先天免疫反应数学建模及动力学分析
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摘要
先天免疫系统是一个复杂的通路互联网络,其中既有反馈或前馈回路,也有通路交谈以及包括转录和翻译后修饰在内的各种不同调控机制.同时,宿主与各种病原体的相互作用,也增加了系统的复杂性,这包括先天免疫受体信号的诱导和识别,细胞抗病毒反应,以及病毒的逃逸机制等.传统的研究方法从某一候选基因出发,分析病毒感染的先天免疫反应,虽然可行,但也存在许多的不足与局限性,限制了我们从系统水平上去分析和理解宿主-病原体的复杂关系,而且也没有办法将已知成分的相关信息结合起来一并考虑.因此,传统的简化方法无法阐明分子系统的整合效应和系统特性,最终只能是孤立地获得一些关于先天免疫反应的个别结论.相比之下,系统生物学方法更关注生物系统组成成分之间复杂的相互作用,以及由这些相互作用所产生的系统行为和生物学功能.因而,从系统论的角度出发,对宿主-病原体相互作用关系进行研究和分析,将更有助于我们全面和“无偏见”地理解和认识先天免疫反应.
本文利用系统生物学方法对先天免疫反应建立数学模型,从系统水平上以干扰素为核心,对先天免疫相关通路进行描述,开展动力学行为分析,无疑将加深对先天免疫反应各主要参与组分之间相互作用关系的理解,从而使得在系统水平上为揭示先天免疫系统抗病毒的作用原理成为可能.为此,本文的主要工作主要集中在以下几个方面：
首先,为了便于把握先天免疫反应的整个过程,我们将干扰素诱导和效应两个阶段结合起来一并考虑,化繁就简,针对其中的关键环节,根据质量作用原理,提出一个三阶的多时滞微分方程模型.依照希尔系数n2=1及n2>1分两种情况,对模型的稳定性进行理论分析,讨论了先天免疫系统抗病毒完全起作用、部分起作用、以及作用失效时所对应的参数空间,同时,研究了系统具有不同时滞情形下其行为如何.我们发现,如果干扰素的相对强度超过一定的阈值,免疫系统就能将病毒逐步消除.而干扰素自反馈环节的协同效应有助于诱发系统的双稳现象,病毒降解率的增加能够引起系统发生霍夫分岔产生周期振荡,以及某些环节的时滞不仅可以诱导系统振荡而且在一定范围内还能镇定失稳系统,将系统从一个不稳定或振荡的状态切换到一个稳定状态.这说明,在先天免疫反应中的某些时滞对于减少由病毒感染引起的病理损伤是有益的,这个有趣的现象并没有被以前文献所描述的.为了说明理论分析的正确性,我们对所有结果进行了数值模拟,并结合生物学实验验证了模型的有效性.
其次,如果忽略模型中的特定生物学背景,本模型可以被看作是一个含有正、负自反馈环路的耦合系统,我们会发现,该系统可以在单稳态,双稳态和振荡之间进行切换.为了更好地阐明正、负反馈如何诱导了系统复杂的动力学行为,我们对系统进行双参数分岔分析和数值模拟.结果表明,系统展现出丰富的分岔现象(鞍结分岔、跨临界分岔、超临界霍夫分岔、亚临界霍夫分岔等),并且,自反馈、正反馈以及负反馈强度在诱导系统复杂动力学行为的方面具有重要作用,而其它系统参数,如相对活性系数K以及相对降解率α2和α4对系统行为没有明显的影响.同时,进一步说明了正反馈仅仅只是诱导双稳的必要条件,协同效应(希尔系数n2≥2)是有助于加强系统的非线性从而易于诱导系统产生双稳.而负反馈只有在适当的正反馈强度下才能导致系统的振荡,并且可以通过调节正、负反馈的强度调节系统振荡的振幅和周期.该模型表现出来的复杂动力学行为可以用来设计一个具有特定生物学功能的生物网络.
最后,本文还利用最优控制理论建立适当的性能函数,讨论在先天免疫系统防护失效情形下如何采取最优的控制策略取得最好的抗病毒治疗效果.我们发现,在基本情形下,虽然三种控制策略各自所产生的最优控制都能有效杀灭病毒,但控制策略1所产生的最优控制不仅产生的费用最小,而且能够使得干扰素以及抗病毒蛋白较快地恢复正常水平.当权重发生变化时,策略1或策略2都将成备选方案,其中,当控制律权重变小或病毒状态权重变大时,策略1下的最优控制将形成最好的治疗方案,与之相反,当控制律权重变大或病毒状态权重变小时,策略2下的最优控制将成为最好的治疗措施.当治疗效率因子变小时,策略2所产生的最优控制不仅产生的费用最低,而且在整个治疗期间的控制律波动性小,可操作性强,因此,通过增强干扰素活性(策略2)将成为最好治疗方案.当治疗效率因子变大时,其中阻断病毒复制(策略1)所产生的最优控制重新成为最好的选择.在所有的讨论情形下,控制策略3都无法形成最好的治疗方案,这说明在疾病治疗时,有时独立的治疗措施或许在费用以及可操作性方面更具有优势.同时,提高治疗效率因子对于实际的疾病治疗不仅将有效降低费用,而且还能增加治疗方案的选择余地.
Innate immune system is a complex network of passages which include not only feed-back or feedforward circuits, but also cross-talks or transcriptional and post-translational modifications. On the other hand, the interactions of host and various pathogens, in-volved the induction and identification of innate immune receptor signaling, cell antiviral responses and viral escape mechanism, also increase the complexity of the system. The traditional method, analyzing the innate immune responses of viral infection from a candidate gene, is feasible, but no doubt restricts understanding of host-pathogen com-plex relationship from system level and has no way to integrate the known information. Thus, the simple method can not elucidate the integration effect and system charac-teristics about the molecular system, and we can only get a few isolated conclusions on the innate immune responses. To the contrast, systems biology is more concerned about the complex interactions between biological system components, as well as the system behaviors and biological functions arising from the interactions. Therefore, to understand the innate immune responses comprehensively, through the analysis of the host-pathogen interactions is helpful from the perspective of system theory.
By systems biology methods, the dissertation establishes the mathematical model of the innate immune responses based interferon as core component, describing the innate immune related pathways and carrying out dynamic behavior analysis. This will undoubtedly enhance understanding the interactions between main components in the innate immune responses, so that it is possible to reveal the innate immune system antiviral mechanism from the system level. Therefore, the main works of the dissertation are focused on the following aspects:
First, in order to understand the whole process of the innate immune responses, we propose a model with three order delays differential equations about virus, interferon and antiviral protein based the mass action law, considering the generation phase and the effect phase of interferon together. In accordance with the Hill coefficient of n2=1and n2>1, we analyze the stability of the model, discuss the parameter space when the innate immune system clears all virus, plays part role or fails antiviral ability and investigate the behaviors under different time delays. We found that the innate immune system can guarantee to remove virus gradually if the relative strength of interferon exceeds a certain threshold. The synergistic effect of interferon self-feedback can induce bistability and increasing the viral fatality rate can cause oscillation by a Hopf bifurcation. Some delays can not only induce the oscillation of the system but also calm instable system within a certain range, switching the system from an unstable or oscillatory state to a stable state. These results show that some delays of innate immune responses are beneficial to reduce the pathological injury caused by viral infection and this interesting phenomenon has not been previously described in the literatures. This helps us to understand the antiviral mechanism of innate immune system. In order to illustrate the correctness of the theoretical analysis, we carry out numerical simulation for all results and validate the model combined with the biological experiments.
Second, the model can be viewed as a regulatory system with a negative feedback coupled with two positive auto-feedback loops if ignoring the specific biological back-ground, which can switch in a single stable, bistable or oscillation state. In order to better illustrate the positive and negative feedback how to induce complex dynamics, we carry out two-parameter bifurcation analysis and numerical simulation and we find that the system exhibits rich bifurcation phenomena (for example, saddle node bifur-cation, transcritical bifurcation, and supercritical or subcritical Hopf bifurcation). And the auto-positive feedback and negative feedback strength (σ1and σ2) plays an impor-tant role in the induction of complicated dynamic behaviors. However, other system parameters, such as relative reactive coefficient K and the relative degradation rates of α2and α4, have no significant effects on the system behaviors. At the same time, we again confirm that the positive feedback is just the necessary condition for bistability and synergistic effect (the Hill coefficient n2≥2) is helpful to induce bistability by strengthening the nonlinear of system. Negative feedback can cause oscillation under appropriate positive feedback strength and we can adjust the amplitude and period of oscillation by regulating strength of positive or negative feedback. The model with the complex dynamic behaviors can be used to design a network with specific biological function.
Finally, establishing the mathematical model not only can be used to predict the behaviors of the system, but also can help us to find a proper method to intervene and control the behaviors of the system. Therefore, in the last part of the dissertation, we discuss how to adopt optimal control strategy to obtain a good therapeutic effect when innate immune system failure in protection by using the optimal control theory. We find that in the basic case, the three control strategies can effectively kill all viruses, but Strategy1will be the best treatment option, which not only lead to the smallest cost but also make the antiviral protein and interferon quickly returning to normal levels. When the weight changes, Strategy1or Strategy2will be the best optional control. When the control law weights are small or virus-weight becomes large, Strategy1will be the best treatment option. On the contrast, when the weight of the control law weights increase or virus-weight decreases, the optimal control will be Strategy2. When the treatment efficiency factor decreases, the control u2by enhancing interferon activity in Strategy2, becoming the best treatment options, not only minimizes the cost, but also reduces control volatility throughout the treatment period. When treatment efficiency factor increases, Strategy1again becomes the best choice. In all our discussions cases, Strategy3is unable to become the best treatment, indicating that a separate treatment sometimes has more advantages, including the smallest cost and operability, in the treatment of the disease. And improving efficiency factor for treatment will not only reduce the cost, but also provide more the treatment choices in the actual treatment of diseases.
本文利用系统生物学方法对先天免疫反应建立数学模型,从系统水平上以干扰素为核心,对先天免疫相关通路进行描述,开展动力学行为分析,无疑将加深对先天免疫反应各主要参与组分之间相互作用关系的理解,从而使得在系统水平上为揭示先天免疫系统抗病毒的作用原理成为可能.为此,本文的主要工作主要集中在以下几个方面：
首先,为了便于把握先天免疫反应的整个过程,我们将干扰素诱导和效应两个阶段结合起来一并考虑,化繁就简,针对其中的关键环节,根据质量作用原理,提出一个三阶的多时滞微分方程模型.依照希尔系数n2=1及n2>1分两种情况,对模型的稳定性进行理论分析,讨论了先天免疫系统抗病毒完全起作用、部分起作用、以及作用失效时所对应的参数空间,同时,研究了系统具有不同时滞情形下其行为如何.我们发现,如果干扰素的相对强度超过一定的阈值,免疫系统就能将病毒逐步消除.而干扰素自反馈环节的协同效应有助于诱发系统的双稳现象,病毒降解率的增加能够引起系统发生霍夫分岔产生周期振荡,以及某些环节的时滞不仅可以诱导系统振荡而且在一定范围内还能镇定失稳系统,将系统从一个不稳定或振荡的状态切换到一个稳定状态.这说明,在先天免疫反应中的某些时滞对于减少由病毒感染引起的病理损伤是有益的,这个有趣的现象并没有被以前文献所描述的.为了说明理论分析的正确性,我们对所有结果进行了数值模拟,并结合生物学实验验证了模型的有效性.
其次,如果忽略模型中的特定生物学背景,本模型可以被看作是一个含有正、负自反馈环路的耦合系统,我们会发现,该系统可以在单稳态,双稳态和振荡之间进行切换.为了更好地阐明正、负反馈如何诱导了系统复杂的动力学行为,我们对系统进行双参数分岔分析和数值模拟.结果表明,系统展现出丰富的分岔现象(鞍结分岔、跨临界分岔、超临界霍夫分岔、亚临界霍夫分岔等),并且,自反馈、正反馈以及负反馈强度在诱导系统复杂动力学行为的方面具有重要作用,而其它系统参数,如相对活性系数K以及相对降解率α2和α4对系统行为没有明显的影响.同时,进一步说明了正反馈仅仅只是诱导双稳的必要条件,协同效应(希尔系数n2≥2)是有助于加强系统的非线性从而易于诱导系统产生双稳.而负反馈只有在适当的正反馈强度下才能导致系统的振荡,并且可以通过调节正、负反馈的强度调节系统振荡的振幅和周期.该模型表现出来的复杂动力学行为可以用来设计一个具有特定生物学功能的生物网络.
最后,本文还利用最优控制理论建立适当的性能函数,讨论在先天免疫系统防护失效情形下如何采取最优的控制策略取得最好的抗病毒治疗效果.我们发现,在基本情形下,虽然三种控制策略各自所产生的最优控制都能有效杀灭病毒,但控制策略1所产生的最优控制不仅产生的费用最小,而且能够使得干扰素以及抗病毒蛋白较快地恢复正常水平.当权重发生变化时,策略1或策略2都将成备选方案,其中,当控制律权重变小或病毒状态权重变大时,策略1下的最优控制将形成最好的治疗方案,与之相反,当控制律权重变大或病毒状态权重变小时,策略2下的最优控制将成为最好的治疗措施.当治疗效率因子变小时,策略2所产生的最优控制不仅产生的费用最低,而且在整个治疗期间的控制律波动性小,可操作性强,因此,通过增强干扰素活性(策略2)将成为最好治疗方案.当治疗效率因子变大时,其中阻断病毒复制(策略1)所产生的最优控制重新成为最好的选择.在所有的讨论情形下,控制策略3都无法形成最好的治疗方案,这说明在疾病治疗时,有时独立的治疗措施或许在费用以及可操作性方面更具有优势.同时,提高治疗效率因子对于实际的疾病治疗不仅将有效降低费用,而且还能增加治疗方案的选择余地.
Innate immune system is a complex network of passages which include not only feed-back or feedforward circuits, but also cross-talks or transcriptional and post-translational modifications. On the other hand, the interactions of host and various pathogens, in-volved the induction and identification of innate immune receptor signaling, cell antiviral responses and viral escape mechanism, also increase the complexity of the system. The traditional method, analyzing the innate immune responses of viral infection from a candidate gene, is feasible, but no doubt restricts understanding of host-pathogen com-plex relationship from system level and has no way to integrate the known information. Thus, the simple method can not elucidate the integration effect and system charac-teristics about the molecular system, and we can only get a few isolated conclusions on the innate immune responses. To the contrast, systems biology is more concerned about the complex interactions between biological system components, as well as the system behaviors and biological functions arising from the interactions. Therefore, to understand the innate immune responses comprehensively, through the analysis of the host-pathogen interactions is helpful from the perspective of system theory.
By systems biology methods, the dissertation establishes the mathematical model of the innate immune responses based interferon as core component, describing the innate immune related pathways and carrying out dynamic behavior analysis. This will undoubtedly enhance understanding the interactions between main components in the innate immune responses, so that it is possible to reveal the innate immune system antiviral mechanism from the system level. Therefore, the main works of the dissertation are focused on the following aspects:
First, in order to understand the whole process of the innate immune responses, we propose a model with three order delays differential equations about virus, interferon and antiviral protein based the mass action law, considering the generation phase and the effect phase of interferon together. In accordance with the Hill coefficient of n2=1and n2>1, we analyze the stability of the model, discuss the parameter space when the innate immune system clears all virus, plays part role or fails antiviral ability and investigate the behaviors under different time delays. We found that the innate immune system can guarantee to remove virus gradually if the relative strength of interferon exceeds a certain threshold. The synergistic effect of interferon self-feedback can induce bistability and increasing the viral fatality rate can cause oscillation by a Hopf bifurcation. Some delays can not only induce the oscillation of the system but also calm instable system within a certain range, switching the system from an unstable or oscillatory state to a stable state. These results show that some delays of innate immune responses are beneficial to reduce the pathological injury caused by viral infection and this interesting phenomenon has not been previously described in the literatures. This helps us to understand the antiviral mechanism of innate immune system. In order to illustrate the correctness of the theoretical analysis, we carry out numerical simulation for all results and validate the model combined with the biological experiments.
Second, the model can be viewed as a regulatory system with a negative feedback coupled with two positive auto-feedback loops if ignoring the specific biological back-ground, which can switch in a single stable, bistable or oscillation state. In order to better illustrate the positive and negative feedback how to induce complex dynamics, we carry out two-parameter bifurcation analysis and numerical simulation and we find that the system exhibits rich bifurcation phenomena (for example, saddle node bifur-cation, transcritical bifurcation, and supercritical or subcritical Hopf bifurcation). And the auto-positive feedback and negative feedback strength (σ1and σ2) plays an impor-tant role in the induction of complicated dynamic behaviors. However, other system parameters, such as relative reactive coefficient K and the relative degradation rates of α2and α4, have no significant effects on the system behaviors. At the same time, we again confirm that the positive feedback is just the necessary condition for bistability and synergistic effect (the Hill coefficient n2≥2) is helpful to induce bistability by strengthening the nonlinear of system. Negative feedback can cause oscillation under appropriate positive feedback strength and we can adjust the amplitude and period of oscillation by regulating strength of positive or negative feedback. The model with the complex dynamic behaviors can be used to design a network with specific biological function.
Finally, establishing the mathematical model not only can be used to predict the behaviors of the system, but also can help us to find a proper method to intervene and control the behaviors of the system. Therefore, in the last part of the dissertation, we discuss how to adopt optimal control strategy to obtain a good therapeutic effect when innate immune system failure in protection by using the optimal control theory. We find that in the basic case, the three control strategies can effectively kill all viruses, but Strategy1will be the best treatment option, which not only lead to the smallest cost but also make the antiviral protein and interferon quickly returning to normal levels. When the weight changes, Strategy1or Strategy2will be the best optional control. When the control law weights are small or virus-weight becomes large, Strategy1will be the best treatment option. On the contrast, when the weight of the control law weights increase or virus-weight decreases, the optimal control will be Strategy2. When the treatment efficiency factor decreases, the control u2by enhancing interferon activity in Strategy2, becoming the best treatment options, not only minimizes the cost, but also reduces control volatility throughout the treatment period. When treatment efficiency factor increases, Strategy1again becomes the best choice. In all our discussions cases, Strategy3is unable to become the best treatment, indicating that a separate treatment sometimes has more advantages, including the smallest cost and operability, in the treatment of the disease. And improving efficiency factor for treatment will not only reduce the cost, but also provide more the treatment choices in the actual treatment of diseases.
引文
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