COMPUTATIONAL PREDICTIONS FOR THE NUCLEATION

MASS AND LAG TIMES INVOLVED IN A42 PEPTIDE

AGGREGATION

Preetam Ghosh

, Bhaswati Datta

and Vijayaraghavan Rangachari

Department of Computer Science, Virginia Commonwealth University, Richmond, VA, U.S.A.

School of Computing,

Department of Chemistry & Biochemistry, University of Southern Mississippi

Hattiesburg, MS, U.S.A.

Keywords: Alzheimer’s Disease, Protein self-assembly, Ordinary differential equation, Mass-kinetics.

Abstract: The aggregates of amyloid-β (Aβ) peptide are the primary neurotoxic species in the brains of Alzheimer’s

patients. We study the molecular-level dynamics of this process employing chemical kinetic simulations by

dissecting the aggregation pathway into pre-nucleation, post-nucleation and protofibril elongation stages.

Here, we discuss how our earlier identified rate constants for protofibril elongation were incorporated into a

simplified simulation of the complete aggregation process to understand the lag-times in the sigmoidal fibril

growth curves of fibril formation. We also present some initial findings on the rate constants and possible

hypotheses on the nucleation mass involved in the pre-nucleation stage.

1 INTRODUCTION

In Alzheimer’s disease (AD), the aggregates of a

protein called, Aβ are strongly believed to be the

cause for neuronal death and cognitive decline

(Selkoe, 2003). Aβ aggregates to form large fibrillar

deposits that follows a sigmoidal growth pattern

involving a ‘lag-phase’ prior to fibril growth. The

lag-phase is generated due to an initial rate-limiting

step of nucleation (Jarrett, 1993); (Harper, 1997).

However, the precise mechanism of nucleation and

size of the nucleus are not known. Accurate in vitro

analyses of the process is difficult as the

intermediate oligomers are difficult to isolate and

characterize. However, one intermediate from the

post-nucleation phase, called protofibrils were

identified (Walsh, 1997) that show propensities to

both elongate as well as laterally associate to grow

into mature fibrils. However, many previous works

on Aβ aggregation kinetics have not incorporated the

pre-nucleation events that constitute a critical step of

the aggregation process, more likely due to the

difficulty in doing so for stochastic processes.

It is important to identify the nucleation mass

and the kinetic rate constants involved in all the

different phases of Aβ aggregation: pre-nucleation,

post-nucleation and fibril elongation. Various

models on Aβ aggregation reviewed in (Morris,

2009), use curve fitting without considering the pre-

nucleation events. Recently, (Lee, 2007) reported a

molecular-level model of insulin aggregation that

forms the basis for the model presented here. Earlier,

we modelled the protofibril elongation and lateral

association stages to report the kinetic rate constants

involved (Ghosh, 2010). Here, our contributions are

summarized as follows: (i) use the rate constants

from protofibril elongation into the biophysically

similar post-nucleation phase; (ii) create a model to

estimate the lag-times and nucleation mass of Aβ42;

(iii) report in vitro Aβ42 aggregation experiments

that motivate our nucleation mass estimates; (iv)

discuss the problems in directly comparing the

simulated lag-times to those from experiments.

2 A PROCESS SIMULATION

2.1 In Vitro Results on Aβ42

Aggregation

We monitored Aβ42 aggregation in five different

concentrations, 10, 25, 50, 75 and 100



M by

thioflavin-T (ThT) fluorescence (lag-times shown in

Fig 1). Since Aβ aggregation is nucleation-

312

Ghosh P., Datta B. and Rangachari V..

COMPUTATIONAL PREDICTIONS FOR THE NUCLEATION MASS AND LAG TIMES INVOLVED IN Aβ42 PEPTIDE AGGREGATION.

DOI: 10.5220/0003782603120316

In Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms (BIOINFORMATICS-2012), pages 312-316

ISBN: 978-989-8425-90-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

1,,

);

fbifbnuinu



dependent, increase in concentration decreases the

lag-time besides increasing the rate of aggregation.

Hence, we observed the least lag-time for 100



followed by 75, 50, 25 and 10



M concentrations

respectively. In addition, there was an inverse linear

correlation between the logarithm of Aβ42

concentration and the corresponding lag-time as

shown in Fig 1. This observation was later used to

accurately characterize the Aβ42 nucleation mass

based on the simulated lag-times.

Figure 1: Lag-times from in vitro experiments.

Figure 2: Reactions towards fibril formation.

2.2 Modified Model on Aβ Aggregation

Here, we adapt the insulin aggregation model in

(Lee, 2007) for the Aβ42 system. We characterize

the pathway using biochemical reactions, compute

the reaction fluxes and formulate the differential

equations for each oligomer concentration as a

function of time. Solving the set of homogeneous

ODEs allow us to study the temporal dynamics of

each oligomer. Fig 2 shows the modified set of

reactions considered in our simulation.

Here, A

’s denote i-mers, n is the nucleation mass

and F is a fibril. The following assumptions were

made: a) monomer adds to i-mers until fibril

formation; b) nucleation involves monomer addition

as well as a structural change in the oligomer A

(this conformational change is implicit); c) post-

nucleation events are faster, as the forward rate

constants for post-nucleation are much higher than

those in pre-

nucleation (i.e., k

nu,n+i

>>k

nu,i

) (a ~10

fold difference

;

1,...,2;

1,...,1;

111,

,,1,

1,,1,

11,,

FkAAk

niJJJ

JJJ

niFkFAkJ

niAkAAkJ

nunnnu

ifbinuinu

fbinunu

fbiifbifb

inuiinuinu



























was reported in (Lee, 2007)); d) the reverse reaction

rate constants are assumed to be independent of size

i, and abbreviated as k

nu-

and k

fb-

. e) since agitation

drastically shortens the lag-times, k

nu,i

and k

fb,i

are

assumed to be diffusion-limited; using the Stokes-

Einstein equation, the diffusivity is proportional to

the inverted cubic root of i, resulting in:

Hence, the reaction fluxes and differential

equations can be derived as follows:

2.3 Integrating Protofibril Elongation

A complete simulation of the Aβ system requires an

estimate of the following six parameters: k

nu,1

, k

fb,1

nu-

, k

fb-

, n and b, where, b is the constant that maps

ThT fluorescence to concentration estimates. It is

impossible to try out different values for each of

these variables to properly match the experimental

plots due to the huge solution space. Hence we

dissected the sigmoidal fibril-growth curve in

(Ghosh, 2010) into: (i) pre-nucleation stage (ii) post-

nucleation stage and (iii) protofibril elongation

stage. The pre- and post-nucleation stages are well-

approximated by the set of reactions shown in Fig 2.

However, protofibril elongation stage needs to

combine reactions from both post-nucleation and

lateral association. This requires the estimation of

two more rate constants: the forward and backward

rate constants for the lateral association stage

denoted by k

and k

la-

respectively. In our previous

report (Ghosh et al., 2010), we estimated the post-

nucleation rate constants (k

fb,1

, k

fb-

, k

and k

la-

)

separately and verified them with in vitro

experiments as follows: k

fb,1

=9.0 × 10

-1

), k

fb-

=4.5 × 10

-1

), k

=9.0 × 10

-1

), k

la-

=6.0 ×

-3

-1

). We next directly substitute the fibril

elongation rate constants into our modified model to

predict the lag-times.

COMPUTATIONAL PREDICTIONS FOR THE NUCLEATION MASS AND LAG TIMES INVOLVED IN Aβ42

PEPTIDE AGGREGATION

313

3 RESULTS AND ANALYSIS

Our model makes all possible oligomers in the pre-

nucleation stage mathematically tractable due to the

abstraction that any post-nucleation stage aggregate

(starting from the nucleation mass itself, i.e., A

) is

treated as a fibril (i.e., F). However, this model does

not consider the length of the fibrils as variables and

hence cannot match the plateaus of the ThT

fluorescence curves generated by experiments. This

is because the fibrils of differing length will have

different contributions on ThT intensity which

cannot be directly captured using this model. Indeed,

in Fig 4, we have mapped the concentration of F to

ThT intensity for different initial Aβ concentrations,

and each curve saturates at the same peak. This

problem was circumvented by assuming different

mapping constants in (Lee, 2007) to separate the

peaks for different Aβ initial concentrations, which

is not a biophysically correct assumption (as

discussed in (Ghosh, 2010)). Thus, this model

cannot implement an entirely accurate simulation of

the pathway. In this paper, however, our main goal

is to study the lag times in the pathway, and hence

predict a working range for the nucleation mass. As

our model in Fig 2 can accurately study the pre-

nucleation stage oligomers, we will henceforth use it

to study only the lag times in the aggregation

pathway generated for different values of the

nucleation mass (n).

Table 1: Lag-times (in hrs) from our simulation for

various estimates of nucleation mass.

3.1 Lag Time Predictions

In Table 1, we show the simulated lag times for

different nucleation mass and initial Aβ

concentration. In order to find the pre-nucleation

rate constants along with the nucleation mass, we

use the following scheme: estimate the rate constants

that give the maximum lag times for each value of

the nucleation mass. Note that, changing the rate

constants further to achieve higher lag-times render

the system of differential equations unstable.

Interestingly, the simulation shows 4 distinctly

different regimes of lag times corresponding to 4

different pairs of rate constants in pre-nucleation

(highlighted using different colors in Table 1). At

the same time, this also characterizes four different

regimes of nucleation masses associated with Aβ

aggregation summarized as follows: Regime 1:

n=7,8,9,10,11; Regime 2: n=12,13,14; Regime 3: n=

15,16,17; Regime 4: n= 18,19,20,21.

The rate constants for each of these regimes are

shown in Table 2. Note that the forward rate

constants were fixed for each nucleation mass, while

the backward rate constant were varied to achieve

the highest lag times as reported in Table 1. The

problem here is that each of the nucleation masses

does allow us to find a pair of rate constants for the

pre-nucleation stage. It is however, not possible to

match the simulated lag times to that observed

experimentally (as reported in Fig 1). We will

discuss this problem in the next section.

Table 2: Rate constants for prenucleation stage for various

estimates of nucleation mass.

Figure 3: Close to linear semi-log plots.

However, as seen in Fig 1 (and also from other

experiments in our lab consistently), the semi-log

plot of the lag-times against initial concentration of

Aβ is linear. So we used this property to figure out

what values of nucleation mass are most feasible for

the Aβ42 pathway. Note that n=10, 11 (in Regime 1)

and n=12 (in Regime 2) are close to linear and

BIOINFORMATICS 2012 - International Conference on Bioinformatics Models, Methods and Algorithms

314

hence may serve as good approximations of the

nucleation mass (Fig 3). It was also observed in

course of the simulations that an initial concentration

of 10 µM made the simulation erratic for a wide

range of rate constants (because of increased

dynamism and stochasticity in the system with lower

molecular count of the species rendering the ODEs

unstable). So, we generated these semi-log plots for

the different regimes (data not shown) by removing

the data points for 10 µM. Indeed, these curves show

a more stable relationship between the lag times and

the initial concentrations, and we find close to linear

behavior for n=10,11 (in Regime 1), n=12,13 (in

Regime 2) and n=15,16,17 (in Regime 3).

The next question is whether 10,11,…,17 is the

right range for the nucleation mass, or can we

further reduce it? Fig 4 shows the concentration

curve for F against time and different initial

concentrations. One requirement for the rate

constants reported above is that these curves must

saturate to the same peak as expected

mathematically. So we considered this to be another

constraint that reduced the range of feasible

nucleation masses to n=10, 11…,14. Note that n=15,

16, 17 did not allow the concentration curves to

saturate (data not shown), and hence were ruled out

as possible candidates for the nucleation mass.

Figure 4: Simulated fluorescence change curves for

different initial concentrations with n=12.

3.2 Can we Compare Simulated and

Experimentally Observed Lag

Times?

The experimental ThT fluorescence plots show the

cumulative effect of all oligomers of a certain size

(and beyond). The results shown above plot the

concentration of F which model the cumulative

effect of all the nucleated oligomers in the pathway.

However, it is assumed that all nucleated oligomers

show up on the ThT curves (this is generally not the

case from actual experiments). Hence, the lag times

estimated from our model are lower than that seen

experimentally. Also, it is not yet known what size

of oligomers actually show up ThT positive and

hence the experimental estimates are at best the

maximum limits of the lag times for each initial Aβ

concentration. To get around this problem, we varied

the rate constants to estimate the maximum possible

lag times for each value of the nucleation mass. This

is still an approximation of the actual system and

needs further study. Ideally, we need to know what

sizes of oligomers are considered ThT positive such

that the experimental curves can be meaningfully

compared to the simulated plots. The present paper,

however, gives us a feasible range of nucleation

masses to work with in order to build a complete

simulation of the on-pathway. The rate constants

estimated in this exercise can serve as a guidance for

the complete simulation where we will need a more

detailed model (with separate parameters for each

post-nucleation oligomer) to properly model their

effects on the system.

4 CONCLUSIONS

In this paper, we have studied the lag times in the

sigmoidal Aβ fibril formation pathway. We also

reported that the nucleation mass can potentially be

in the range 10,11,…, 14 mers. In order to reduce

the complexity of the entire fibril formation

pathway, we used the rate constants that we have

earlier estimated for the post-nucleation stage into a

modified model that can approximately characterize

the complete pathway. These estimates will serve as

the basis for implementing a complete and accurate

simulation of the pathway wherein we have

approximately estimated all the 6 variables involved.

Such a simulation will pave the path to study the

complete system dynamics of Aβ aggregation

leading to a better understanding of AD in general.

ACKNOWLEDGEMENTS

This work was supported by NSF-1158608.

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PEPTIDE AGGREGATION

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