红茶提取物茶黄素酸和尼莫地平通过线粒体途径凋亡介导的协同神经保护作用模型的研究

Modeling the neuro-protection of theaflavic acid from black tea and its synergy with nimodipine via mitochondria apoptotic pathway 红茶提取物茶黄素酸和尼莫地平通过线粒体途径凋亡介导的协同神经保护作用模型的研究 first-author corresp Tingzhe SUN, confucian007@126.com Tingzhe SUN, confucian007@126.com confucian007@126.com https://orcid.org/0000-0002-9575-4993 School of Life Sciences, the Province Key Laboratory of the Biodiversity Study and Ecology Conservation in Southwest Anhui, Anqing Normal University, Anqing246133, China School of Life Sciences, the Province Key Laboratory of the Biodiversity Study and Ecology Conservation in Southwest Anhui, Anqing Normal University, Anqing246133, China Ischemic stroke presents a leading cause of mortality and morbidity worldwide. Theaflavic acid (TFA) is a theaflavin isolated from black tea that exerts a potentially neuro-protective effect. However, the dynamic properties of TFA-mediated protection remain largely unknown. In the current study, we evaluated the function of TFA in the mitochondria apoptotic pathway using mathematical modeling. We found that TFA-enhanced B-cell lymphoma 2 (Bcl-2) overexpression can theoretically give rise to bistability. The bistability is highly robust against parametric stochasticity while also conferring considerable variability in survival threshold. Stochastic simulations faithfully match the TFA dose response pattern seen in experimental studies. In addition, we identified a dose- and time-dependent synergy between TFA and nimodipine, a clinically used neuro-protective drug. This synergistic effect was enhanced by bistability independent of temporal factors. Precise application of pulsed doses of TFA can also promote survival compared with sustained TFA treatment. These data collectively demonstrate that TFA treatment can give rise to bistability and that synergy between TFA and nimodipine may offer a promising strategy for developing therapeutic neuro-protection against ischemic stroke. 目的 2 利用数学建模方法来研究茶黄素酸和尼莫地平在线粒体凋亡途径中的协同神经保护作用。 创新点 2 茶黄素酸诱导的Bcl-2表达在理论上可以引起系统性的双稳态。此双稳态对模型参数变化具有很强的鲁棒性，但同时又对生存状态阈值的变化具有相当的敏感性。通过随机模拟，我们可以很好地拟合实验中测定的茶黄素酸剂量反应。此外，我们确定了茶黄素酸和尼莫地平（一种临床应用的神经保护药物）之间的剂量和时间依赖性协同作用，这种协同效应通过双稳态得到增强。 方法 2 采用微分方程描述分子间相互作用；运用局部敏感性分析探讨系统参数敏感性；运用Bliss和Loewe联合指数计算茶黄素酸和尼莫地平的协同性；使用MATCONT工具箱揭示系统的分岔性质。 结论 2 与持续的茶黄素酸处理相比，对细胞给予脉冲样茶黄素酸处理可以显著提高细胞生存率。这些数据表明，茶黄素酸可以诱导系统产生双稳态，同时茶黄素酸与尼莫地平的协同作用可能为缺血性中风的治疗提供了一种有效的方式。 Theaflavic acid (TFA) Nimodipine Ischemic stroke Apoptosis Synergy 茶黄素酸 尼莫地平 缺血性中风 凋亡 协同性 1 Introduction 1 Ischemic stroke is characterized by induction of inflammatory and hypoxic responses owing to deprivation of glucose and oxygen in the brain ( Yang et al., 2019 Yang et al., 2019 ). Apoptosis will be triggered in the penumbra, the area around the ischemic core without restored blood flow, possibly because of massive production of reactive oxygen species (ROS) and reduced adenosine triphosphate (ATP) levels in these tissues ( Mehta et al., 2007 Mehta et al., 2007 ). However, reperfusion of blood flow will also exacerbate the brain injury via uncontrolled oxidative stress ( Thompson et al., 2012 Thompson et al., 2012 ). B-cell lymphoma 2 (Bcl-2) family members have been shown to actively participate in ischemia-induced apoptosis ( Koubi et al., 2005 Koubi et al., 2005 ). Under oxidative stress, overexpression of the anti-apoptotic member Bcl-2 can reduce ROS production below harmful levels and inhibit apoptosis ( Seidman et al., 2002 Seidman et al., 2002 , 2004 2004 ; Chong et al., 2014 Chong et al., 2014 ). Therefore, precise regulation of Bcl-2 expression can exert a cytoprotective effect for brain tissue under oxygen and glucose deprivation/restoration (OGD/R) ( Koubi et al., 2005 Koubi et al., 2005 ). Bcl-2 family members contribute significantly to apoptosis and can be categorized into three major classes: anti-apoptotic, pro-apoptotic, and Bcl-2 homology 3 (BH3)-only proteins ( Youle and Strasser, 2008 Youle and Strasser, 2008 ). The complex interactions among Bcl-2 family proteins confer bistability and determination of cell fate ( Sun et al., 2010 Sun et al., 2010 ; Yin et al., 2017 Yin et al., 2017 ). Recently, Li et al. (2020) Li et al. (2020) identified the role of theaflavic acid (TFA), a phytochemical extracted from black tea, in inhibiting mitochondrial apoptosis in pheochromacytoma 12 (PC12) cells. TFA promotes translocation of the nuclear factor erythroid 2-related factor 2 (Nrf2) into the nucleus and activates anti-oxidant response element (ARE) signaling to up-regulate the expression of anti-apoptotic Bcl-2 proteins ( Niture and Jaiswal, 2012 Niture and Jaiswal, 2012 ; Li et al., 2020 Li et al., 2020 ). Therefore, TFA may regulate the "Bcl-2 apoptotic switch" ( Sun et al., 2010 Sun et al., 2010 ) by adjusting the expression of anti-apoptotic proteins. Bistability is characterized as a typical switch-like transition between alternative states ( Tyson et al., 2003 Tyson et al., 2003 ). However, it seems that the protective effect of TFA exhibits a gradient pattern ( Li et al., 2020 Li et al., 2020 ). Therefore, understanding of the dynamics and detailed mechanisms of TFA-mediated cytoprotective effects and of the means by which TFA tips the balance between alternative cell fates remains largely elusive. Previous studies have demonstrated that TFA can induce Nrf2 nuclear translocation and promote production of Bcl-2 ( Niture and Jaiswal, 2012 Niture and Jaiswal, 2012 ; Li et al., 2020 Li et al., 2020 ). Furthermore, it is also possible that enhanced production of the anti-apoptotic Bcl-2 induces bistability ( Sun et al., 2010 Sun et al., 2010 ). The aim of the current study is to investigate the theoretical possibility of the ability of TFA to potentiate a bistable response in OGD/R. Ordinary differential equations (ODEs) were constructed to delineate the intricate interactions among Bcl-2 family members and TFA. Local sensitivity coefficients were calculated to determine sensitive parameters. Stochastic simulations were performed to evaluate robustness and patterns of bistability. Nimodipine has been identified as a neuroprotective drug ( Roine et al., 1990 Roine et al., 1990 ). In the present work, the synergistic effects between TFA and nimodipine treatment in neuroprotection were also evaluated. Bliss and Loewe combination indexes were used to quantify the synergy. 2 Materials and methods 1 2.1 Model construction 2 We have previously constructed a model that describes the mitochondrial apoptotic pathway, with two independent positive feedback loops (the complex interactions among Bcl-2 family members and Bcl-2-associated X/Bcl-2 antagonist/killer (Bax/Bak) autoactivation); this model displays bistable responses in the levels of Bcl-2 family members ( Sun et al., 2010 Sun et al., 2010 ). Bcl-2 family members contribute significantly to intrinsic apoptosis ( Legrand et al., 2019 Legrand et al., 2019 ). Briefly, the pro-apoptotic proteins include Bax group proteins (e.g., Bax, Bak, or Bcl-2-related ovarian killer (Bok)) and BH3-only members (e.g., BH3-interacting domain death agonist (Bid), Bcl-2-associated agonist of cell death (Bad), Bcl-2-like 11 (Bim), Bcl-2-modifying factor (Bmf), p53-up-regulated modulator of apoptosis (Puma), and nicotinamide adenine dinucleotide phosphate (NADPH) oxidase heavy chain subunit A (Noxa)), which contain only the BH3-only domain ( Legrand et al., 2019 Legrand et al., 2019 ). The anti-apoptotic members comprise Bcl-2 and B-cell lymphoma-extra large (Bcl-x L ) ( Legrand et al., 2019 Legrand et al., 2019 ). In response to damage signals, Bax monomers and BH3-only members can translocate from the cytosol to the mitochondrial outer membrane (MOM) and relocation of Bcl-2 family members can be viewed as the initial phase of MOM permeabilization (MOMP) and apoptosis ( Tsujimoto, 2003 Tsujimoto, 2003 ). Selected members of BH3-only proteins are "activators," which enable conformational change and Bax/Bak (hereafter dubbed "Bax" for simplicity) pore formation, whereas other BH3-only members are designated as "enablers" by competitive interaction with anti-apoptotic members ( Letai et al., 2002 Letai et al., 2002 ). Anti-apoptotic members (labeled "Bcl-2" for simplicity) can serve as inhibitors to counteract the effects of proapoptotic proteins ( Letai et al., 2002 Letai et al., 2002 ). Activated Bax/Bak monomers can aggregate and induce formation of the mitochondrial apoptosis channel (MAC) ( Dejean et al., 2005 Dejean et al., 2005 ). Numerous pro-apoptotic signals can trigger apoptosis by modulating expression of anti- and pro-apoptotic Bcl-2 family members ( Willis and Adams, 2005 Willis and Adams, 2005 ); therefore, we considered the production rate of diverse Bcl-2 family members instead of cytotoxic stress as bifurcation parameters as previously described ( Cui et al., 2008 Cui et al., 2008 ). We only considered the effect of Bcl-2 family members on apoptosis; non-canonical functions (e.g., autophagy, inflammation, and senescence) which indirectly affect apoptosis ( Chong et al., 2020 Chong et al., 2020 ) were not considered in this study. In our model, we aimed to investigate the effects of TFA from black tea using mathematical modeling ( Li et al., 2020 Li et al., 2020 ). Li et al. (2020) Li et al. (2020) found that TFA can protect PC12 cells from apoptosis induced by OGD/R. Mechanistic study has suggested that TFA up-regulates the expression of an antiapoptotic Bcl-2 protein by promoting the nuclear translocation of Nrf2 and ARE signaling ( Li et al., 2020 Li et al., 2020 ). Indeed, Nrf2 can bind the Bcl-2 ARE and induce Bcl-2 gene expression under stressed conditions ( Niture and Jaiswal, 2012 Niture and Jaiswal, 2012 ). Therefore, we assumed that total Nrf2 centration is constant ( Liu et al., 2017 Liu et al., 2017 ) and TFA can increase the nuclear fraction of Nrf2 (Table S1) ( Li et al., 2020 Li et al., 2020 ). Mathematically, Nrf2 nuc =[ k basal +(1- k basal )×TFA× k in /(TFA× k in + k out )]×Nrf2 T , where Nrf2 nuc is the nuclear Nrf2 concentration, Nrf2 T is the total Nrf2 concentration, k in and k out characterize the nuclear import and export of Nrf2 induced by TFA, respectively, and the parameter k basal represents the basal nuclear Nrf2 fraction according to the experiments of Li et al. (2020) Li et al. (2020) (i.e., under OGD/R without TFA treatment in PC12 cells, there exists a significant nuclear Nrf2 fraction). Promoter activity of Bcl-2 ARE was then expressed as Nrf2 nuc /(Nrf2 nuc + K Nrf2 ), where K Nrf2 is the threshold concentration for Nrf2 nuc . Parameters K Nrf2 , k basal , k in , and k out were fitted to the TFA dose response for transcriptional activity of ARE-bound Nrf2 from the experiments of Li et al. (2020) Li et al. (2020) ( Fig. 1 Fig. 1 f). Transcriptional induction of Bcl-2 by nuclear Nrf2 was modeled as p Bcl-2 = p TFA ×Nrf2 nuc /(Nrf2 nuc + K Nrf2 ), where p represents production (Tables S1 and S2). To evaluate the potential synergistic effects of TFA and nimodipine co-treatment, the regulatory role of nimodipine was incorporated into our model. Liu et al. (2004) Liu et al. (2004) found that nimodipine can up-regulate Bcl-2 messenger RNA (mRNA) expression while down-regulating Bax mRNA; thus, nimodipine can affect the production of both Bcl-2 and Bax proteins. To reflect our assumption that nimodipine increases the production rate of Bcl-2, we introduced an additional term for Bcl-2 production (Table S1). In the model with TFA and nimodipine co-treatment, p Bcl-2 was modified as: p Bcl-2 =p TFA ×Nrf2 nuc /(Nrf2 nuc +K Nrf2 )+ p Nimo ×Nimo/( Ec+ Nimo), where Nimo denotes the concentration of nimodipine and Ec is the threshold concentration for nimodipine. The term p Nimo ×Nimo/( Ec+ Nimo) represents the effect of nimodipine-enhanced Bcl-2 production. In addition, the Bax production rate p 1 was further modified as p 1 ', in which p 1 '= p 1 × Ec /( Ec+ Nimo). This term characterizes the effect of nimodipine-induced Bax suppression. For simplicity, we assumed that enhancement of Bcl-2 production and inhibition of Bax production had an identical threshold value Ec . Since treatment of PC12 cells with 50 μmol/L nimodipine alone had similar effects as in cells treated with 50 μmol/L TFA ( Li et al., 2020 Li et al., 2020 ), the Ec value was estimated to yield temporal profiles in dynamics of cells treated with 50 μmol/L TFA. All kinetic interactions were formulated using ODEs (Tables S1 and S2) and parameter values are shown in Table S3. Parameter values for the Bcl-2 activation switch ( k 1 to k 17 , u 1 to u 9 , p 1 and p 3 ) were assigned based on our previous work ( Sun et al., 2009 Sun et al., 2009 ). 2.2 Stochastic simulation 2 Stochastic simulations were implemented using the Poisson τ -leap method to characterize intrinsic noise and improve computational efficiency in well-stirred systems ( Tian and Burrage, 2006 Tian and Burrage, 2006 ). Briefly, suppose that there are N species in the system [ S 1 , S 2 , …, S N ] with molecular number x = X i ( t ) for species S i at time t . These species interact dynamically with each other via reaction channels [ R 1 , R 2 , …, R M ]. For R j , the propensity function a j ( x ) for a given state inside a volume Ω . Therefore, a j d t denotes the probability of occurrence for reaction R j , in the infinitesimal time interval [ t , t+ d t ). The element ν ij state vector ν j represents the change in molecular numbers for species S i owing to the reaction R j . In the Poisson τ -leap method, the number ( M ) of reactions for channel R j firing at time interval [ t , t+τ ) obeys Poisson distribution ( P ) with mean a j τ . The system is updated following the rule: <math display="block" id="M1"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="bold-italic">ν</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608974&type= 54.27133560 8.72066689 Extrinsic noise was also incorporated by perturbing the parameters by multiplying them with lognormally distributed random variables ( μ =1, σ 2 =0.02) ( Keizer et al., 2019 Keizer et al., 2019 ). Therefore, in our stochastic simulations, both intrinsic noise and extrinsic noise were considered. 2.3 Bifurcation analysis 2 Bifurcation data were obtained using the MATLAB package MATCONT ( Dhooge et al., 2003 Dhooge et al., 2003 ). 2.4 Synergy 2 Synergy was evaluated using the Loewe and Bliss combination indices (CIs) ( Straetemans et al., 2005 Straetemans et al., 2005 ; Fitzgerald et al., 2006 Fitzgerald et al., 2006 ). The Loewe combination index CI Loewe is defined as follows: <math display="block" id="M2"><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">w</mi><mi mathvariant="normal">e</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608977&type= 32.42733383 8.88999939 where d 1 (TFA) and d 2 (nimodipine) are the agent doses in the combination isobologram with respect to x (10% or 40%) percentage decrease in total levels of activated Bax, C x (1) and C x (2) denote the concentrations of a single TFA and nimodipine dose, respectively, leading to x (10% or 40%) percentage decrease in total levels of activated Bax. CI Loewe < 1, CI Loewe =1, and CI Loewe > 1 indicate Loewe synergy, additivity, and antagonism, respectively. The Bliss independence or combination index CI Bliss is defined as: <math display="block" id="M3"><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi mathvariant="normal">B</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608979&type= 70.01933289 9.05933285 where R 1 ( x ), R 2 ( y ), and R 12 ( x , y ) denote the relative changes in total levels of activated Bax to TFA (at a dose of x ), nimodipine (at a dose of y ), and TFA-nimodipine combination treatment (at a dose combination of [ x , y ]), respectively. As a result, CI Bliss < 1, CI Bliss =1, and CI Bliss > 1 indicate synergistic, additive, and antagonistic effects, respectively. 2.5 Local sensitivity coefficient 2 Local sensitivity can evaluate the dynamic responses to infinitesimal changes in kinetic parameters. A dynamic system can be defined by x ' =f ( x , θ ), where x and θ represent the state and parameter vectors, respectively. Local sensitivity coefficient S is defined by <math display="block" id="M4"><mi mathvariant="bold-italic">S</mi><mo>=</mo><mfrac><mrow><mrow><mrow><mo>∂</mo><mi mathvariant="normal">S</mi><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>/</mo><mrow><mi mathvariant="normal">S</mi><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow><mrow><mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow><mo>/</mo><mrow><mi mathvariant="bold-italic">θ</mi></mrow></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">S</mi><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=27608981&type= 41.31733322 8.88999939 where SN i ( i =1 or 2) is the saddle node in bifurcation diagram. 2.6 Model simulation 2 The integration of ODEs and numerical simulations was performed using MATLAB (R2018b). 3 Results 1 3.1 Potential bistability induced by TFA-enhanced Bcl-2 production 2 TFA can accelerate production of Bcl-2 via increased Nrf2 nuclear translocation ( Li et al., 2020 Li et al., 2020 ). We incorporated this effect into our model of Bax activation switch (see Section 2; Fig. 1 Fig. 1 ). After OGD/R, excess ROS will lead to mitochondrial damage and induce apoptosis in neuronal cells ( Luo et al., 2013 Luo et al., 2013 ; Li et al., 2020 Li et al., 2020 ). Therefore, the initial state of cells after OGD/R should be a pro-apoptotic state with high level of activated Bax or low levels of anti-apoptotic proteins. After TFA treatment, expression of anti-apoptotic Bcl-2 is reinforced and cells may revert to a survival state if overexpressed Bcl-2 can effectively antagonize pro-apoptotic proteins (e.g., Bax) (Figs. S1a‒S1d). We observed that the system displayed bistability characterized by the folded region in the two-parameter bifurcation diagram ( Fig. 1 Fig. 1 b). As the Bax production rate ( p 1 ) increased, we noted that a higher TFA-induced Bcl-2 production rate ( p TFA ) was required to "neutralize" activated Bax and maintain a state with extremely low levels of the pro-apoptotic Bax proteins (i.e., survival state; Fig. 1 Fig. 1 b). As expected, higher doses of TFA were needed when p 1 was elevated to maintain high levels of Bcl-2 and cellular survival ( Fig. 1 Fig. 1 c). We could identify CUSP bifurcation points in the two-parameter bifurcation diagrams and the bistable regions were enlarged at higher p 1 (Figs. 1 1 d and 1 1 e). We also noted that the simulated dose response of TFA for Bcl-2 promoter activity was consistent with the findings of Li et al. (2020) Li et al. (2020) ( Fig. 1 Fig. 1 f). These results suggest that TFA treatment may reshape the bistability of the intrinsic apoptotic pathway. 3.2 Sensitivity analysis for bistability 2 We then evaluated parametric sensitivity for bistable responses. When p TFA is significantly increased, Bcl-2 overexpression may effectively counteract activated Bax ( Fig. 2 Fig. 2 a). In the bifurcation diagram for activated Bax with reference to p TFA , two saddle nodes (SN 1 and SN 2 ) can be identified and SN 2 indicates a deterministic threshold for survival after OGD/R ( Fig. 2 Fig. 2 a). Not surprisingly, p 1 positively affected both saddle nodes (Figs. 1 1 b and 2 2 b). Production rate of pro-apoptotic BH3-only proteins also markedly increased the threshold for survival ( Fig. 2 Fig. 2 b). As expected, the degradation rates for inactive Bax and BH3-only members played a negative role ( Fig. 2 Fig. 2 b). Two parameters k basal and k Nrf2 showed the highest sensitivities for SN 1 and SN 2 , respectively, since they were directly associated with Bcl-2 production ( Fig. 2 Fig. 2 b). Notably, activator ( u 3 ) and MAC ( u 9 ) degradation rates specifically lowered the threshold for survival (SN 2 ) (blue bars, Fig. 2 Fig. 2 b). From sensitivity analysis, we could identify universal and specific factors to regulate cellular threshold for survival and death. 3.3 Robustness features of TFA-mediated bistability 2 Since TFA-induced bistability is an important contributor to cellular survival after OGD/R, we further investigated the property of robustness. From bifurcation theory, the steady state connecting SN 1 and SN 2 is unstable ( Tyson et al., 2003 Tyson et al., 2003 ). Once a "cell" overcomes the barrier height between the unstable steady state and the higher (upper) steady state for activated Bax after OGD/R, the lower steady state will be reached and a "cell" will survive within the bistable region ( Fig. 3 Fig. 3 a). We defined the maximal barrier height as the largest difference in activated Bax levels between the high and unstable steady state ( Fig. 3 Fig. 3 a). First, we varied all parameters around their reference values by multiplying them with a lognormally distributed random variable (i.e., μ =1, σ 2 =0.1, 0.2, or 0.3). Interestingly, we noted that TFA-induced bistability was highly robust and more than 90% of simulations retained bistability even when the variance reached 0.3 (90.1%, Fig. 3 Fig. 3 b). Under parametric stochasticity, we noted that SN 1 was relatively less variable, while SN 2 showed higher variation ( Fig. 3 Fig. 3 c). Notably, TFA induces Bcl-2 expression; once the productive effect of Bcl-2 exceeds the SN 2 threshold, the levels of active Bax will dramatically decrease and cells may revert to a survival state after OGD/R ( Fig. 2 Fig. 2 a). Therefore, a highly variable SN 2 implies that the transition between alternative cell fates (survival and death) might frequently occur after TFA treatment. Importantly, we found only a minimal correlation between maximal barrier height and SN 1 ( Fig. 3 Fig. 3 d, left), but a strong correlation between maximal barrier height and SN 2 ( Fig. 3 Fig. 3 d, middle). The correlation was further improved if the bistable range (SN 2 ‒SN 1 ) was used as the metric ( Fig. 3 Fig. 3 d, right). Taken together, although the bistability is robust, the threshold for survival is highly variable. 3.4 TFA dose-dependent responses via stochastic simulations 2 We then performed stochastic simulations to evaluate TFA dose-dependent responses. Under 5 μmol/L TFA treatment, stochastic simulations identified that specific runs retained high levels of activated Bax, whereas some runs led to collapse in levels of activated Bax ( Fig. 4 Fig. 4 a). If "cells" could revert to survival state after OGD/R, we then evaluated their recovery rate by measuring the half recovery time (Figs. 4 4 a and 4 4 b). We found that increasing TFA doses accelerated recovery and survival ( Fig. 4 Fig. 4 b). Average half recovery time changed from 3.40, 3.18 to 2.92 h as the TFA dose increased from 5, 10 to 50 μmol/L ( Fig. 4 Fig. 4 b). Not surprisingly, higher TFA doses decreased the average levels of activated Bax while raising levels of Bcl-2 ( Fig. 4 Fig. 4 c), thus qualitatively aligning with experimental data ( Li et al., 2020 Li et al., 2020 ). Li et al. (2020) Li et al. (2020) found that about (48.9±5.1)% of PC12 cells are viable when treated without TFA after OGD/R, and treatment with TFA dosages may increase the percentage of viable cells. Our stochastic simulations assumed a hypothetical threshold of activated Bax which differentiates survival from death ( Fig. 4 Fig. 4 d, dashed lines). Varying the threshold level of activated Bax yielded a final viability fraction from 43.8% (i.e., 48.9%‒5.1%) to 54.0% (i.e., 48.9%+5.1%); we evaluated whether our simulations matched the experimentally determined dose response of TFA ( Li et al., 2020 Li et al., 2020 ). When the threshold is defined as indicated (46.9% viability; Fig. 4 Fig. 4 d, top), the TFA dose response most closely matches the experimental data ( Li et al., 2020 Li et al., 2020 ). However, consistency was identified after a 4-h TFA treatment ( Figs. 4 Figs. 4 d (top) and 4e), which is in accordance with the experimental setting by Li et al. (2020) Li et al. (2020) (viability in their experiments was measured after 4 h TFA treatment). When viability was evaluated for a 12-h TFA treatment after OGD/R, the deviation in simulated TFA dose responses from experimental data was significant ( Figs. 4 Figs. 4 d (bottom) and 4f). Notably, the bimodal response, which is a notable characteristic of bistability ( Tyson et al., 2003 Tyson et al., 2003 ), could be observed at later time points (e.g., 12 h TFA treatment; Fig. 4 Fig. 4 d, bottom). These results suggested that the TFA dose response is time-dependent and bimodality can be detected only at later time points. 3.5 Optimal modulation of synergistic responses between TFA and nimodipine 2 Previous work has shown that the neuro-protective drug nimodipine can up-regulate Bcl-2 expression while simultaneously inhibiting Bax expression ( Liu et al., 2004 Liu et al., 2004 ). Applying nimodipine to PC12 cells, Li et al. (2020) Li et al. (2020) also found that 50 μmol/L nimodipine has an effect similar to that of 50 μmol/L TFA in viability and apoptosis assays. We then incorporated the effect of nimodipine in our model (see Section 2; Fig. S2) to investigate the potential synergy for combined TFA-nimodipine treatment. We found that nimodipine treatment could give rise to systematic bistability ( Fig. 5 Fig. 5 a). The two-parameter bifurcation diagram also shows that specific combinations of TFA and nimodipine guaranteed bistable responses ( Fig. 5 Fig. 5 b, grey). We then examined whether TFA and nimodipine could function cooperatively during recovery from OGD/R using the Bliss combination index (see Section 2). Evaluating the responses for reduction in active Bax monomer (indicative of transition from death to survival) at 4 or 12 h after TFA and nimodipine co-treatment, we found that at 4 h, combined TFA-nimodipine treatment usually had synergistic effects ( Fig. 5 Fig. 5 c). Bistable regions in the two-parameter diagrams indeed increased synergy (Figs. 5 5 c and 5 5 d). At 12 h after TFA-nimodipine co-treatment, however, we could observe both synergy (CI Bliss < 1) and antagonism (CI Bliss > 1) depending on the doses of TFA and nimodipine ( Fig. 5 Fig. 5 d). Indeed, the regions of highest synergy roughly coincided with the bistable regions (Figs. 5 5 b and 5 5 d). We also analyzed the isobologram of apoptotic responses after combined TFA-nimodipine treatment using the Loewe combination index. Isobolograms (10% and 40%) were evaluated ( Fig. 5 Fig. 5 e). Simulations demonstrated that TFA and nimodipine displayed differential synergistic effects for different conditions. Specifically, TFA and nimodipine showed slight synergy or near-additivity (10%; Fig. 5 Fig. 5 e, left two panels). However, TFA and nimodipine exhibited remarkable synergy in 40% isobologram ( Fig. 5 Fig. 5 e, right two panels). These data suggest that TFA and nimodipine show objective- and dose-dependent synergism during anti-apoptotic responses. 3.6 Evaluation of impulse responses for TFA 2 Li et al. (2020) Li et al. (2020) used a continuous TFA dosage to treat PC12 cells; we further extended their findings and evaluated the response to pulsed doses of TFA. Pulses were reshaped with different durations and terminal heights (the maximum was fixed as 50 μmol/L; Fig. 6 Fig. 6 a). Note that we applied a super-threshold TFA dose 50 μmol/L (Fig. S1) as the pulse height. However, if the pulse duration did not persist for a sufficiently long time, the level of active Bax monomer could not overcome the threshold and would be maintained at high levels ( Fig. 6 Fig. 6 b). If the duration was longer, however, we observed that Bax dropped to low levels and the cell would commit to survival ( Fig. 6 Fig. 6 c). We further measured the steady state of active Bax monomers and the levels of active Bax at 4 or 12 h after termination of TFA impulse with varying impulse duration and terminal height. We identified a gradual response surface if measurement was performed at 4 h after termination of TFA impulse ( Fig. 6 Fig. 6 d, left). However, a "switch-like" response pattern could be observed when we quantified active Bax monomers at 12 h ( Fig. 6 Fig. 6 d, center). If TFA terminal height was slightly more than 10 μmol/L, even a TFA pulse of small duration would lead to strongly decreased active Bax levels ( Fig. 6 Fig. 6 d, center). A definite "switch-like" pattern was seen if the steady states were evaluated ( Fig. 6 Fig. 6 d, right). A duration of TFA pulse of more than approximately 400 min would irreversibly cut down active Bax expression irrespective of terminal height ( Fig. 6 Fig. 6 d, right). Similarly, a minor 50 μmol/L TFA pulse with a terminal height of approximately 11.5 μmol/L would always lead to survival ( Fig. 6 Fig. 6 d, right). Collectively, these results demonstrate that appropriate application of TFA pulses can tip the balance between survival and death. 4 Discussion 1 Since ischemic stroke remains a major threat to health worldwide, this study modeled the impact of TFA on protection of OGD/R-induced apoptosis ( Li et al., 2015 Li et al., 2015 ). We investigated a potential mechanism involving the mitochondria apoptotic pathway. Bistability, which arises from complex interactions in the apoptotic pathway, has been identified in numerous studies ( Bagci et al., 2006 Bagci et al., 2006 ; Legewie et al., 2006 Legewie et al., 2006 ; Chen et al., 2007 Chen et al., 2007 ; Albeck et al., 2008 Albeck et al., 2008 ; Cui et al., 2008 Cui et al., 2008 ; Ho and Harrington, 2010 Ho and Harrington, 2010 ; Sun et al., 2010 Sun et al., 2010 ; Yin et al., 2017 Yin et al., 2017 ). TFA, a theaflavin previously isolated from black tea ( Li et al., 2020 Li et al., 2020 ), has been shown to promote translocation of Nrf2 into the nucleus and to activate ARE signaling to increase Bcl-2 expression ( Niture and Jaiswal, 2012 Niture and Jaiswal, 2012 ). Therefore, TFA can regulate the expression of anti-apoptotic proteins and modulate bistability in the apoptotic switch. The modulatory role of TFA is reminiscent of resveratrol-induced apoptosis ( Jiang et al., 2005 Jiang et al., 2005 ) and suggests that the apoptotic switch can be regulated. The ischemic core cannot access sufficient ATP and oxygen because of occlusion of blood flow in cerebral tissues ( Pellegrini-Giampietro et al., 2009 Pellegrini-Giampietro et al., 2009 ). Following ischemia and restoration of blood flow, oxygen supply is largely restored, leading to overproduction of ROS and oxidative stress ( Saito et al., 2005 Saito et al., 2005 ). The deleterious effects of excessive ROS include activation of the mitochondrial apoptotic pathway and cell death after cerebral ischemia and reperfusion ( Chong et al., 2014 Chong et al., 2014 ). Excess ROS can also enhance deletion of mitochondrial DNA and induce hearing loss ( Seidman, 2000 Seidman, 2000 ; Seidman et al., 2000 Seidman et al., 2000 ; Coling et al., 2003 Coling et al., 2003 ). The anti-apoptotic member Bcl-2 proteins have a defined role in restraining mitochondria ROS overproduction by facilitating the mitochondrial incorporation of glutathione under oxidative stress ( Chong et al., 2014 Chong et al., 2014 ). Therefore, the protective function of TFA in reducing ROS after OGD/R can be mediated by TFA-induced Bcl-2 overexpression ( Chong et al., 2014 Chong et al., 2014 ; Li et al., 2020 Li et al., 2020 ). For the sake of simplicity, we did not incorporate ROS or calcium signaling in our model, since Bcl-2 expression may inversely reflect the intrinsic ROS levels; there is also well-characterized positive crosstalk between ROS and calcium signaling ( Görlach et al., 2015 Görlach et al., 2015 ). Model fitting to some experimental data reported by Li et al. (2020) Li et al. (2020) (e.g., ROS or calcium levels, even superoxide dismutase activity) can be easily achieved if the fit to cellular viability and Bcl-2 promoter activity is acceptable (Figs. 1 1 f and 4 4 e) ( Chong et al., 2014 Chong et al., 2014 ; Görlach et al., 2015 Görlach et al., 2015 ). Therefore, our model closely aligns with the experimental data from Li et al. (2020) Li et al. (2020) and can be used to make testable predictions. The bistable system allows co-existence of two stable steady states ( Spencer and Sorger, 2011 Spencer and Sorger, 2011 ). Bistability can be used to explain the binary decisions on cell fate (survival or death) ( Spencer and Sorger, 2011 Spencer and Sorger, 2011 ). A bistable system is insensitive to minor perturbations and exhibits "all-or-none" transitions. From our simulation, we argue that TFA-mediated bistable responses are robust to large parametric stochasticity. However, we further note that the deterministic threshold for survival vs. death (i.e., SN 2 in Fig. 3 Fig. 3 a) is highly variable. TFA enhances Bcl-2 production and overexpressed anti-apoptotic Bcl-2 counteracts pro-apoptotic effectors such as BH3-only members and activated Bax to overcome the barriers to apoptosis (i.e., moves beyond SN 2 ). The maximum barrier height is also positively correlated with the survival threshold. These properties collectively imply that even there is robust bistability at system level; cell fate after TFA-treatment might vary with large variations in the survival threshold. This effect may partially explain the "gradient" pattern in TFA dose responses seen in Li et al. (2020) Li et al. (2020) and may also shed light on the phenomenon of "fractional killing" ( Roux et al., 2015 Roux et al., 2015 ). Specifically, we found that the effects of nimodipine and TFA may be increased when used in combination. These synergistic effects may be a salient feature of the protective roles of combined TFA-nimodipine treatment against OGD/R. Notably, the synergy between TFA and nimodipine was time- and dose-dependent (Figs. 5 5 c and 5 5 d). At early time points, TFA and nimodipine showed consistent synergy. However, if the measurements were performed at later time points, the combined effects may shift from synergism to antagonism depending on the TFA and nimodipine doses. Optimal synergy has been achieved approximately at the bistable regions in TFA and nimodipine space irrespective of temporal dependence. A monostable domain far from the bistable regions will mitigate the synergistic effects or even lead to antagonism. These findings argue that bistability may augment the synergistic effects and that the optimal use of TFA and nimodipine may offer favorable synergy for protecting neuronal cells against OGD/R-induced apoptosis. Since nimodipine is a neuro-protective drug already used in clinical practice ( Roine et al., 1990 Roine et al., 1990 ) and TFA is a phytochemical extracted from black tea, the synergy between these two compounds suggests a potentially promising strategy. The synergy between TFA and nimodipine may possibly be more effective against drug-induced side effects compared with parenteral edaravone and nimodipine co-treatment ( Li et al., 2018 Li et al., 2018 ). In summary, we investigated the protective role of TFA in the mitochondria apoptotic pathway. TFA-mediated bistability displays both robust and variable features. The synergistic effects of TFA and nimodipine may provide crucial insight into neuronal protection against ischemia stroke. References Albeck JG , Burke JM , Spencer SL , et al. , 2008 . Modeling a snap-action, variable-delay switch controlling extrinsic cell death . PLoS Biol , 6 ( 12 ): 2831 - 2852 . https://doi.org/10.1371/journal.pbio.0060299 https://doi.org/10.1371/journal.pbio.0060299 Modeling a snap-action, variable-delay switch controlling extrinsic cell death Bagci EZ , Vodovotz Y , Billiar TR , et al. , 2006 . Bistability in apoptosis: roles of Bax, Bcl-2, and mitochondrial permeability transition pores . Biophys J , 90 ( 5 ): 1546 - 1559 . https://doi.org/10.1529/biophysj.105.068122 https://doi.org/10.1529/biophysj.105.068122 Bistability in apoptosis: roles of Bax, Bcl-2, and mitochondrial permeability transition pores Chen C , Cui J , Lu HZ , et al. , 2007 . Modeling of the role of a Bax-activation switch in the mitochondrial apoptosis decision . Biophys J , 92 ( 12 ): 4304 - 4315 . https://doi.org/10.1529/biophysj.106.099606 https://doi.org/10.1529/biophysj.106.099606 Modeling of the role of a Bax-activation switch in the mitochondrial apoptosis decision Chong SJF , Low ICC , Pervaiz S , 2014 . Mitochondrial ROS and involvement of Bcl-2 as a mitochondrial ROS regulator . Mitochondrion , 19 : 39 - 48 . https://doi.org/10.1016/j.mito.2014.06.002 https://doi.org/10.1016/j.mito.2014.06.002 Mitochondrial ROS and involvement of Bcl-2 as a mitochondrial ROS regulator Chong SJF , Marchi S , Petroni G , et al. , 2020 . Noncanonical cell fate regulation by Bcl-2 proteins . Trends Cell Biol , 30 ( 7 ): 537 - 555 . https://doi.org/10.1016/j.tcb.2020.03.004 https://doi.org/10.1016/j.tcb.2020.03.004 Noncanonical cell fate regulation by Bcl-2 proteins Coling DE , Yu KCY , Somand D , et al. , 2003 . Effect of SOD1 overexpression on age- and noise-related hearing loss . Free Radic Biol Med , 34 ( 7 ): 873 - 880 . https://doi.org/10.1016/s0891-5849(02)01439-9 https://doi.org/10.1016/s0891-5849(02)01439-9 Effect of SOD1 overexpression on age- and noise-related hearing loss Cui J , Chen C , Lu HZ , et al. , 2008 . Two independent positive feedbacks and bistability in the Bcl-2 apoptotic switch . PLoS ONE , 3 ( 1 ): e1469 . https://doi.org/10.1371/journal.pone.0001469 https://doi.org/10.1371/journal.pone.0001469 Two independent positive feedbacks and bistability in the Bcl-2 apoptotic switch Dejean LM , Martinez-Caballero S , Guo L , et al. , 2005 . Oligomeric Bax is a component of the putative cytochrome c release channel MAC, mitochondrial apoptosis-induced channel . Mol Biol Cell , 16 ( 5 ): 2424 - 2432 . https://doi.org/10.1091/mbc.E04-12-1111 https://doi.org/10.1091/mbc.E04-12-1111 Oligomeric Bax is a component of the putative cytochrome c release channel MAC, mitochondrial apoptosis-induced channel Dhooge A , Govaerts W , Kuznetsov YA , 2003 . MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs . ACM Trans Math Softw , 29 ( 2 ): 141 - 164 . https://doi.org/10.1145/779359.779362 https://doi.org/10.1145/779359.779362 MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs Fitzgerald JB , Schoeberl B , Nielsen UB , et al. , 2006 . Systems biology and combination therapy in the quest for clinical efficacy . Nat Chem Biol , 2 ( 9 ): 458 - 466 . https://doi.org/10.1038/nchembio817 https://doi.org/10.1038/nchembio817 Systems biology and combination therapy in the quest for clinical efficacy Görlach A , Bertram K , Hudecova S , et al. , 2015 . Calcium and ROS: a mutual interplay . Redox Biol , 6 : 260 - 271 . https://doi.org/10.1016/j.redox.2015.08.010 https://doi.org/10.1016/j.redox.2015.08.010 Calcium and ROS: a mutual interplay Ho KL , Harrington HA , 2010 . Bistability in apoptosis by receptor clustering . PLoS Comput Biol , 6 ( 10 ): e1000956 . https://doi.org/10.1371/journal.pcbi.1000956 https://doi.org/10.1371/journal.pcbi.1000956 Bistability in apoptosis by receptor clustering Jiang H , Zhang LJ , Kuo J , et al. , 2005 . Resveratrol-induced apoptotic death in human U251 glioma cells . Mol Cancer Ther , 4 ( 4 ): 554 - 561 . https://doi.org/10.1158/1535-7163.MCT-04-0056 https://doi.org/10.1158/1535-7163.MCT-04-0056 Resveratrol-induced apoptotic death in human U251 glioma cells Keizer EM , Bastian B , Smith RW , et al. , 2019 . Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise . Phys Rev E , 99 ( 5 ): 052417 . https://doi.org/10.1103/PhysRevE.99.052417 https://doi.org/10.1103/PhysRevE.99.052417 Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise Koubi D , Jiang H , Zhang LJ , et al. , 2005 . Role of Bcl-2 family of proteins in mediating apoptotic death of PC12 cells exposed to oxygen and glucose deprivation . Neurochem Int , 46 ( 1 ): 73 - 81 . https://doi.org/10.1016/j.neuint.2004.06.006 https://doi.org/10.1016/j.neuint.2004.06.006 Role of Bcl-2 family of proteins in mediating apoptotic death of PC12 cells exposed to oxygen and glucose deprivation Legewie S , Blüthgen N , Herzel H , 2006 . Mathematical modeling identifies inhibitors of apoptosis as mediators of positive feedback and bistability . PLoS Comput Biol , 2 ( 9 ): e120 . https://doi.org/10.1371/journal.pcbi.0020120 https://doi.org/10.1371/journal.pcbi.0020120 Mathematical modeling identifies inhibitors of apoptosis as mediators of positive feedback and bistability Legrand AJ , Konstantinou M , Goode EF , et al. , 2019 . The diversification of cell death and immunity: Memento mori . Mol Cell , 76 ( 2 ): 232 - 242 . https://doi.org/10.1016/j.molcel.2019.09.006 https://doi.org/10.1016/j.molcel.2019.09.006 The diversification of cell death and immunity: Memento mori Letai A , Bassik MC , Walensky LD , et al. , 2002 . Distinct BH3 domains either sensitize or activate mitochondrial apoptosis, serving as prototype cancer therapeutics . Cancer Cell , 2 ( 3 ): 183 - 192 . https://doi.org/10.1016/S1535-6108(02)00127-7 https://doi.org/10.1016/S1535-6108(02)00127-7 Distinct BH3 domains either sensitize or activate mitochondrial apoptosis, serving as prototype cancer therapeutics Li LX , Yiin GS , Geraghty OC , et al. , 2015 . Incidence, outcome, risk factors, and long-term prognosis of cryptogenic transient ischaemic attack and ischaemic stroke: a population-based study . Lancet Neurol , 14 ( 9 ): 903 - 913 . https://doi.org/10.1016/S1474-4422(15)00132-5 https://doi.org/10.1016/S1474-4422(15)00132-5 Incidence, outcome, risk factors, and long-term prognosis of cryptogenic transient ischaemic attack and ischaemic stroke: a population-based study Li QJ , Ji ZL , Li K , 2018 . Effect of combination of parenteral edaravone and nimodipine on ischemic cerebral injury following cerebral hemorrhage . Trop J Pharm Res , 17 ( 5 ): 955 - 960 . https://doi.org/10.4314/tjpr.v17i5.27 https://doi.org/10.4314/tjpr.v17i5.27 Effect of combination of parenteral edaravone and nimodipine on ischemic cerebral injury following cerebral hemorrhage Li Y , Shi J , Sun XT , et al. , 2020 . Theaflavic acid from black tea protects PC12 cells against ROS-mediated mitochondrial apoptosis induced by OGD/R via activating Nrf2/ARE signaling pathway . J Nat Med , 74 : 238 - 246 . https://doi.org/10.1007/s11418-019-01333-4 https://doi.org/10.1007/s11418-019-01333-4 Theaflavic acid from black tea protects PC12 cells against ROS-mediated mitochondrial apoptosis induced by OGD/R via activating Nrf2/ARE signaling pathway Liu CQ , Zhou RX , Sun SG , 2004 . Nimodipine modulates Bcl-2 and Bax mRNA expression after cerebral ischemia . J Huazhong Univ Sci Technol Med Sci , 24 ( 2 ): 170 - 172 . https://doi.org/10.1007/BF02885421 https://doi.org/10.1007/BF02885421 Nimodipine modulates Bcl-2 and Bax mRNA expression after cerebral ischemia Liu L , Huang WW , Wang JH , et al. , 2017 . Anthraquinone derivative exerted hormetic effect on the apoptosis in oxygen-glucose deprivation-induced PC12 cells via ERK and Akt activated Nrf2/HO-1 signaling pathway . Chem Biol Interact , 262 : 1 - 11 . https://doi.org/10.1016/j.cbi.2016.12.001 https://doi.org/10.1016/j.cbi.2016.12.001 Anthraquinone derivative exerted hormetic effect on the apoptosis in oxygen-glucose deprivation-induced PC12 cells via ERK and Akt activated Nrf2/HO-1 signaling pathway Luo YG , Yang XF , Zhao ST , et al. , 2013 . Hydrogen sulfide prevents OGD/R-induced apoptosis via improving mitochondrial dysfunction and suppressing an ROS-mediated caspase-3 pathway in cortical neurons . Neurochem Int , 63 ( 8 ): 826 - 831 . https://doi.org/10.1016/j.neuint.2013.06.004 https://doi.org/10.1016/j.neuint.2013.06.004 Hydrogen sulfide prevents OGD/R-induced apoptosis via improving mitochondrial dysfunction and suppressing an ROS-mediated caspase-3 pathway in cortical neurons Mehta SL , Manhas N , Raghubir R , 2007 . Molecular targets in cerebral ischemia for developing novel therapeutics . Brain Res Rev , 54 ( 1 ): 34 - 66 . https://doi.org/10.1016/j.brainresrev.2006.11.003 https://doi.org/10.1016/j.brainresrev.2006.11.003 Molecular targets in cerebral ischemia for developing novel therapeutics Niture SK , Jaiswal AK , 2012 . Nrf2 protein up-regulates antiapoptotic protein Bcl-2 and prevents cellular apoptosis . J Biol Chem , 287 ( 13 ): 9873 - 9886 . https://doi.org/10.1074/jbc.M111.312694 https://doi.org/10.1074/jbc.M111.312694 Nrf2 protein up-regulates antiapoptotic protein Bcl-2 and prevents cellular apoptosis Pellegrini-Giampietro DE , Mannaioni G , Bagetta G , 2009 . Post-ischemic brain damage: the endocannabinoid system in the mechanisms of neuronal death . FEBS J , 276 ( 1 ): 2 - 12 . https://doi.org/10.1111/j.1742-4658.2008.06765.x https://doi.org/10.1111/j.1742-4658.2008.06765.x Post-ischemic brain damage: the endocannabinoid system in the mechanisms of neuronal death Roine RO , Kaste M , Kinnunen A , et al. , 1990 . Nimodipine after resuscitation from out-of-hospital ventricular fibrillation: a placebo-controlled, double-blind, randomized trial . JAMA , 264 ( 24 ): 3171 - 3177 . https://doi.org/10.1001/jama.1990.03450240073043 https://doi.org/10.1001/jama.1990.03450240073043 Nimodipine after resuscitation from out-of-hospital ventricular fibrillation: a placebo-controlled, double-blind, randomized trial Roux J , Hafner M , Bandara S , et al. , 2015 . Fractional killing arises from cell-to-cell variability in overcoming a caspase activity threshold . Mol Syst Biol , 11 ( 5 ): 803 . https://doi.org/10.15252/msb.20145584 https://doi.org/10.15252/msb.20145584 Fractional killing arises from cell-to-cell variability in overcoming a caspase activity threshold Saito A , Maier CM , Narasimhan P , et al. , 2005 . Oxidative stress and neuronal death/survival signaling in cerebral ischemia . Mol Neurobiol , 31 ( 1-3 ): 105 - 116 . https://doi.org/10.1385/MN:31:1-3:105 https://doi.org/10.1385/MN:31:1-3:105 Oxidative stress and neuronal death/survival signaling in cerebral ischemia Seidman MD , 2000 . Effects of dietary restriction and antioxidants on presbyacusis . Laryngoscope , 110 ( 5 ): 727 - 738 . https://doi.org/10.1097/00005537-200005000-00003 https://doi.org/10.1097/00005537-200005000-00003 Effects of dietary restriction and antioxidants on presbyacusis Seidman MD , Khan MJ , Bai U , et al. , 2000 . Biologic activity of mitochondrial metabolites on aging and age-related hearing loss . Am J Otol , 21 ( 2 ): 161 - 167 . https://doi.org/10.1016/s0196-0709(00)80003-4 https://doi.org/10.1016/s0196-0709(00)80003-4 Biologic activity of mitochondrial metabolites on aging and age-related hearing loss Seidman MD , Khan MJ , Tang WX , et al. , 2002 . Influence of lecithin on mitochondrial DNA and age-related hearing loss . Otolaryngol Head Neck Surg , 127 ( 3 ): 138 - 144 . https://doi.org/10.1067/mhn.2002.127627 https://doi.org/10.1067/mhn.2002.127627 Influence of lecithin on mitochondrial DNA and age-related hearing loss Seidman MD , Ahmad N , Joshi D , et al. , 2004 . Age-related hearing loss and its association with reactive oxygen species and mitochondrial DNA damage . Acta Otolaryngol Suppl , ( 552 ): 16 - 24 . https://doi.org/10.1080/03655230410017823 https://doi.org/10.1080/03655230410017823 Age-related hearing loss and its association with reactive oxygen species and mitochondrial DNA damage Spencer SL , Sorger PK , 2011 . Measuring and modeling apoptosis in single cells . Cell , 144 ( 6 ): 926 - 939 . https://doi.org/10.1016/j.cell.2011.03.002 https://doi.org/10.1016/j.cell.2011.03.002 Measuring and modeling apoptosis in single cells Straetemans R , O'Brien T , Wouters L , et al. , 2005 . Design and analysis of drug combination experiments . Biom J , 47 ( 3 ): 299 - 308 . https://doi.org/10.1002/bimj.200410124 https://doi.org/10.1002/bimj.200410124 Design and analysis of drug combination experiments Sun TZ , Chen C , Wu YY , et al. , 2009 . Modeling the role of p53 pulses in DNA damage-induced cell death decision . BMC Bioinformatics , 10 : 190 . https://doi.org/10.1186/1471-2105-10-190 https://doi.org/10.1186/1471-2105-10-190 Modeling the role of p53 pulses in DNA damage-induced cell death decision Sun TZ , Lin XZ , Wei YN , et al. , 2010 . Evaluating bistability of Bax activation switch . FEBS Lett , 584 ( 5 ): 954 - 960 . https://doi.org/10.1016/j.febslet.2010.01.034 https://doi.org/10.1016/j.febslet.2010.01.034 Evaluating bistability of Bax activation switch Thompson JW , Narayanan SV , Perez-Pinzon MA , 2012 . Redox signaling pathways involved in neuronal ischemic preconditioning . Curr Neuropharmacol , 10 ( 4 ): 354 - 369 . https://doi.org/10.2174/157015912804143577 https://doi.org/10.2174/157015912804143577 Redox signaling pathways involved in neuronal ischemic preconditioning Tian TH , Burrage K , 2006 . Stochastic models for regulatory networks of the genetic toggle switch . Proc Natl Acad Sci USA , 103 ( 22 ): 8372 - 8377 . https://doi.org/10.1073/pnas.0507818103 https://doi.org/10.1073/pnas.0507818103 Stochastic models for regulatory networks of the genetic toggle switch Tsujimoto Y , 2003 . Cell death regulation by the Bcl-2 protein family in the mitochondria . J Cell Physiol , 195 ( 2 ): 158 - 167 . https://doi.org/10.1002/jcp.10254 https://doi.org/10.1002/jcp.10254 Cell death regulation by the Bcl-2 protein family in the mitochondria Tyson JJ , Chen KC , Novak B , 2003 . Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell . Curr Opin Cell Biol , 15 ( 2 ): 221 - 231 . https://doi.org/10.1016/S0955-0674(03)00017-6 https://doi.org/10.1016/S0955-0674(03)00017-6 Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell Willis SN , Adams JM , 2005 . Life in the balance: how BH3-only proteins induce apoptosis . Curr Opin Cell Biol , 17 ( 6 ): 617 - 625 . https://doi.org/10.1016/j.ceb.2005.10.001 https://doi.org/10.1016/j.ceb.2005.10.001 Life in the balance: how BH3-only proteins induce apoptosis Yang CJ , Hawkins KE , Doré S , et al. , 2019 . Neuroinflammatory mechanisms of blood-brain barrier damage in ischemic stroke . Am J Physiol Cell Physiol , 316 ( 2 ): C135 - C153 . https://doi.org/10.1152/ajpcell.00136.2018 https://doi.org/10.1152/ajpcell.00136.2018 Neuroinflammatory mechanisms of blood-brain barrier damage in ischemic stroke Yin ZY , Qi H , Liu LL , et al. , 2017 . The optimal regulation mode of Bcl-2 apoptotic switch revealed by bistability analysis . Biosystems , 162 : 44 - 52 . https://doi.org/10.1016/j.biosystems.2017.09.011 https://doi.org/10.1016/j.biosystems.2017.09.011 The optimal regulation mode of Bcl-2 apoptotic switch revealed by bistability analysis Youle RJ , Strasser A , 2008 . The BCL-2 protein family: Opposing activities that mediate cell death . Nat Rev Mol Cell Biol , 9 ( 1 ): 47 - 59 . https://doi.org/10.1038/nrm2308 https://doi.org/10.1038/nrm2308 The BCL-2 protein family: Opposing activities that mediate cell death Dan MU and Tingzhe SUN conceptualized the study, performed project administration, and acquired the funding; Dan MU, Huaguang QIN, and Tingzhe SUN developed the methodology and wrote the manuscript; Dan MU, Huaguang QIN, Mengjie JIAO, Shaogui HUA, and Tingzhe SUN validated the study, performed the data analysis and curation, and reviewed the draft; Tingzhe SUN supervised the study. All authors have read and agreed to the published version of the manuscript, and therefore, have full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. This research was supported by the National Natural Science Foundation of China (Nos. 31971185 and 31800316), the Key Projects of Support Program for Outstanding Young Talents in Colleges and Universities of Anhui Province (Nos. gxyqZD2020031 and gxyq2018034), and the Key Project of the Education Department of Anhui Province, China (No. KJ2017A359). We thank Dr. Liangliang ZHU (School of Life Sciences, the Province Key Laboratory of the Biodiversity Study and Ecology Conservation in Southwest Anhui, Anqing Normal University, Anhui, China) for critical reading of the manuscript. 10.1631/jzus.B2000540 1673-1581（2021）02-0123-13 MU Dan,QIN Huaguang,JIAO Mengjie et al., 2021. Modeling the neuro-protection of theaflavic acid from black tea and its synergy with nimodipine via mitochondria apoptotic pathway. Journal of Zhejiang University-SCIENCE B(Biomedicine & Biotechnology), 22(2):123-135. MU Dan,QIN Huaguang,JIAO Mengjie et al., 2021. Modeling the neuro-protection of theaflavic acid from black tea and its synergy with nimodipine via mitochondria apoptotic pathway. Journal of Zhejiang University-SCIENCE B(Biomedicine & Biotechnology), 22(2):123-135. 2020-09-11 2020-11-08 2021-02-15 2022-05-09 © Zhejiang University Press 2021 Dan MU, Huaguang QIN, Mengjie JIAO, Shaogui HUA, and Tingzhe SUN declare that they have no conflict of interest. This article does not contain any studies with human or animal subjects performed by any of the authors. Compliance with ethics guidelines Supplementary information 1 Figs. S1 and S2; Tables S1‒S3 浙江大学学报（英文版）（B辑：生物医学和生物技术） 2 22