Modeling the dynamics of innate immune response to Parkinson disease with therapeutic approach

This paper aims to mathematically model the dynamics of Parkinson's disease with therapeutic strategies. The constructed model consists of five state variables: healthy neurons, infected neurons, extracellular α-syn, active microglia, and resting microglia. The qualitative analysis of the model produced an unstable free equilibrium point and a stable endemic equilibrium point. Moreover, these results are validated by numerical experiments with different initial values. Two therapeutic interventions, reduction of extracellular α-syn and reduction of inflammation induced by activated microglia in the central nervous system, are investigated. It is observed that the latter has no apparent effect in delaying the deterioration of neurons. However, treatment to reduce extracellular α-syn preserves neurons and delays the onset of Parkinson's disease, whether alone or in combination with another treatment.

Parkinson's disease (PD) is one of the most common neurodegenerative diseases after Alzheimer's disease. An estimated 6.9 million people were diagnosed with PD worldwide in 2015. By 2040, the number of patients may reach 14.2 million [1].

Symptoms of PD appear as movement disorders, such as imbalance, tremors, and stiffness. It may be preceded by non-motor symptoms such as lack of smell and constipation. The appearance of symptoms develops over time. So far, there is no cure for the disease, but there are treatments that relieve symptoms but may cause many side effects [2, 3].

The pathological feature of PD is the death of dopaminergic neurons that secrete dopamine in a brain region known as Substantia Nigra pars Compacta, which may occur in other areas of the brain. Studies have also demonstrated the presence of Lewy bodies at autopsies of Parkinson's patients. These bodies are an aggregation of misfolded alpha-synuclein (α-syn). It has also been established in other neurodegenerative diseases such as Lewy body dementia (LBD) [2, 47].

Neuronal death occurs when a cell's autophagy is disrupted by aggregating α-syn protein within a neuron. This leads to protein accumulation within neurons [8, 9], and it may contribute to the release of the protein outside the neuron. Extracellular α-syn can occur as oligomers, protofibrils, or fibrils. They lead to endogenous protein accumulation, stimulation of neuroinflammation, exacerbation of synaptic pathology, and neuronal loss [2, 1012]. Several studies indicate that pathological α-syn may act as a prion-like protein and spread from cell to cell [1315]. Pathogenic protein forms may confer toxicity on recipient cells [7, 9, 16, 17], leading to the spread of pathology in the brains of patients with Parkinson's and other α-synucleinopathies.

The immune system protects the body from any foreign antigen. It is divided into the innate and adaptive immune systems. The innate response is rapid but not selective; however, the adaptive response is slower but selective with memory against future exposures to the same pathogen [18].

Microglia are a type of innate immune cell in the central nervous system (CNS). Usually, they are in a quiescent phase but are constantly checking the brain's environment [19, 20]. They become active very quickly if there exists any change from bacteria, viruses, or injury to the CNS [19, 20]. As a result, they release anti-inflammatory cytokines for protection before any damage occurs [21], and to alert neighboring cells to action. Microglia also initiate cell death when any of the neurons are defective [21]. They are the fastest to absorb and dispose of α-syn aggregates, then other types of innate immune cells [2224]. The role of microglia autophagy is to prevent neurodegeneration by clearing the protein released from neurons. However, during a severe inflammation, activated microglia can be upregulated over time and fail to be controlled, causing damage and killing of healthy neurons [21].

Mathematical models facilitate the representation of complex dynamics of the immune response in the CNS in the fight against diseases. They contribute to understanding disease behavior and response to treatments to predict future therapeutic strategies [25, 26]. Alqarni et al [27] introduced two mathematical models to describe the resulting interactions between microglia and neural stem cells and their effect on brain cells in stroke patients during the initial stage of injury, that is, within 72 h, and after that in the recovery stage. The first model aims to study the impact of inflammatory damage by microglia on brain cells during a stroke. While the second model determines the extent to which microglia affect the formation, production, and proliferation of neural stem cells in the recovery phase to improve the brain after injury. They concluded that immune cells help get rid of dead cells and generate stem cells, but their overactivation may occur, which results in damage to the affected area. Puri and Li [28] presented a model describing the interaction between astrocytes (quiescent and proliferating), microglia (normal and activated), neurons (surviving and dead), and amyloid-β to search for the essential element that plays a vital role in Alzheimer's disease. They concluded that the production of active microglia is a significant factor in the formation of neuropathy. In [29], Hao and Friedman formed a mathematical model to simulate the effect of drugs for Alzheimer's disease both under clinical trials or failed in the experiment. The model is described by a system of nonlinear partial differential equations consisting of neurons, astrocytes, microglia, macrophages, amyloid-β aggregation, and hyperphosphorylated tau proteins. They concluded with a therapeutic suggestion to slow the progression of Alzheimer's disease. As for models on PD, Kuznetsova and Kuznetsova [30] developed a model to describe the transfer of α-syn protein inside the neuron from the soma to the synapse. The model consisted of four nonlinear differential equations composed of α-syn protein in monomer and polymeric states in both soma and synapse. They found that the abnormal accumulation of α-syn protein occurs due to the failure of the degradation mechanisms. In [31], they demonstrated two mathematical models to study the effect of two types of transport, active and diffusive, on α-syn transmission in both healthy and diseased axons. They found that the accumulation of α-syn aggregation in Lewy bodies occurs in diseased axons. Sneppen et al [32] studied the relationship between α-syn aggregation and proteasome activity, forming a mathematical model consisting of three compartments: fibrils, fibrils-proteasome complex, and the proteasome. They concluded that α-syn aggregation increases if the ratio between the proteasome and α-syn is reduced below a certain threshold level.

The previous studies on PD [3032] modeled the aggregation and degradation of α-syn inside the neuron cell. In this study, we address the extracellular α-syn and their impact on the progression of PD. We introduce a model that describes the interaction of extracellular α-syn with neurons and the arise response from the innate immune system in the CNS. In particular, we examine the effect of activated microglia in eliminating extracellular α-syn and their impact on healthy and infected neurons. To our knowledge, this is the first mathematical model that displays this relationship. The study is expected to give a better understanding of PD and assist in therapeutic interventions.

The paper is detailed as follows. In section 2, we formulate the model and present a description of the variables and parameters. The feasible region of the model is demonstrated in section 3. The stability of the equilibrium points are investigated in section 4. Numerical simulations adjacent to parameters analysis are illustrated in section 5. Finally, we give a concise conclusion in section 6.

The model describes the relationship between the extracellular α-syn protein and its transmission from infected neurons to healthy neurons. It also includes the innate immune cells' reaction to the presence of protein outside the neurons. In particular, the response of microglia that leads to their activation, which, in chronic activation, may yield to neuronal death due to the cytokines they secrete. We investigate the efficacy of treatments that may reduce the presence of extracellular α-syn or reduce the activation of microglia. The model consists of five separate classes: the density of healthy brain neurons, N(t); the density of infected brain neurons, I(t); the density of extracellular α-syn, αS(t); and the density of rested and activated microglia, Mr(t) and Ma(t), respectively.

The dynamics of the model are represented in figure 1 which is based on the following assumptions. New neurons are formulated at a rate of τ1 due to neurogenesis in the striatum [33] and substantia nigra [3437]. Similarly, microglia proliferate at a rate τ2 [38, 39]. In addition, apoptosis of neurons and microglia is at a rate of μ1 and μ3, respectively. Extracellular α-syn acts in a prion-like manner and transfers into healthy neuron cells at a rate β; as a result, the neuron becomes infected. The aggregation of α-syn within the infected neuron leads to its death at a rate d1. The lysis of infected neurons releases a percentage e of α-syn in the extracellular region. Resting microglia are activated at a rate a due to the behavior of infected neurons and the appearance of α-syn. After the stimulus disappears, activated microglia are restrained back to the resting state at a rate μ2. Treatments for PD depend on either reducing the uptake of extracellular α-syn or minimizing inflammation caused by activated microglia in the CNS. In [40], the review discusses some research to inhibit the cell's uptake of extracellular α-syn under endocytosis. One method was blocking α-syn receptors, particularly the immune receptor LAG3, by directed antibodies. Another method was to restrain the endocytosis of extracellular α-syn by interfering with heparan sulfate proteoglycans on the cell's surface, which arranges the uptake of α-syn. The other therapeutic approach is maintaining neuroprotectivity by limiting inflammation caused by activated microglia. A pharmacological target is to regulate receptors on microglia, such as TLR2 and CB2, to inhibit the activation of microglia and suppress inflammation [41]. We incorporate both treatments in the model by introducing the parameters σ1 and σ2. The parameters represent the effective ratio of the therapeutic drug aiming to overcome extracellular α-syn (σ1) or to control activated microglia (σ2).

Figure 1. The dynamics of model (1).

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The model is represented by a system of nonlinear ordinary differential equations as follows:

Equation (1)

The parameters in model (1) belong to the interval (0, 1]. A summary of all variables and parameters is displayed in table 1.

Table 1. Definition of symbols in model (1).

SymbolDefinitionUnits t Timeday N Density of healthy neurons in the braing/ml I Density of infected neurons in the braing/ml αS Density of extracellular α-syn in the braing/ml MaDensity of activated microgliag/ml MrDensity of resting microgliag/ml τ1Density of new neurons per day due to neurogenesisg/ml/day τ2Density of new microglia per day due to neurogenesisg/ml/day β Neuron infection rateml/g/day μ1Apoptosis rate of neuronsday−1 μ2Annihilation rate of activated microgliaday−1 μ3Apoptosis rate of microgliaday−1 d1Death rate of infected neurons due to α-syn aggregationsday−1 e Percentage of α-syn survival from death of infected neurons. σ1Effective ratio of therapeutic approach to target extracellular α-syn. σ2Effective ratio of therapeutic approach to control activated microglia. a Activation rate of microglia due to extracellular α-syn and infected neuronsml/g/day

In this section, we examine the well-posedness of model (1) by showing the positivity and boundedness of the feasible region for the model. Also, we obtain the steady-state solutions (equilibrium points) of the model.

3.1. Positivity and boundedness

Theorem 1. If the initial values of model (1) are non-negative, N(0) ⩾ 0, I(0) ⩾ 0, αS(0) ⩾ 0, Ma(0) ⩾ 0, and Mr(0) ⩾ 0, then the solutions of the model, N(t), I(t), αS(t), Ma(t), and $_}(t)\left.\right)$, are non-negative for all t > 0.

Proof. Let N(0) ⩾ 0, according to the first equation in model (1), we have,

It can be written as:

Thus,

Integration gives,

Similarly, by applying the same approach to the rest of the equations in system (1), we obtain

Since N(0), I(0), αS(0), Ma(0) and Mr(0) are non-negative and the exponential function is always positive, then the solution (N(t), I(t), αS(t), Ma(t), Mr(t)) is non-negative for all t > 0. □

Theorem 2. The feasible region of model (1),

is positively invariant, where αS0 = αS(0), z ∈ [0, T], and T is a positive number.

Proof. From the first equation in (1), we get

Thus,

Integration gives,

Therefore, lim supt→  N(t) ⩽ τ1/μ1. Since the infected neurons I are a part from the healthy neurons, then I ⩽ τ1/μ1.

From the third equation in (1), we have

Integration gives,

where z ∈ [0, T], and T is a positive number. Similarly, from (1), Mr ⩽ τ2/μ3. Accordingly, Ma ⩽ τ2/μ3.

Next, we prove that Ω is a positively invariant set. Let (N(0), I(0), αS(0), Ma(0), Mr(0)) be in Ω, from (1), we have

At the boundary,

Thus, the solutions (N(t), I(t), αS(t), Ma(t), Mr(t)) stay in Ω. Hence Ω is positively invariant. □

3.2. Equilibria of the model

Equilibrium points of a dynamical model are found by obtaining the steady-state solutions of the model. Thus, we set all the rates in (1) to zero, that is,

Equation (2)Equation (3)Equation (4)Equation (5)Equation (6)

Then solving the system (2)–(6) for the state variables. If αS = 0, then from (3), I = 0. Accordingly, from (5), Ma = 0. By substituting for αS = I = Ma = 0 in (2) and (6), we obtain N = τ1/μ1 and Mr = τ2/μ3, respectively. Hence, the free equilibrium point is

which exists always with no conditions. If αS ≠ 0, then by adding the equations in (5) and (6), we have

Equation (7)

Substituting (7) into (5), we get

Equation (8)

From (4) we obtain,

Equation (9)

Substituting (9) into (8), we get

Equation (10)

Substituting (9) and (10) into (3),we obtain

Equation (11)

Substituting (10) and (11) into (2), we have

Equation (12)

Equation (12) can be rewritten as a polynomial equation of the fourth degree in Ma:

Equation (13)

where,

By using Descartes' rule of signs [42], we can get the number of possible positive real zeros in (13) based on the signs of its coefficients. The number of positive roots is equal to the changes in coefficients' signs of P(Ma), or less than that number by an even integer. All the coefficients in (13) are positive except for ζ1. However, an inspection of all the possible cases yields only one change in the signs (see table 2); therefore, there exists one positive root for P(Ma), that is, $_}^$.

Table 2. The probabilities of the positive roots of polynomial.

Cases ζ4ζ3ζ2ζ1ζ0No. of changes in signNo. of + roots1+ + + −−112+ + + +−11

Hence, model (1) produces another equilibrium point, E*. We summarize the results in the following theorem.

Theorem 3. Model (1) has two equilibrium points:

Free equilibrium point, $^=(^,0,0,0,_}^)$, where $^=\frac_}_}$ and $_}^=\frac_}_}$.Endemic equilibrium point, $^=(^,^,\alpha ^,_}^,_}^)$, where $_}^=_/_$ and N*, I*, and αS* are given in terms of $_}^$ in equations (11), (9) and (10), respectively, and $_}^$ satisfies equation (13).

We investigate the local stability of the equilibrium points of model (1) using the linearization method [43].

Theorem 4. The free equilibrium point, $^=(^,0,0,0,_}^)$, of model (1) is unstable.

Proof. The Jacobian matrix of model (1) evaluated at E0 is:

Then, the characteristic equation, |J − λI| = 0, yield

Equation (14)

Clearly λ1 = −μ1, λ2 = −μ2 and λ3 = −μ3. As for λ4,5, they satisfy the equation:

Equation (15)

where

The general roots of (15) have the form:

Since α1 > 0 and α2 < 0, then $_< \sqrt_^-4_}$. Thus, one sign of the eigenvalues λ4,5 is negative and the other is positive. Consequently, not all eigenvalues are negative; hence, E0 is unstable. □

Theorem 5. The endemic equilibrium point, $^=(^,^,\alpha ^,_}^,_}^)$, of model (1) is locally asymptotically stable.

Proof. The proof is demonstrated in the appendix. □

Numerical simulations of model (1) are demonstrated using MATLAB software. First, we run some numerical experiments to reinforce the previous qualitative analysis. Then, we explore the impact of the therapeutic intervention parameters on the dynamics of PD.

5.1. Numerical experiments

We perform three experiments by solving model (1) numerically. The values of the parameters are presented in table 3. The simulations are illustrated in figure 2 for different initial conditions. The figure shows that in all three experiments, the solution curves tend to the equilibrium, E* = (0.0037, 0.1333, 0.0486, 0.0205, 0.0333). Hence, the numerical solution of model (1) agrees with the qualitative results.

Table 3. Parameter values.

SymbolValueUnitsReference τ10.0001g/ml/dayEstimated τ20.0005g/ml/dayEstimated β 0.5ml/g/dayEstimated μ11.9 × 10−4day−1[29] μ26 × 10−3day−1[29] μ30.015day−1[29] d13.4 × 10−4day−1[29] e 0.5 Estimated σ10.01 Estimated σ20.01 Estimated a 2

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