非线性生物系统中的时空噪声动力学
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摘要
本学位论文以非线性动力学和统计物理的随机过程理论为基本理论,研究了肿瘤体系、被捕食者与捕食者生态体系、互利共生生态体系等生物系统中的时空噪声动力学行为。本论文的研究对深入认识肿瘤生长扩散规律以及在传统化疗和放疗基础上探讨新的免疫疗法具有前瞻性,对临床治疗有一定指导意义;同时对保持涨落环境下的生态系统的平衡、稳定与发展也给出了重要启示。主要包括三大部分:
(一)肿瘤体系的时空噪声效应部分
首先,在Logistic生长模型中引入了加性和乘性关联噪声,通过求解其对应的Fokker-Planck方程,揭示了乘性噪声使肿瘤生长发生相变,发现肿瘤细胞数目以及生长规律随乘性噪声的强度增加,表现出随机共振特征。在一定的乘性噪声强度下,噪声非但不能使肿瘤细胞灭绝,反而促使其增长。噪声关联对随机共振特征起弱化作用,同源强噪声使肿瘤远离衰亡而稳定生长和扩张。
其次,扩展到酶动力学的范畴,分别引入乘性噪声、免疫效应及治疗性外力等综合作用,考察此时肿瘤细胞的定量演化规律。结果表明:(1)在治疗强度与治疗占空比组成的参数空间中存在一个区分有效治疗与无效治疗的临界治疗边界,其变化规律具有指数衰减特征;(2)引入不同占空比和波形的周期性外场模拟不同临床治疗方案,对肿瘤细胞数量的增长和衰减进行定量计算,结果发现治疗强度的减小使临界治疗占空比显著增加;(3)免疫作用增强使临界治疗强度下降。初步的临床观察结果与动力学理论模拟计算结果在变化趋势上基本一致;(4)对于受周期性调节作用的肿瘤体系,发现乘性噪声可以诱导肿瘤治疗出现随机共振效应。
最后,研究了一维和二维肿瘤细胞的生长扩散问题,求出相应的平均场理论解。同时,仿真计算二维离散空间中具有不同扩散能力的肿瘤体系的生长形貌和相应的时空斑图,通过引入结构因子对肿瘤的空间形态进行量化表征。结果表明,乘性时空噪声对肿瘤生长起“双刃剑”作用,即非但抑制激发态肿瘤,而且唤醒休眠态肿瘤。均匀时空可能导致肿瘤在空间呈现膨胀性生长,而非均匀时空却导致肿瘤呈浸润性生长,其取决于肿瘤细胞生长的时空环境的涨落程度。
(二)生态系统的关联噪声研究部分
分别考虑了被捕食者与捕食者生态系统和互利共生生态系统,通过引入关联的加性噪声和乘性噪声来表现生态系统的自然性涨落和环境外力涨落。结果发现,这些涨落一方面对被捕食者与捕食者生态系统的平衡出现破坏性作用,另一方面对共生生态系统的个体同步发展起到促进作用。在被捕食与捕食生态系统中,涨落的环境总是在催生和淘汰着捕食者,但对被捕食者的影响很小。在共生生态系统中,由于共生个体间的密切利益关系,它们在千变万化的自然中总是同步演化、共同进退。
(三)关联噪声诱导有序现象部分
以Ising双稳系统为例,阐述关联噪声在非线性系统中的响应问题,如噪声诱导相变、多参数空间的随机共振和重入(Reentrance)现象等等,探讨处理关联噪声问题的统计物理学方法以及相应的动力学理论,为生物系统中的关联噪声问题的解决提供随机方法,同时也为生物系统的类似噪声诱导有序现象提供启发性思维。
In this dissertation, using the methods of nonlinear dynamics and stochastic process in statistical physics, I have studied spatiotemporal noise dynamics in selected biosystems, e.g., tumor systems, prey-predator ecosystems and mutualism systems. This research has looked insight into the growth law of tumor cells and provided many illuminations to the traditional cancer therapy. At the same time, it is also suggested that the environmental protection is quite necessary in order to keep the balance and the stability of ecosystems. It includes three sections:
1. The spatiotemporal noise effects on tumor systems
Firstly, I have studied the effect of additive and multiplicative noises on the growth of a tumor based on a logistic growth model. The steady-state probability distribution and the average population of the tumor cells were given to explain the important roles of correlated noises in the tumor growth. It had been reported that multiplicative noise induces a phase transition of the tumor growth from an uni-stable state to a bi-stable state; the relationship between the intensity of multiplicative noise and the population of the tumor cells shows a stochastic resonance-like characteristic. It was also confirmed that additive noise weakened rather than extinguished the tumor growth. Homologous noises, however, promoted the growth of a tumor. I also discussed about the relationship between the tumor treatment and the model.
Secondly, considering the growth of tumor cells modeled by an enzyme dynamic process under an immune surveillance, I have studied in anti-tumor immunotherapy the single-variable growth dynamics of tumor cells subjected to a multiplicative noise and an external therapy intervention simultaneously. There are four main findings: (1) Two simulative parameters of therapy, i.e., therapy intensity and therapy duty-cycle, were introduced to characterize a treatment process similar to a tumor clinic therapy. There exists a critical therapy boundary which, in an exponent-decaying form, divides the parameter region of therapy into an invalid and a valid treatment zone, respectively. (2) A greater critical therapy duty-cycle is necessary to achieve a valid treatment for a lower therapy intensity while the critical therapy intensity decreases accordingly with an enhancing immunity. (3) The primary clinic observation of the patients with the typical non-hodgekin’s lymphoma basically agreed with the dynamic simulations. (4) In an anti-tumor system modulated by a seasonal external field, for optimally selected values of the multiplicative noise intensity, stochastic resonance can be observed, which is manifested by the quasi-symmetry of two potential minima.
Finally, the studies of one and two dimensional models of spatially extended anti-tumor system with a fluctuation in growth rate was carried out. It is suggested that the spatiotemporal noise, assumed to reflect the environmental fluctuation in a spatially extended tumor system, can induce nonequilibrium phase transition. In this thesis I introduce the structure factor to reveal the invasive tumor growth quantitatively. The multiplicative noise is found to have opposite effects: the positive effect on a non-excited tumor and the negative effect on an excited tumor. The homogenous environment can lead the tumor cells to expansive growth, while the inhomogenous environment may result in the infiltrative growth of the tumor cells. The different responses were determined by the level of environmental fluctuation.
2. Correlated noises in ecosystems
I have investigated a Volterra ecosystem driven by correlated noises. The fluctuation in the death rate of the predator induces an increase in population densities of the predators. The fluctuation in the growth rate of the prey, however, leads the predators to decay. It is reported that the predators undergo sensitivity to a random environment, whereas the preys exhibit a surprising endurance to the same stochastic factor. The predators are of better stability under strong correlation of noises.
Understanding the cause of the synchronization of population evolution is an important issue for ecological improvement. Here I present a Lotka-Volterra-type model driven by two correlated environmental noises and show, via theoretical analysis and direct simulation, that noise correlation can induce a synchronization of the mutualists. The time series of mutual species exhibit a chaotic-like fluctuation, which is independent of the noise correlation, however, the chaotic fluctuation of mutual species ratio decreases with the noise correlation. A quantitative parameter defined for characterizing chaotic fluctuation provides a good approach to measure when the complete synchronization happens.
3. Correlated noises induced order phenomena
Here I report the phenomenon of the nonequilibrium dynamical phase transition (NDPT) appearing in a kinetic Ising spin system (ISS) subjected to a joint application of a determinant external field and the stochastic mutually correlated noises simultaneously. For example, within the multi-parameter space, the dynamic order parameter takes on a stochastic resonance with reentrant trend against noise intensity. This method is expected to be applied in the biosystems with correlated noises.
(一)肿瘤体系的时空噪声效应部分
首先,在Logistic生长模型中引入了加性和乘性关联噪声,通过求解其对应的Fokker-Planck方程,揭示了乘性噪声使肿瘤生长发生相变,发现肿瘤细胞数目以及生长规律随乘性噪声的强度增加,表现出随机共振特征。在一定的乘性噪声强度下,噪声非但不能使肿瘤细胞灭绝,反而促使其增长。噪声关联对随机共振特征起弱化作用,同源强噪声使肿瘤远离衰亡而稳定生长和扩张。
其次,扩展到酶动力学的范畴,分别引入乘性噪声、免疫效应及治疗性外力等综合作用,考察此时肿瘤细胞的定量演化规律。结果表明:(1)在治疗强度与治疗占空比组成的参数空间中存在一个区分有效治疗与无效治疗的临界治疗边界,其变化规律具有指数衰减特征;(2)引入不同占空比和波形的周期性外场模拟不同临床治疗方案,对肿瘤细胞数量的增长和衰减进行定量计算,结果发现治疗强度的减小使临界治疗占空比显著增加;(3)免疫作用增强使临界治疗强度下降。初步的临床观察结果与动力学理论模拟计算结果在变化趋势上基本一致;(4)对于受周期性调节作用的肿瘤体系,发现乘性噪声可以诱导肿瘤治疗出现随机共振效应。
最后,研究了一维和二维肿瘤细胞的生长扩散问题,求出相应的平均场理论解。同时,仿真计算二维离散空间中具有不同扩散能力的肿瘤体系的生长形貌和相应的时空斑图,通过引入结构因子对肿瘤的空间形态进行量化表征。结果表明,乘性时空噪声对肿瘤生长起“双刃剑”作用,即非但抑制激发态肿瘤,而且唤醒休眠态肿瘤。均匀时空可能导致肿瘤在空间呈现膨胀性生长,而非均匀时空却导致肿瘤呈浸润性生长,其取决于肿瘤细胞生长的时空环境的涨落程度。
(二)生态系统的关联噪声研究部分
分别考虑了被捕食者与捕食者生态系统和互利共生生态系统,通过引入关联的加性噪声和乘性噪声来表现生态系统的自然性涨落和环境外力涨落。结果发现,这些涨落一方面对被捕食者与捕食者生态系统的平衡出现破坏性作用,另一方面对共生生态系统的个体同步发展起到促进作用。在被捕食与捕食生态系统中,涨落的环境总是在催生和淘汰着捕食者,但对被捕食者的影响很小。在共生生态系统中,由于共生个体间的密切利益关系,它们在千变万化的自然中总是同步演化、共同进退。
(三)关联噪声诱导有序现象部分
以Ising双稳系统为例,阐述关联噪声在非线性系统中的响应问题,如噪声诱导相变、多参数空间的随机共振和重入(Reentrance)现象等等,探讨处理关联噪声问题的统计物理学方法以及相应的动力学理论,为生物系统中的关联噪声问题的解决提供随机方法,同时也为生物系统的类似噪声诱导有序现象提供启发性思维。
In this dissertation, using the methods of nonlinear dynamics and stochastic process in statistical physics, I have studied spatiotemporal noise dynamics in selected biosystems, e.g., tumor systems, prey-predator ecosystems and mutualism systems. This research has looked insight into the growth law of tumor cells and provided many illuminations to the traditional cancer therapy. At the same time, it is also suggested that the environmental protection is quite necessary in order to keep the balance and the stability of ecosystems. It includes three sections:
1. The spatiotemporal noise effects on tumor systems
Firstly, I have studied the effect of additive and multiplicative noises on the growth of a tumor based on a logistic growth model. The steady-state probability distribution and the average population of the tumor cells were given to explain the important roles of correlated noises in the tumor growth. It had been reported that multiplicative noise induces a phase transition of the tumor growth from an uni-stable state to a bi-stable state; the relationship between the intensity of multiplicative noise and the population of the tumor cells shows a stochastic resonance-like characteristic. It was also confirmed that additive noise weakened rather than extinguished the tumor growth. Homologous noises, however, promoted the growth of a tumor. I also discussed about the relationship between the tumor treatment and the model.
Secondly, considering the growth of tumor cells modeled by an enzyme dynamic process under an immune surveillance, I have studied in anti-tumor immunotherapy the single-variable growth dynamics of tumor cells subjected to a multiplicative noise and an external therapy intervention simultaneously. There are four main findings: (1) Two simulative parameters of therapy, i.e., therapy intensity and therapy duty-cycle, were introduced to characterize a treatment process similar to a tumor clinic therapy. There exists a critical therapy boundary which, in an exponent-decaying form, divides the parameter region of therapy into an invalid and a valid treatment zone, respectively. (2) A greater critical therapy duty-cycle is necessary to achieve a valid treatment for a lower therapy intensity while the critical therapy intensity decreases accordingly with an enhancing immunity. (3) The primary clinic observation of the patients with the typical non-hodgekin’s lymphoma basically agreed with the dynamic simulations. (4) In an anti-tumor system modulated by a seasonal external field, for optimally selected values of the multiplicative noise intensity, stochastic resonance can be observed, which is manifested by the quasi-symmetry of two potential minima.
Finally, the studies of one and two dimensional models of spatially extended anti-tumor system with a fluctuation in growth rate was carried out. It is suggested that the spatiotemporal noise, assumed to reflect the environmental fluctuation in a spatially extended tumor system, can induce nonequilibrium phase transition. In this thesis I introduce the structure factor to reveal the invasive tumor growth quantitatively. The multiplicative noise is found to have opposite effects: the positive effect on a non-excited tumor and the negative effect on an excited tumor. The homogenous environment can lead the tumor cells to expansive growth, while the inhomogenous environment may result in the infiltrative growth of the tumor cells. The different responses were determined by the level of environmental fluctuation.
2. Correlated noises in ecosystems
I have investigated a Volterra ecosystem driven by correlated noises. The fluctuation in the death rate of the predator induces an increase in population densities of the predators. The fluctuation in the growth rate of the prey, however, leads the predators to decay. It is reported that the predators undergo sensitivity to a random environment, whereas the preys exhibit a surprising endurance to the same stochastic factor. The predators are of better stability under strong correlation of noises.
Understanding the cause of the synchronization of population evolution is an important issue for ecological improvement. Here I present a Lotka-Volterra-type model driven by two correlated environmental noises and show, via theoretical analysis and direct simulation, that noise correlation can induce a synchronization of the mutualists. The time series of mutual species exhibit a chaotic-like fluctuation, which is independent of the noise correlation, however, the chaotic fluctuation of mutual species ratio decreases with the noise correlation. A quantitative parameter defined for characterizing chaotic fluctuation provides a good approach to measure when the complete synchronization happens.
3. Correlated noises induced order phenomena
Here I report the phenomenon of the nonequilibrium dynamical phase transition (NDPT) appearing in a kinetic Ising spin system (ISS) subjected to a joint application of a determinant external field and the stochastic mutually correlated noises simultaneously. For example, within the multi-parameter space, the dynamic order parameter takes on a stochastic resonance with reentrant trend against noise intensity. This method is expected to be applied in the biosystems with correlated noises.
引文
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