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**|**PLoS One**|**v.5(12); 2010**|**PMC3001872

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PLoS One. 2010; 5(12): e15176.

Published online 2010 December 14. doi: 10.1371/journal.pone.0015176

PMCID: PMC3001872

Vladimir Brusic, Editor^{}

Dana-Farber Cancer Institute, United States of America

Conceived and designed the experiments: LL IKP. Performed the experiments: LL IKP. Analyzed the data: IKP LL. Contributed reagents/materials/analysis tools: IKP LL. Wrote the paper: LL IKP.

Received 2010 September 13; Accepted 2010 October 27.

Copyright Puri, Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

This article has been cited by other articles in PMC.

Despite extensive research, the pathogenesis of neurodegenerative Alzheimer's disease (AD) still eludes our comprehension. This is largely due to complex and dynamic cross-talks that occur among multiple cell types throughout the aging process. We present a mathematical model that helps define critical components of AD pathogenesis based on differential rate equations that represent the known cross-talks involving microglia, astroglia, neurons, and amyloid-β (Aβ). We demonstrate that the inflammatory activation of microglia serves as a key node for progressive neurodegeneration. Our analysis reveals that targeting microglia may hold potential promise in the prevention and treatment of AD.

Alzheimer's disease (AD) is one of the most prevalent neurodegenerative disorders associated with aging, causing dementia and related severe public health concerns [1]. Despite extensive research effort and progress, the pathogenesis of AD remains incompletely understood, partly due to highly complex and intertwined intercellular cross-talks taking place throughout the aging process [2]. Consequently, despite limited treatment options to manage and slow the progression of AD, no effective cure is available.

Although the deposition of amyloid-β (Aβ) peptides and formation of senile plaques in the brain is the cardinal morphological feature identifying the clinical phenotype of AD [3], [4], increasing clinical and basic studies suggest that inflammatory activation of microglia may play an equally important role during the initiation and progression of the disease [5]. Microglia are resident innate immune macrophages within brain tissues, capable of expressing pro-inflammatory mediators and reactive oxygen species when activated by inflammatory signals including amyloid-β (Aβ) [6]. In healthy brains, together with quiescent astroglia (Aq), resting microglia may adopt an anti-inflammatory state (M2) and in turn foster neuron survival (Ns) and prevent astroglia proliferation (Ap) [7], [8]. As inflammatory signals (e.g. Aβ) gradually build, microglia may adopt an activated pro-inflammatory state (M1), leading to A_{p} proliferation and neuron death (Nd) [9], [10], [11]. Neuronal debris, amyloid-β (Aβ), and/or proliferating astroglia (Ap) may in turn further exacerbate the inflammatory phenotype of M1 macroglia [12], [13]. The multiple positive and negative feedbacks among these cells are thus crucial for neurodegeneration that eventually alters the neuronal structure and function during the pathogenesis of AD (Figure 1).

Due to its multi-cellular components and complex nature, conventional experimental approaches have failed to identify critical underlying causes for AD, contributing to the lack of an effective therapeutic treatment. Mathematical models can serve as powerful tools to understand the molecular and cellular processes that control complex diseases [14], [15]. Indeed, there have been several attempts to model the process of senile plaque formation [16], [17], [18], [19]. Specifically, these approaches focused on a nucleation step that is coupled with rates for the irreversible binding of Aβ monomers to the fibril ends, the lateral aggregation of filaments into fibrils, and fibril elongation through end-to-end association. Other modeling efforts examined the signaling cascade responsible for microglia migration and activation in response to an initial inflammation-provoking stimulus involving Aβ [16], [20].

However, no systematic modeling approaches have been reported to examine the network cross-talks among microglia, neuron, and astroglia, and the corresponding pathological consequence. Here, we evaluate the dynamic network involving multiple cross-talks among distinct states of microglia, astroglia, and neurons through a mathematical model. Our approach has led to an intriguing insight suggesting that microglia activation in addition to a threshold for Aβ may be the critical initiator for the pathogenesis of AD.

We propose a sixteen pathway AD mechanism involving seven species that is shown schematically in Fig. 1. The paths have rates α_{i} that implicitly represent the influences of intercellular signaling along them. The mechanism is based on an assumption of constant risk of neuronal death, i.e., a single event randomly initiates cell death independently of the state of any other neuron at any instant [21]. The spatiotemporal influence of diffusion is neglected since local cell events are assumed to occur on a slower timescale than signal dispersion through chemotaxis.

The seven rate equations for the cell populations and the number of Aβ molecules in an arbitrary local volume can be written through seven coupled rate equations, namely,

(1)

(2)

(3)

(4)

(5)

(6)

(7)

These relate the change in each cell population or the number of Aβ molecules at any instant to the values of all species at that time. For instance, Eq. (1) relates the rate of change in N_{s} to the A_{q}, A_{p}, and M_{1} populations with the pathway weights α_{1}, α_{2}, and α_{3}, respectively. Whereas A_{q} increases the rate of change of N_{s}, A_{p} and M_{1} decrease it. Equation (5) for the rate of change of the M_{2} population is the most complex, since it involves nine pathway weights, and five cell populations and Aβ. The conversion of N_{s} into N_{d} is irreversible, whereas those of A_{q} and M_{2} into A_{p} and M_{1} are reversible.

The rates for each α_{i} are specified, as shown in presented in Table 1 for each pathway. Since the literature points to the path N_{s} → Aβ being dominant, we assume that it is also the fastest. Its rate is set at 1/year, i.e., each year every N_{s} cell stimulates the formation of a sustaining an Aβ molecule. Likewise, since neuronal survival decreases significantly once disease progresses, we assume that the overall path M_{2} → A_{q} → N_{s} is slow so that the associated rates α_{1} and α_{4} are also relatively the smallest. The other rates are similarly specified in terms of their relative abilities to facilitate or inhibit the formation of a cell or Aβ molecule according to the particular pathway. Next, we specify the initial composition of the volume under consideration. These initial conditions for the seven species are presented in Table 2.

Our objective is to be able to describe neuropathogenesis during AD in terms of the N_{s} and N_{d} populations. Hence, we first determine the sensitivities of these cells to changes in the rates α_{i}, using the usual definition of the sensitivity coefficient,

(8)

The sensitivity coefficients for N_{s}, N_{d}, M_{1} and M_{2} cells, presented in Table 1, are determined after 20 years for ±2.5% perturbations in each α_{i} value. A cell population is more sensitive to a change in a rate that produces a larger value of |S(N_{j})|. Positive values for S(N_{j}) imply that a rate contributes to an increase in N_{j} while a negative value implies that its influence leads to a corresponding population decrease. The sensitivity analysis shows that the N_{s} and N_{d} populations are most sensitive to the path A_{q} → N_{s}, which increases neuronal survival and decreases neuron death. Important paths that inhibit neuropathogenesis include A_{q} → M_{2}, A_{q} M_{1} and M_{2} M_{1}, while those that enhance disease involve M_{1} → N_{d}, Aβ M_{2} and Aβ → M_{1}.

A similar analysis that perturbs the initial cell populations and the number of Aβ molecules tenfold is presented in Table 2. It shows that, in comparison to the other species, the M_{1} population is most sensitive to these substantial perturbations in the initial amount of any species while N_{d} is only sensitive to the initial amounts of M_{1} and M_{2}. This implies an important role for microglia during AD progression. The sensitivity coefficients S(M_{1}) and S(M_{2}), also presented in Table 1, show that, as for N_{s} and N_{d}, the dominant paths that inhibit neuropathogenesis by affirming M_{2} and decreasing M_{1} are also A_{q} → M_{2}, A_{q} M_{1} and M_{2} M_{1}. Once again, paths 7 and 14 involving Aβ, i.e., Aβ M_{2} and Aβ → M_{1} promote AD progression. The model suggests that interventions aimed at decreasing α_{8} and α_{13}, which involve M_{1}, M_{2} and Aβ and contribute to AD progression, are the ones more likely to diminish neuropathogenesis. This intuitive result emphasizes that decreasing the number of reactive microglia and ensuring a sufficient population of quiescent astroglia is important in treating AD.

The temporal variation in various species for the rates in Table 1 is illustrated in Fig. 2. Figure 2(a) presents the N_{s}, M_{1} and Aβ populations over 20 years, and Fig. 2(b) the corresponding values for N_{d} and A_{p}. Most notable is the influence of the removal rate α_{r}, which stabilizes the number of Aβ molecules after three years. Following that period, there is only a gradual increase in N_{d} that is coupled with a corresponding decline in N_{s}. Consequently, the microglia populations are also relatively stable. Therefore, the rates in Table 1 should be considered as being representative of a healthy population.

We examine the influence of varying α_{r} on neuropathogeneis in Fig. 3, which presents the M_{1}, A_{p}, Aβ and N_{d} populations over 20 years for three values of α_{r}. As α_{r} decreases, there is an increasing neuronal death. Thus, all four populations, which are associated with AD progression, increase. While microglia play an important role in AD, Fig. 3 shows how the local Aβ concentration plays a critical role in initiating and promoting AD.

We investigate this further by varying α_{8} and α_{13}. Figure 4 presents results for the M_{1}, N_{d} and Aβ populations over 20 years for three values of α_{13}. As α_{13} increases, the M_{1}, A_{p} and N_{d} populations also increase, leading to an associated decrease in neuronal survival, as illustrated through Eqs. (1) and (2) of the mathematical model. A tenfold increase in α_{13} leads to a near doubling in N_{d} after 20 years. As N_{s} decreases so does Aβ, but the smaller protein concentration is still sufficient to promote neuropathogenesis among the smaller N_{s} population. Identical results are obtained for similar variations in the rate α_{8} for Aβ M_{2}, since the sensitivity coefficients for each of M_{1} and N_{d} towards paths 8 and 13 are identical.

We present a mathematical model for neuropathogenesis during AD that involves neurons, normal and reactive glial cells, and Aβ. It uses neuronal death as a surrogate for senile plaque formation. By monitoring neuronal health, we are able to identify intuitive strategies for interventions. In particular, the model suggests that the most effective intervention is one that improves the inhibition of reactive microglia and Aβ by normal microglia, and ensuring a sufficient population of quiescent astroglia. Overall, neuropathogenesis proceeds through the production of reactive microglia.

Our analysis is consistent with experimental data that indicate that inflammation may be an early initiator for AD, long before the apparent senile plaque formation [22], [23]. It further reinforces the notion that additional studies should be directed at examining earlier inflammatory signals and alterations involving microglia as a key node so as to better define AD initiation and understand mechanisms for effective prevention and treatment of the disease.

We realize that our mathematical analysis is an initial attempt to examine AD and may not fully account for the associate intertwined cellular communication pathways. Nevertheless, it serves as a hypothesis provoking and building process that should encourage integrated analyses of AD pathogenesis. Future experimental data examining the cross-talks among microglia, astroglia, and neurons will allow us to better refine our model and implement realistic parameters in the rate equations.

The authors would like to thank members of the Li laboratory for critical discussion.

**Competing Interests: **The authors have declared that no competing interests exist.

**Funding: **Internal funding from Virginia Tech supported this work. Virginia Tech had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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