model of trisomy 21.
We kept the model as simple as possible and so all the processes that are included in the model are essential to reproduce the results shown here. Stochastic simulation was necessary in order to investigate the cellular variability in plaque formation. Each simulation run represents the outcome for one brain cell and we carried out 500 simulations for each parameter set. The simulation output for 500 cells usually contain cells that do not develop any association with plaques over a 100-year period (depending on the parameter set) and in cells where plaques form, the time of formation is very variable. Our models predict that dimerisation of Aβ, which is the first step in the aggregation process, has a large effect on both the timing of plaque formation and the probability of a plaque forming.
We used the model to examine the effects of immunization against dimers by simulating the addition of antibodies at different ages followed by continued provision of antibodies. This more closely resembles passive immunization with continual provision of antibodies. Ideally, a model of active immunization would be preferable as this procedure would act much better clinically. However, active immunization is a complicated process and would require a more complex model than we have attempted here. Although we used a simple method to mimic the process, the findings are still valuable and show that it would be worthwhile to modify the model for active immunization. An important finding from the simulations is that the use of antibodies against dimers could greatly reduce plaque burden and that they would be much more beneficial if used in mid-life rather than in old age, but that benefit is provided even if administered at later ages provided the patient did not already have a high burden of plaques. An exciting possibility is the development of a vaccine for babies with Down’s syndrome to prevent the early onset of AD. The current model has focussed on the role of amyloid plaques in AD although the actual role of plaques in the disease process is still not clear. On the one hand some studies have shown a lack of correlation between plaque burden and the duration or severity of disease [40, 41]. Conversely, another study showed an increase in plaques in the neocortex which correlated with severity of dementia and the authors concluded that plaques are not a consistent feature of ageing . Plaques are found in regions of the brain that are affected in AD and they are usually associated with neuronal processes causing synaptic loss and so may disrupt communication between neurons. However, the exact role of plaques in synaptic loss is still not clear . Support for the role of plaques in AD comes from post-mortem studies of brain specimens from participants in the Phase 2a active immunization trial that there was a significant reduction in amyloid pathology which correlated with an improvement in neurite abnormalities in the hippocampus and an amelioration of tau pathology . Our model does not currently include any pathogenic role for plaques, rather we just used them as a measured outcome. A future development of the model would be to extend it to include detail of synaptic function and include possible mechanisms for how this could be disrupted by plaques.
Our models investigated only the role of monomers and dimers in the aggregation process but could be extended to include further steps such as larger oligomers and protofibrils. This could be achieved by adding further species to the model and replacing the reaction of two dimers forming a plaque with a series of reactions. This would allow the model to be used to examine the effects of interventions on soluble oligomers. Since soluble oligomers may be more toxic than the fibrillar form of amyloid, this would be very desirable in order to use the model to test interventions prior to clinical trials. However, it should be noted that this will make the model more complex and slower to simulate, and we would need further information on how the different amyloid components adversely affect cellular processes. As the model is encoded in SBML, any future developments will be straightforward. Although we focus here on the role of amyloid in AD, we have previously modelled the aggregation of both Aβ and tau in a more complex model . However, the previous model could only be used to mimic cellular models of aggregation over time periods of days rather than years due to the greatly increased computational load. A potential compromise in future studies might be to take some of the key steps involved in tau aggregation and add them to the model of Aβ turnover and aggregation. We could also extend the model to include microglia and then use the model to examine the effect of immunotherapy against amyloid plaques. Another important extension to the model would be to add apolipoprotein E (APOE) which exists as three different isoforms (APOE2, APOE3 and APOE4) with APOE4 being known to have a pathogenic role in AD . A particularly relevant role of APOE in AD is that the APOE-lipoprotein binds to Aβ and is involved in Aβ clearance. However APOE4 is less efficient than the other isoforms in this role, which could contribute to its pathogenicity . Finally we could extend the model to examine the interaction between protein aggregation and mitochondrial dysfunction in AD. For example we could add the interaction between APP, Aβ and mitochondria (Figure 1) as there is evidence that the accumulation of full-length APP and Aβ in the mitochondrial compartment has a causative role in impairing mitochondrial physiological functions . We could also use the model to examine other possible therapies such as promoting degradation of Aβ by administering agents that enhance the activity of Aβ-degrading enzymes . Our model predicts that lower levels of neprilysin leads to an increase in plaque formation and so our model supports that the use of such therapies could be beneficial. We could use our model to test this by simulating the addition of extra neprilysin at different time-points. This might require inserting more detail of the mechanisms involved in the decline in neprilysin activity with age. This approach could also be relevant for other diseases such as cerebral amyloid angiopathy since it has recently been shown that neprilysin protects cerebrovascular smooth muscle cells against Aβ induced degeneration .
Progressive ageing of populations around the world, has been accompanied by alarming growth in the numbers of patients diagnosed with AD and other dementia-like illnesses. The large number of studies showing a correlation between synaptotoxic amyloid species, synapse loss and cognitive deterioration which clinically characterises AD patients, strongly suggests that targeting abnormal levels of Aβ in the brain might provide a potentially powerful preventative strategy. Nevertheless, there remain important questions about how such interventions might act upon the normal mechanisms for formation and degradation of protein aggregates within the brain. It is important to understand the kinetics of these processes, which are significantly influenced by the actions of chance at the levels of the underlying molecular interactions. By employing stochastic mathematical modelling, the present study served to provide evidence for the validity of following such an approach. Moreover, Aß dimer formation was highlighted as a particularly important contributing molecular element underlying AD pathology, in agreement with recently published experimental studies. In conclusion, the study served to provide better insight into the kinetic processes underlying eventual fibril/plaque formation, which might help pinpoint where and at which time-point in such processes is the most therapeutically promising targets for therapeutic intervention.
We began by building a model of Aβ turnover, which includes reactions for Aβ production and degradation but does not consider the aggregation process. The aim of this model, which we refer to as Model 1, was to see the effect of decreased clearance on levels of Aβ over time. We then extended this model to include details of Aβ aggregation (Model 2) to examine the effects of decreased clearance of Aβ on the aggregation process. We also examined a variation of Model 2 in which a different assumption was made concerning the growth of plaques. Details of all three models are given below.
Model 1: Aβ production and degradation
We used mass action kinetics which means that the rate of a reaction is directly proportional to the number of molecules (note that we use stochastic simulation; in a deterministic model number of molecules would be replaced by concentration) of each reactant raised to the power of its stoichiometry. In the first reaction equation, Source is a constant dummy variable and so this is a zeroth-order reaction with a constant production rate and k prod is the hazard of this reaction. The second reaction equation is of first-order and the hazard of this reaction is k deg Abeta. Details of how the model was coded and simulated are given later in the Methods section under the heading “Tools used for model construction and simulation”.
In order to simulate the model, we need to choose values for the parameters and the initial amount of Abeta. We used the measured Aβ production rate (k prodAbeta = 1.86e-5 molecules s-1) from Mawuenyega and colleagues . The degradation rate depends on the concentration of Aβ in the cell (e.g. k degAbeta * Abeta). Therefore we need to know the steady state level of Aβ. The concentration of Aβ42 was measured in CSF and was found to be 500pg/ml for patients with normal Aβ clearance rates  and gives no indication about the level of soluble Aβ42 at the cell surface where it is produced. Usually, Aβ is rapidly cleared and therefore we would expect the level of Aβ in the vicinity of the membrane to be low, especially as APP is found throughout the cellular membrane and is not localised to a particular position. Therefore we assume that the local level of Aβ is initially given by Abeta = 1, and assume k degAbeta = 2.1e-5 s-1 for the normal degradation rate and k degAbeta = 1.5e-5 s-1 for the degradation rate in AD. The model was simulated 500 times for a particular set of parameters over a time period of 100 years and the level of Aβ was output to a file (500 time-points were chosen). The results were analysed and plotted in R (Figures 1234).
Model 2: Aβ turnover and aggregation
Two Aβ monomers may interact to form a dimer.
Dimers may disassociate into two monomers.
Two dimers may interact to form a small aggregate.
Aggregates may grow by the addition of monomers only (although we also consider growth by dimers in Model 3).
Aggregates may disaggregate to release a monomer.
When aggregates reach a certain threshold size, they continue to grow until they reach a maximum size. Once the aggregate has passed the threshold size we consider it to be a plaque.
Since the actual size of the threshold is not fixed, i.e. it varies in a stochastic fashion, we do not have separate species for small aggregates and plaques and use the name AbP for all aggregates. Therefore, in our model output analysis we assume that if AbP is greater than a certain level then it is counted as a plaque. We chose 30 for this threshold size as plaques of this size, or larger, rarely completely disaggregated.
Rate laws and parameter values
Parameters for Aβ turnover
We used the same values for Aβ production and degradation as in Model 1. However, once we added the aggregation steps, we found that the model predicted plaques could occur at early ages when we set the degradation rate at the AD level. Originally we assumed that Aβ clearance was impaired throughout life. However, with this assumption it was not possible to find a parameter set where levels of plaques are very low early in life but increase only after the age of 60 years (Figure 5B). There are 3 possible ways that we could modify the model to provide more realistic output. Firstly, the parameters which affect lag time could be adjusted. Secondly, further steps could be added to the aggregation process such as formation of trimers, oligomers and protofibrils, in order to increase the lag time. Finally, we assumed that the degradation rate for AD is lower than normal throughout life but it is more likely that the rate declines with age. There are no parameters in the model which affect only lag time (discussed in further detail in the Results section). We did not wish to add further complexity to the model, so did not consider the second possibility. The assumption of a constant low degradation rate is probably not correct since if this was the case, then we would expect to see cases of sporadic AD at earlier ages. Therefore we chose to amend the model so that the degradation rate declines with age. We added one of the Aβ degrading enzymes to the model, namely neprilysin, since it is known that its level decreases with age. We assumed that the Aβ degradation rate depended on the level of neprilysin and that the level of neprilysin declined with age in AD. Therefore we needed an additional parameter, k degNep , for the degradation rate of neprilysin. We chose the value for this so that neprilysin values declined by about 30% by age 60 which resulted in Aβ degradation rates similar to those observed in AD patients at late ages.
Parameters for Aβ aggregation
List of reactions for Model 2
Kinetic rate law
k deg Abeta (Normal rate)
k deg Abeta * Nep (AD rate)
k dimer Abeta*(Abeta-1)*0.5
k dedimer AbDim
k pf AbDim*(AbDim-1)*0.5
k pg Abeta*(AbP2/(k pghalf 2 + AbP2))
k disagg AbP
k degNep *Nep
We classify aggregates containing more than 30 monomers as plaques (i.e. AbP >30), as our model output shows that once aggregates reach this size, they are unlikely to disaggregate completely but continue to grow in size. Therefore, our use of the term “plaque” incorporates all aggregates of Aβ which have exceeded a critical threshold size and have a high probability of forming the amyloid deposits which are termed “plaques” in the disease state.
The rate of fibril formation and fibril growth has been measured experimentally and it has been reported that fibrils grow at a rate of 0.5 monomers per minute and that new fibril nucleation takes place approximately once in 5 days per micelle [13, 50]. We chose values for k pf and k pg based on these measurements. We do not have data for the rates of dimerisation, disassociation of dimers or disaggregation of plaques and so we used a global parameter scan in order to find parameter sets which predict very low levels of plaques before the age of 50 years and then a gradual increase in the number of plaques up to age 100 years (the maximum age used in our simulations). In addition, we assumed that the parameters need to satisfy the condition that levels of plaques do not exceed 10% by age 100 years when Aβ clearance is normal, but that for impaired Aβ clearance, levels exceed 25% by the age of 100 calendar years. We do not have data on the percentage of cells associated with plaques in normal ageing and AD, but these values seemed reasonable. In future, these values may be measurable and then it would be possible to fit parameter sets to clinical data.
List of species for Model 2
Model 3: Aβ turnover and aggregation with plaque growth via dimers
Although the current consensus holds that plaque growth is via monomers, it cannot be ruled out that dimers provide the building blocks for such pathological aggregates . To examine this possibility, we changed the reactions for plaque growth and disaggregation. This model was fitted with different parameter values and did not receive such extensive analysis as shown in Model 2. The parameters are much more sensitive in this model. Apart from k pf all the parameters can take values that result in no cells associated with plaques, all cells associated with plaques or an intermediate level of plaques (Figure 9).
Tools used for model construction and simulation
The models are encoded in the Systems Biology Markup Language (SBML), a computer-readable format for network models . The tool used to create the SBML code was SBML short-hand which uses a Python script to convert a short-hand version of SBML into the full code . The code is available from the Biomodels database (Biomodels ID: MODEL1202290000)[54, 55] and is also provided in Additional file 2. The SBML model can be imported in any software tool which has the facility for stochastic simulation. The stochastic simulations were carried out using the gillespie2 code (available from the SBML website ), which is based on the Gillespie algorithm . Briefly, this algorithm uses random numbers to simulate the time to the next event and to pick the reaction that occurs at this time according to the reaction hazard. The number of molecules of each species is updated according to the reaction which occurred. The procedure is repeated until the simulated time to the next event exceeds the maximum time of the simulation. The model results were analysed and plotted using the R statistical package.
The slopes of the regression lines in Figure 4 were tested for statistical significance using a student’s unpaired t-test. The correlation between two variables was calculated using Pearson’s r statistic. The 95% confidence interval error bars for the percentage of cells associated with plaques were calculated using ±1.96 x s.e., where s.e. = √(p(100-p)/N), p is the percentage and N is the number of simulated cells. All calculations were carried out in the R statistical package.
Amyloid precursor protein
Systems biology markup language
Sodium dodecyl sulphate.
CJP built the model whilst funded by Alzheimer Scotland and Alzheimer’s Research UK. The work was taken forward within the research context of the Centre for Integrated Systems Biology of Ageing and Nutrition (initially funded by BBSRC) and the UK NIHR Biomedical Research Centre for Ageing and Age-related Disease award to the Newcastle upon Tyne Foundation Hospitals NHS Trust. We are also grateful to funding support from a Research Council UK. RCUK Academic fellowship (JLE).
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