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| | ==生物化学== | | ==生物化学== |
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| | [[File:Stimuli.pdf|thumb|Three-dimensional invariant measure for cellular-differentiation featuring a two-stable mode. | | [[File:Stimuli.pdf|thumb|Three-dimensional invariant measure for cellular-differentiation featuring a two-stable mode. |
| | The axes denote cell counts for three types of cells: progenitor (<math>z</math>), osteoblast (<math>y</math>), and chondrocyte (<math>x</math>). Pro-osteoblast stimulus promotes P→O transition.<ref name=CME>{{cite journal | last1 = Kryven| first1 = I.| last2 = Röblitz| first2 = S.| last3 = Schütte| first3 = Ch.| year =2015| title = Solution of the chemical master equation by radial basis functions approximation with interface tracking| journal = BMC Systems Biology | volume = 9| issue = 1| pages = 67| doi = 10.1186/s12918-015-0210-y| pmid = 26449665| pmc= 4599742}} {{open access}}</ref>]] | | The axes denote cell counts for three types of cells: progenitor (<math>z</math>), osteoblast (<math>y</math>), and chondrocyte (<math>x</math>). Pro-osteoblast stimulus promotes P→O transition.<ref name=CME>{{cite journal | last1 = Kryven| first1 = I.| last2 = Röblitz| first2 = S.| last3 = Schütte| first3 = Ch.| year =2015| title = Solution of the chemical master equation by radial basis functions approximation with interface tracking| journal = BMC Systems Biology | volume = 9| issue = 1| pages = 67| doi = 10.1186/s12918-015-0210-y| pmid = 26449665| pmc= 4599742}} {{open access}}</ref>]] |
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| − | 使用双稳性视角有助于理解细胞的基础功能，比如细胞周期中的决策过程、细胞分化<ref name=Ghaffarizadeh>{{cite journal |vauthors=Ghaffarizadeh A, Flann NS, Podgorski GJ |year = 2014 |title = Multistable switches and their role in cellular differentiation networks |journal = BMC Bioinformatics |volume = 15 |pages = S7+ |doi = 10.1186/1471-2105-15-s7-s7 |pmid = 25078021 |pmc = 4110729}}</ref>和细胞凋亡。双稳性还能解释癌症早期的<font color="#ff8000">细胞内稳态cellular homeostasis</font>失调、朊病毒疾病以及<font color="#ff8000">物种形成speciation</font><ref name=Wilhelm>{{cite journal |author = Wilhelm, T |year = 2009 |title = The smallest chemical reaction system with bistability |journal = BMC Systems Biology |volume = 3 |pages = 90 |doi = 10.1186/1752-0509-3-90 |pmid = 19737387 |pmc = 2749052}}</ref>。
| + | 若想合理利用双稳性，生物化学系统需要具备三个必要条件：正反馈机制、约束机制和稳定机制。<ref name=Wilhelm>{{cite journal |author = Wilhelm, T |year = 2009 |title = The smallest chemical reaction system with bistability |journal = BMC Systems Biology |volume = 3 |pages = 90 |doi = 10.1186/1752-0509-3-90 |pmid = 19737387 |pmc = 2749052}}</ref>。 |
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| − | 正反馈调节可以产生双稳性。正反馈回路(比如 X 激活 Y、Y 激活 X)将输出信号与输入信号耦合在一起，是细胞信号转导中的重要调节机制，它可以作为<font color="#ff8000">全或无All-or-none</font>信号开关<ref name="O. Brandman, J. E 2005">O. Brandman, J. E. Ferrell Jr., R. Li, T. Meyer, Science 310, 496 (2005)</ref>。许多生物系统（如非洲爪蟾卵“Xenopus”母细胞的成熟过程<ref>{{cite journal|author1=Ferrell JE Jr. |author2=Machleder EM |s2cid=34863795 |title=The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes.|journal=Science|date=1998|volume=280|issue=5365|pages=895–8|pmid=9572732|doi=10.1126/science.280.5365.895|bibcode=1998Sci...280..895F }}<!--|accessdate=20 March 2015--></ref>、哺乳动物的钙信号转导过程和芽殖酵母“budding yeast”的极化）都包含时序正反馈回路，或者多时间尺度的反馈回路<ref name="O. Brandman, J. E 2005"/> 。具有多时间尺度反馈回路（或称为”双时间开关dual-time switches”）能够(a)增加调节: 两个开关具有独立可变的激活和失活时间，或(b)过滤噪声<ref name="O. Brandman, J. E 2005"/>。
| + | 受'''约束'''的'''正反馈'''过程能够产生双稳性。正反馈机制(比如 X 激活 Y、Y 激活 X)将输出信号与输入信号耦合在一起，使系统向特定方向持续演化。约束机制防止正反馈过程无止境地进行。它们协同作用可以产生'''全或无All-or-none'''信号开关[7]。许多生物化学系统（如非洲爪蟾卵“Xenopus”母细胞的成熟过程[8]、哺乳动物的钙信号转导过程和芽殖酵母“budding yeast”的极化）都包含或慢或快的时序正反馈回路，有时二者兼而有之，称为“双时间开关dual-time switches”。dual-time switches能够增加调节能力（每个开关具有独立可变的激活和失活时间）并过滤噪声<ref name="O. Brandman, J. E 2005">O. Brandman, J. E. Ferrell Jr., R. Li, T. Meyer, Science 310, 496 (2005)</ref>。 |
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| − | Bistability can also arise in a biochemical system only for a particular range of parameter values, where the parameter can often be interpreted as the strength of the feedback. In several typical examples, the system has only one stable fixed point at low values of the parameter. A [[saddle-node bifurcation]] gives rise to a pair of new fixed points emerging, one stable and the other unstable, at a critical value of the parameter. The unstable solution can then form another saddle-node bifurcation with the initial stable solution at a higher value of the parameter, leaving only the higher fixed solution. Thus, at values of the parameter between the two critical values, the system has two stable solutions. An example of a dynamical system that demonstrates similar features is
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| − | 在生物化学系统中，往往只有在特定的生化参数范围内才能产生双稳性，这些参数可以被解释为反馈强度。以单一参数 <math>r</math> 描述反馈强度的系统为例。当 <math>r</math> 比较小的时候，系统只有一个稳定不动点。当 <math>r</math> 增长到临界值 <math>r_0</math> 附近时，一个鞍结分岔引起一对新的不动点出现，一个是稳定的，一个是不稳定的。当 <math>r</math> 继续增长时，不稳定解与初始稳定解形成另一个鞍结分岔，只留下较高的固定解，在两个临界值之间的参数值，系统有两个稳定的解。
| + | 生化参数处于特定范围内时才能产生双稳性，这些参数共同影响着反馈强度。以单一参数 r 调节反馈强度的系统为例：（1）当 r <r1时，系统只有一个稳定不动点x1。（2）当 r1<r<r2时，一个'''鞍结分岔saddle-node bifurcation'''产生一对新的不动点：不稳定点x2和稳定点x3，且x1<x2<x3。它们构成双稳态系统。（3）当 r2<r时 ，x1与x2作为鞍结分岔逆过程融合消失，只留下x3。 |
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| | + | 具有上述特点的数学模型样例如下<ref name="Angeli 2003">{{cite journal| author1 = Angeli, David| author2=Ferrell, JE| author3=Sontag, Eduardo D| title=Detection of multistability, bifurcations, and hysteresis in a large calss of biological positive-feedback systems| journal=PNAS| year=2003| volume=101| issue=7| doi=10.1073/pnas.0308265100| pmid=14766974| pmc=357011| pages=1822–7| bibcode=2004PNAS..101.1822A| doi-access=free}}</ref>： |
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| − | where <math>x</math> is the output, and <math>r</math> is the parameter, acting as the input.<ref name="Angeli 2003">{{cite journal| author1 = Angeli, David| author2=Ferrell, JE| author3=Sontag, Eduardo D| title=Detection of multistability, bifurcations, and hysteresis in a large calss of biological positive-feedback systems| journal=PNAS| year=2003| volume=101| issue=7| doi=10.1073/pnas.0308265100| pmid=14766974| pmc=357011| pages=1822–7| bibcode=2004PNAS..101.1822A| doi-access=free}}</ref>
| + | '''稳定机制'''（比如额外的激活子activator和抑制子inhibitor）能够提升系统的'''鲁棒性robustness'''，使系统能够容忍更剧烈的生化参数变化，保持“开关”特性。例如，在细胞生物学中， CDK1(Cyclin Dependent Kinase 1)激活 Cdc25（激活子activator），同时使 Wee1（inactivator）失活，让细胞进入有丝分裂。如果没有这种双重反馈，系统仍然是双稳态的，但是不能容忍如此广泛的浓度范围。<ref>{{cite journal|author=Ferrell JE Jr.|title=Feedback regulation of opposing enzymes generates robust, all-or-none bistable responses|journal=Current Biology|year=2008|volume=18|issue=6|doi=10.1016/j.cub.2008.02.035|pages=R244–R245|pmid=18364225|pmc=2832910}}</ref>双稳态在'''黑腹果蝇Drosophila melanogaster'''的胚胎发育中也被描述过，例如'''前后轴anterior-posterior axis'''和'''背腹轴dorso-ventral axis'''<ref>{{cite journal|last=Wang|first=Yu-Chiun|author2=Ferguson, Edwin L.|title=Spatial bistability of Dpp–receptor interactions during Drosophila dorsal–ventral patterning|journal=Nature|date=10 March 2005|volume=434|issue=7030|pages=229–234|doi=10.1038/nature03318|pmid=15759004|bibcode=2005Natur.434..229W|s2cid=4415152}}</ref><ref>{{cite journal|last=Umulis|first=D. M. |author2=Mihaela Serpe |author3=Michael B. O’Connor |author4=Hans G. Othmer|title=Robust, bistable patterning of the dorsal surface of the Drosophila embryo|journal=Proceedings of the National Academy of Sciences|date=1 August 2006|volume=103|issue=31|pages=11613–11618|doi=10.1073/pnas.0510398103 |pmid=16864795 |pmc=1544218|bibcode=2006PNAS..10311613U |doi-access=free }}</ref>的形成与眼睛的发育。<ref>{{cite journal|last=Graham|first=T. G. W.|author2=Tabei, S. M. A.|author3=Dinner, A. R.|author4=Rebay, I.|title=Modeling bistable cell-fate choices in the Drosophila eye: qualitative and quantitative perspectives|journal=Development|date=22 June 2010|volume=137|issue=14|pages=2265–2278|doi=10.1242/dev.044826|pmid=20570936|pmc=2889600}}</ref> |
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| − | where x is the output, and r is the parameter, acting as the input.
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| − | 其中 x 是输出，r 是参数，作为输入。
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| − | Bistability can be modified to be more robust and to tolerate significant changes in concentrations of reactants, while still maintaining its "switch-like" character. Feedback on both the activator of a system and inhibitor make the system able to tolerate a wide range of concentrations. An example of this in cell biology is that activated CDK1 (Cyclin Dependent Kinase 1) activates its activator Cdc25 while at the same time inactivating its inactivator, Wee1, thus allowing for progression of a cell into mitosis. Without this double feedback, the system would still be bistable, but would not be able to tolerate such a wide range of concentrations.<ref>{{cite journal|author=Ferrell JE Jr.|title=Feedback regulation of opposing enzymes generates robust, all-or-none bistable responses|journal=Current Biology|year=2008|volume=18|issue=6|doi=10.1016/j.cub.2008.02.035|pages=R244–R245|pmid=18364225|pmc=2832910}}</ref>
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| − | Bistability can be modified to be more robust and to tolerate significant changes in concentrations of reactants, while still maintaining its "switch-like" character. Feedback on both the activator of a system and inhibitor make the system able to tolerate a wide range of concentrations. An example of this in cell biology is that activated CDK1 (Cyclin Dependent Kinase 1) activates its activator Cdc25 while at the same time inactivating its inactivator, Wee1, thus allowing for progression of a cell into mitosis. Without this double feedback, the system would still be bistable, but would not be able to tolerate such a wide range of concentrations.
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| − | 可以对双稳态进行修改，使其更加稳定，能够容忍反应物浓度的显著变化，同时仍然保持其“开关式”特性。系统的激活剂和抑制剂的反馈使系统能够容忍广泛的浓度范围。细胞生物学中的一个例子是，激活的 CDK1(细胞周期蛋白依赖性激酶1)激活其激活因子 Cdc25，同时使其失活因子 wee1失活，从而允许细胞进入有丝分裂。如果没有这种双重反馈，系统仍然是双稳态的，但是不能容忍如此广泛的浓度范围。
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| − | Bistability has also been described in the embryonic development of ''Drosophila melanogaster'' (the fruit fly). Examples are anterior-posterior<ref>{{cite journal|last=Lopes|first=Francisco J. P.|author2=Vieira, Fernando M. C.|author3=Holloway, David M.|author4=Bisch, Paulo M.|author5=Spirov, Alexander V.|author6=Ohler, Uwe|title=Spatial Bistability Generates hunchback Expression Sharpness in the Drosophila Embryo|journal=PLOS Computational Biology|date=26 September 2008|volume=4|issue=9|pages=e1000184|doi=10.1371/journal.pcbi.1000184|pmid=18818726|pmc=2527687|bibcode=2008PLSCB...4E0184L}}</ref> and dorso-ventral<ref>{{cite journal|last=Wang|first=Yu-Chiun|author2=Ferguson, Edwin L.|title=Spatial bistability of Dpp–receptor interactions during Drosophila dorsal–ventral patterning|journal=Nature|date=10 March 2005|volume=434|issue=7030|pages=229–234|doi=10.1038/nature03318|pmid=15759004|bibcode=2005Natur.434..229W|s2cid=4415152}}</ref><ref>{{cite journal|last=Umulis|first=D. M. |author2=Mihaela Serpe |author3=Michael B. O’Connor |author4=Hans G. Othmer|title=Robust, bistable patterning of the dorsal surface of the Drosophila embryo|journal=Proceedings of the National Academy of Sciences|date=1 August 2006|volume=103|issue=31|pages=11613–11618|doi=10.1073/pnas.0510398103 |pmid=16864795 |pmc=1544218|bibcode=2006PNAS..10311613U |doi-access=free }}</ref> axis formation and eye development.<ref>{{cite journal|last=Graham|first=T. G. W.|author2=Tabei, S. M. A.|author3=Dinner, A. R.|author4=Rebay, I.|title=Modeling bistable cell-fate choices in the Drosophila eye: qualitative and quantitative perspectives|journal=Development|date=22 June 2010|volume=137|issue=14|pages=2265–2278|doi=10.1242/dev.044826|pmid=20570936|pmc=2889600}}</ref>
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| − | Bistability has also been described in the embryonic development of Drosophila melanogaster (the fruit fly). Examples are anterior-posterior and dorso-ventral axis formation and eye development.
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| − | 双稳态在黑腹果蝇(果蝇)的胚胎发育中也被描述过。例如前后轴和背腹轴的形成和眼睛的发育。
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| − | A prime example of bistability in biological systems is that of [[Sonic hedgehog]] (Shh), a secreted signaling molecule, which plays a critical role in development. Shh functions in diverse processes in development, including patterning limb bud tissue differentiation. The Shh signaling network behaves as a bistable switch, allowing the cell to abruptly switch states at precise Shh concentrations. ''gli1'' and ''gli2'' transcription is activated by Shh, and their gene products act as transcriptional activators for their own expression and for targets downstream of Shh signaling.<ref>Lai, K., M.J. Robertson, and D.V. Schaffer, The sonic hedgehog signaling system as a bistable genetic switch. Biophys J, 2004. 86(5): pp. 2748–57.</ref> Simultaneously, the Shh signaling network is controlled by a negative feedback loop wherein the Gli transcription factors activate the enhanced transcription of a repressor (Ptc). This signaling network illustrates the simultaneous positive and negative feedback loops whose exquisite sensitivity helps create a bistable switch.
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| − | A prime example of bistability in biological systems is that of Sonic hedgehog (Shh), a secreted signaling molecule, which plays a critical role in development. Shh functions in diverse processes in development, including patterning limb bud tissue differentiation. The Shh signaling network behaves as a bistable switch, allowing the cell to abruptly switch states at precise Shh concentrations. gli1 and gli2 transcription is activated by Shh, and their gene products act as transcriptional activators for their own expression and for targets downstream of Shh signaling.Lai, K., M.J. Robertson, and D.V. Schaffer, The sonic hedgehog signaling system as a bistable genetic switch. Biophys J, 2004. 86(5): pp. 2748–57. Simultaneously, the Shh signaling network is controlled by a negative feedback loop wherein the Gli transcription factors activate the enhanced transcription of a repressor (Ptc). This signaling network illustrates the simultaneous positive and negative feedback loops whose exquisite sensitivity helps create a bistable switch.
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| − | 生物系统双稳态的一个典型例子是音速猬，一种分泌型信号分子，在生物系统的发展中起着关键作用。Shh 在不同的发育过程中起作用，包括模式化的肢芽组织分化。Shh 信号网络就像一个双稳态开关，允许细胞在精确的 Shh 浓度下突然切换状态。Gli1和 gli2的转录被 Shh 激活，它们的基因产物作为转录激活因子，用于自身的表达和 Shh 信号的下游目标。赖、 k、 m.j. Robertson 及 d.v。作为双稳态遗传开关的音猬信号系统。2004.86(5) : pp.2748–57.同时，Shh 信号网络由一个负反馈环控制，其中 Gli 转录因子激活一个抑制因子(Ptc)的增强转录。这种信令网络说明了同时存在的正反馈环路和负反馈环路，其灵敏度极高，有助于产生双稳态开关。
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| − | Bistability can only arise in biological and chemical systems if three necessary conditions are fulfilled: positive [[feedback]], a mechanism to filter out small stimuli and a mechanism to prevent increase without bound.<ref name=Wilhelm/>
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| − | Bistability can only arise in biological and chemical systems if three necessary conditions are fulfilled: positive feedback, a mechanism to filter out small stimuli and a mechanism to prevent increase without bound.
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| − | 生物和化学系统中只有满足以下三个必要条件才会出现双稳性: 正反馈、过滤小刺激的机制和无约束增长的防止机制。
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| − | Bistable chemical systems have been studied extensively to analyze relaxation kinetics, [[non-equilibrium thermodynamics]], [[stochastic resonance]], as well as [[climate variability and change|climate change]].<ref name=Wilhelm/> In bistable spatially extended systems the onset of local correlations and propagation of traveling waves have been analyzed.<ref name=Elf>{{cite journal |last1 = Elf |first1 = J. | last2 = Ehrenberg| first2 = M. |s2cid = 17770042 |year = 2004 |title = Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases |journal = Systems Biology|volume = 1 |number = 2| pages = 230–236 |pmid = 17051695 | doi=10.1049/sb:20045021}}</ref><ref name=Kochanzyck>{{cite journal |last1 = Kochanczyk |first1 = M. |last2 = Jaruszewicz |first2 = J. |last3 = Lipniacki |first3 = T. |title = Stochastic transitions in a bistable reaction system on the membrane |journal = Journal of the Royal Society Interface |volume = 10 |number = 84 |pages = 20130151 |pmid = 23635492 |pmc = 3673150 |doi = 10.1098/rsif.2013.0151 |date=Jul 2013}}</ref>
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| − | Bistable chemical systems have been studied extensively to analyze relaxation kinetics, non-equilibrium thermodynamics, stochastic resonance, as well as climate change. In bistable spatially extended systems the onset of local correlations and propagation of traveling waves have been analyzed.
| + | 另一个典型例子是音猬因子Sonic hedgehog（Shh）信号网络，同时存在的正反馈环路和负反馈环路。Shh是一种分泌型信号分子，在发育过程中起着关键作用，比如肢芽limb bud组织分化。当Shh浓度到达阈值时，gli1和 gli2的转录被激活，相应的产物作为转录激活因子进一步增强自身的转录，同时增强Ptc（一种抑制因子）的转录。<ref>Lai, K., M.J. Robertson, and D.V. Schaffer, The sonic hedgehog signaling system as a bistable genetic switch. Biophys J, 2004. 86(5): pp. 2748–57.</ref> |
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| − | 双稳态化学体系已经被广泛研究以分析弛豫动力学，非平衡态热力学，随机共振，以及气候变化。在双稳态空间扩展系统中，分析了局域相关性和行波传播的开始。
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| − | Bistability is often accompanied by [[hysteresis]]. On a population level, if many realisations of a bistable system are considered (e.g. many bistable cells ([[speciation]])<ref name=Nielsen>{{cite journal |author = Nielsen |last2 = Dolganov |first2 = Nadia A. |last3 = Rasmussen |first3 = Thomas |last4 = Otto |first4 = Glen |last5 = Miller |first5 = Michael C. |last6 = Felt |first6 = Stephen A. |last7 = Torreilles |first7 = Stéphanie |last8 = Schoolnik |first8 = Gary K. |editor1-last = Isberg |year = 2010 |editor1-first = Ralph R. |title = A Bistable Switch and Anatomical Site Control Vibrio cholerae Virulence Gene Expression in the Intestine |journal = PLOS Pathogens |volume = 6 |issue = 9 |pages = 1 |doi=10.1371/journal.ppat.1001102
| + | 双稳态化学体系已经被广泛研究，用以分析弛豫动力学，非平衡态热力学，随机共振，以及气候变化<ref name=Wilhelm/>。在'''空间扩展系统spatially extended systems'''中，双稳态被用以分析局域相关性和行波的传播。<ref name=Elf>{{cite journal |last1 = Elf |first1 = J. | last2 = Ehrenberg| first2 = M. |s2cid = 17770042 |year = 2004 |title = Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases |journal = Systems Biology|volume = 1 |number = 2| pages = 230–236 |pmid = 17051695 | doi=10.1049/sb:20045021}}</ref><ref name=Kochanzyck>{{cite journal |last1 = Kochanczyk |first1 = M. |last2 = Jaruszewicz |first2 = J. |last3 = Lipniacki |first3 = T. |title = Stochastic transitions in a bistable reaction system on the membrane |journal = Journal of the Royal Society Interface |volume = 10 |number = 84 |pages = 20130151 |pmid = 23635492 |pmc = 3673150 |doi = 10.1098/rsif.2013.0151 |date=Jul 2013}}</ref> |
| − | |pmid = 20862321 |pmc = 2940755|display-authors=etal}}</ref>), one typically observes [[bimodal distribution]]s. In an ensemble average over the population, the result may simply look like a smooth transition, thus showing the value of single-cell resolution.
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| − | Bistability is often accompanied by hysteresis. On a population level, if many realisations of a bistable system are considered (e.g. many bistable cells (speciation)), one typically observes bimodal distributions. In an ensemble average over the population, the result may simply look like a smooth transition, thus showing the value of single-cell resolution.
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| − | 双稳态常伴有滞后现象。在人口水平上，如果考虑一个双稳态系统的许多现实(例如:。许多双稳态细胞(物种形成) ，人们通常观察双峰分布。在人口的总体均值上，结果可能看起来就像是一个平滑的过渡，从而显示了单细胞分辨率的价值。
| + | 双稳态常伴有'''滞回现象hysteresis'''。在细胞群体水平上，如果内部存在许多种双稳态机制（比如双稳态细胞<ref name=Nielsen>{{cite journal |author = Nielsen |last2 = Dolganov |first2 = Nadia A. |last3 = Rasmussen |first3 = Thomas |last4 = Otto |first4 = Glen |last5 = Miller |first5 = Michael C. |last6 = Felt |first6 = Stephen A. |last7 = Torreilles |first7 = Stéphanie |last8 = Schoolnik |first8 = Gary K. |editor1-last = Isberg |year = 2010 |editor1-first = Ralph R. |title = A Bistable Switch and Anatomical Site Control Vibrio cholerae Virulence Gene Expression in the Intestine |journal = PLOS Pathogens |volume = 6 |issue = 9 |pages = 1 |doi=10.1371/journal.ppat.1001102 |
| | + | |pmid = 20862321 |pmc = 2940755|display-authors=etal}}</ref>），系统状态通常处于双峰分布，其变化过程就像平滑的过渡。这种个体与群体的关联性也体现了单细胞研究的价值。 |
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| − | A specific type of instability is known as ''modehopping'', which is bi-stability in the frequency space. Here trajectories can shoot between two stable limit cycles, and thus show similar characteristics as normal bi-stability when measured inside a Poincare section.
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| − | A specific type of instability is known as modehopping, which is bi-stability in the frequency space. Here trajectories can shoot between two stable limit cycles, and thus show similar characteristics as normal bi-stability when measured inside a Poincare section.
| + | 一种特殊类型的不稳定性被称为'''模式跳变modehopping'''，它是频率空间中的双稳定性。系统的演化轨迹可以在两个稳定极限环stable limit cycle之间跳转，在庞加莱截面Poincare section内呈现出双稳态相似的特性。 |
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| − | 一种特殊类型的不稳定性被称为调制解调器跳变，它是频率空间中的双稳定性。这里的轨迹可以在两个稳定极限环之间发射，因此当在庞加莱截面内测量时，显示出与正常双稳态相似的特性。
| + | 使用双稳性视角有助于理解细胞的基础功能，比如细胞周期中的决策过程、细胞分化<ref name=Ghaffarizadeh>{{cite journal |vauthors=Ghaffarizadeh A, Flann NS, Podgorski GJ |year = 2014 |title = Multistable switches and their role in cellular differentiation networks |journal = BMC Bioinformatics |volume = 15 |pages = S7+ |doi = 10.1186/1471-2105-15-s7-s7 |pmid = 25078021 |pmc = 4110729}}</ref>和细胞凋亡。双稳性还能解释癌症早期的<font color="#ff8000">细胞内稳态cellular homeostasis</font>失调、朊病毒疾病以及<font color="#ff8000">物种形成speciation</font><ref name=Wilhelm/> |
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| | == 机械 == | | == 机械 == |