“双稳性”的版本间的差异

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[[Image:Bistability graph.svg|thumb|upright=1.4|A graph of the [[potential energy]] of a bistable system; it has two local minima <math>x_1</math> and <math>x_2</math>.  A surface shaped like this with two "low points" can act as a bistable system; a ball resting on the surface can only be stable at those two positions, such as balls marked "1" and "2".  Between the two is a local maximum <math>x_3</math>. A ball located at this point, ball 3, is in equilibrium but unstable; the slightest disturbance will cause it to move to one of the stable points.]]
 
  
  
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</math>
 
</math>
  
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该方程有三个平衡点: <math>y=1</math>, <math>y=0</math>, and <math>y=-1</math>。中点 <math>y=0</math> 不稳定，而其他两点是稳定的。<math>y(t)</math>的演化方向和最终状态取决于初始条件 <math>y(0)</math>。若 <math>y(0)>0</math>，则 <math>y(t)</math> 趋向于1，若 <math>y(0)<0</math>，则 <math>y(t)</math> 趋向-1。<ref name="Chong">{{cite journal | author = Ket Hing Chong | author2 = Sandhya Samarasinghe | author3 = Don Kulasiri | author4 = Jie Zheng | name-list-style = amp | year = 2015| title = Computational techniques in mathematical modelling of biological switches | journal = Modsim2015 | pages =  578–584 }} For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in {{cite journal |last1=Collins |first1=James J. |last2=Gardner |first2=Timothy S. |last3=Cantor |first3=Charles R. |title=Construction of a genetic toggle switch in Escherichia coli |journal=Nature |volume=403 |issue=6767 |pages=339–42 |year=2000 |pmid=10659857 |doi=10.1038/35002131 |bibcode=2000Natur.403..339G |s2cid=345059 }}. The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'.</ref>
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该方程有三个平衡点: <math>y=1</math>, <math>y=0</math>, and <math>y=-1</math>。中点 <math>y=0</math> 不稳定，而其他两点是稳定的。<math>y(t)</math>的演化方向和最终状态取决于初始条件 <math>y(0)</math>。若 <math>y(0)>0</math>，则 <math>y(t)</math> 趋向于1，若 <math>y(0)<0</math>，则 <math>y(t)</math> 趋向-1。<ref name="Chong">{{cite journal | author = Ket Hing Chong | author2 = Sandhya Samarasinghe | author3 = Don Kulasiri | author4 = Jie Zheng | year = 2015| title = Computational techniques in mathematical modelling of biological switches | journal = Modsim2015 | pages =  578–584 }} For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in {{cite journal |last1=Collins |first1=James J. |last2=Gardner |first2=Timothy S. |last3=Cantor |first3=Charles R. |title=Construction of a genetic toggle switch in Escherichia coli |journal=Nature |volume=403 |issue=6767 |pages=339–42 |year=2000 |pmid=10659857 |doi=10.1038/35002131 |bibcode=2000Natur.403..339G }}. The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'.</ref>
  
 
更复杂的双稳性系统 <math>\frac{dy}{dt} = y (r-y^2)</math> 具有超临界的'''<font color="#ff8000">叉分岔pitchfork bifurcation</font>'''现象。
 
更复杂的双稳性系统 <math>\frac{dy}{dt} = y (r-y^2)</math> 具有超临界的'''<font color="#ff8000">叉分岔pitchfork bifurcation</font>'''现象。
  
 
==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.
 
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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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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>。
 
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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'''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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'''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}}</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>
  
 
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>
 
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，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>
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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. |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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棘轮棘爪是一种精心设计的工具ーー一种多重稳定的“过中心”系统，用来产生不可逆的运动。当棘爪向前方转动时，它会越过中心。在这种情况下，“过中心”是指棘轮是稳定的，“锁定”在一个给定的位置，直到再次点击向前，这与棘轮无法在反方向转动无关。
 
棘轮棘爪是一种精心设计的工具ーー一种多重稳定的“过中心”系统，用来产生不可逆的运动。当棘爪向前方转动时，它会越过中心。在这种情况下，“过中心”是指棘轮是稳定的，“锁定”在一个给定的位置，直到再次点击向前，这与棘轮无法在反方向转动无关。
  
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[[File:Ratchet example.gif|thumbnail|A ratchet in action. Each tooth in the ratchet together with the regions to either side of it constitutes a simple bistable mechanism.]]
 
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thumbnail|A ratchet in action. Each tooth in the ratchet together with the regions to either side of it constitutes a simple bistable mechanism.
 
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一个行动中的棘轮。棘轮中的每个齿以及棘轮两侧的区域构成一个简单的双稳态机构。
 
  
 
== 参考文献 ==
 
== 参考文献 ==

2022年5月21日 (六) 10:32的版本

此词条暂由彩云小译翻译，翻译字数共1727，正在由用户Llk进行审阅，带来阅读不便，请见谅。


若一个动力学系统有两个稳定的平衡态，则称该系统具有 双稳性Bistability[1]。电灯的开关是一种常见的双稳性机械系统，其中的杠杆设计使开关停留在“开”或“关”的位置，而不是中间位置。机械系统、电子系统、非线性光学系统、化学系统和生物系统都可能具有双稳性。

在保守力场系统中，若势能有两个局部极小值点，则系统具有双稳性[2]。由于势能函数具有连续性，该系统一定存在不稳定的势能极大值点。在静息状态下，粒子将处于某一势能极小值点，极大值点可以视为它们之间的一道屏障。若给予足够的活化能，粒子能够穿过极大值屏障，从一个稳定区域到达另一个稳定区域，并在一段时间之后静止在极小值点上（假定该系统有能量损耗），这段时间称为 驰豫时间relaxation time

双稳性被广泛应用于二进制存储器当中，是 触发器flip-flop的基本特性，可以存储1比特数据，其中一个状态表示“0”，另一个状态表示“1”。它也用于 弛豫振荡器relaxation oscillator、 多谐振荡器multivibrator和 施密特触发器Schmitt trigger等器件中。光学系统中的 光学双稳性Optical bistability表示两种稳定的共振传输状态，依赖于输入内容。生物化学系统的双稳性通常与滞回现象Hysteresis有关。


数学

双稳性动力学系统中的经典数学模型如下

[math]\displaystyle{ \frac{dy}{dt} = y(1-y^2). }[/math]

该方程有三个平衡点: [math]\displaystyle{ y=1 }[/math], [math]\displaystyle{ y=0 }[/math], and [math]\displaystyle{ y=-1 }[/math]。中点 [math]\displaystyle{ y=0 }[/math] 不稳定，而其他两点是稳定的。[math]\displaystyle{ y(t) }[/math]的演化方向和最终状态取决于初始条件 [math]\displaystyle{ y(0) }[/math]。若 [math]\displaystyle{ y(0)\gt 0 }[/math]，则 [math]\displaystyle{ y(t) }[/math] 趋向于1，若 [math]\displaystyle{ y(0)\lt 0 }[/math]，则 [math]\displaystyle{ y(t) }[/math] 趋向-1。[3]

更复杂的双稳性系统 [math]\displaystyle{ \frac{dy}{dt} = y (r-y^2) }[/math] 具有超临界的叉分岔pitchfork bifurcation现象。

生物化学

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。[4]。


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[5]。


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。


具有上述特点的数学模型样例如下[6]：

[math]\displaystyle{ \frac{\mathrm{d}x}{\mathrm{d}t} = r + \frac{x^5}{1+x^5} - x }[/math]


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。[7]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[8][9]的形成与眼睛的发育。[10]

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。[11]


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[4]。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。[12][13]


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[14]），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。


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。

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[15]和细胞凋亡。双稳性还能解释癌症早期的细胞内稳态cellular homeostasis失调、朊病毒疾病以及物种形成speciation[4]

机械

Bistability as applied in the design of mechanical systems is more commonly said to be "over centre"—that is, work is done on the system to move it just past the peak, at which point the mechanism goes "over centre" to its secondary stable position. The result is a toggle-type action- work applied to the system below a threshold sufficient to send it 'over center' results in no change to the mechanism's state.


在机械系统中应用于机械系统设计中的双稳态通常被称为“过中心”ーー也就是说，系统所做的工作使其刚刚超过峰值，在峰值处，机构“过中心”到次稳定位置。结果是一个切换类型的动作-工作应用于系统低于一个阈值足以发送它“过中心”的结果没有改变机制的状态。

Springs are a common method of achieving an "over centre" action. A spring attached to a simple two position ratchet-type mechanism can create a button or plunger that is clicked or toggled between two mechanical states. Many ballpoint and rollerball retractable pens employ this type of bistable mechanism.

弹簧是实现“过中心”动作的常用方法。一个弹簧连接到一个简单的两个位置棘轮式机构，可以创造一个按钮或柱塞点击或切换在两个机械状态。许多圆珠笔和滚珠笔都采用这种双稳态机构。

An even more common example of an over-center device is an ordinary electric wall switch. These switches are often designed to snap firmly into the "on" or "off" position once the toggle handle has been moved a certain distance past the center-point.


一个更常见的过中心装置的例子是一个普通的电动墙开关。这些开关的设计通常是，一旦切换手柄已经移动了一定距离，超过中心点，就可以牢固地扣入“开”或“关”位置。

A ratchet-and-pawl is an elaboration—a multi-stable "over center" system used to create irreversible motion. The pawl goes over center as it is turned in the forward direction. In this case, "over center" refers to the ratchet being stable and "locked" in a given position until clicked forward again; it has nothing to do with the ratchet being unable to turn in the reverse direction.

棘轮棘爪是一种精心设计的工具ーー一种多重稳定的“过中心”系统，用来产生不可逆的运动。当棘爪向前方转动时，它会越过中心。在这种情况下，“过中心”是指棘轮是稳定的，“锁定”在一个给定的位置，直到再次点击向前，这与棘轮无法在反方向转动无关。


参考文献

  1. ↑ Morris, Christopher G. (1992). Academic Press Dictionary of Science and Technology. Gulf Professional publishing. pp. 267. ISBN 978-0122004001. https://books.google.com/books?id=nauWlPTBcjIC&q=bistable+bistability&pg=PA267. 
  2. ↑ Nazarov, Yuli V.; Danon, Jeroen (2013). Advanced Quantum Mechanics: A Practical Guide. Cambridge University Press. pp. 291. ISBN 978-1139619028. https://books.google.com/books?id=w20gAwAAQBAJ&q=bistability+minimum&pg=PA291. 
  3. ↑ Ket Hing Chong; Sandhya Samarasinghe; Don Kulasiri; Jie Zheng (2015). "Computational techniques in mathematical modelling of biological switches". Modsim2015: 578–584. For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in Collins, James J.; Gardner, Timothy S.; Cantor, Charles R. (2000). "Construction of a genetic toggle switch in Escherichia coli". Nature. 403 (6767): 339–42. Bibcode:2000Natur.403..339G. doi:10.1038/35002131. PMID 10659857.. The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'.
  4. ↑ 4.0 4.1 4.2 Wilhelm, T (2009). "The smallest chemical reaction system with bistability". BMC Systems Biology. 3: 90. doi:10.1186/1752-0509-3-90. PMC 2749052. PMID 19737387.
  5. ↑ O. Brandman, J. E. Ferrell Jr., R. Li, T. Meyer, Science 310, 496 (2005)
  6. ↑ Angeli, David; Ferrell, JE; Sontag, Eduardo D (2003). "Detection of multistability, bifurcations, and hysteresis in a large calss of biological positive-feedback systems". PNAS. 101 (7): 1822–7. Bibcode:2004PNAS..101.1822A. doi:10.1073/pnas.0308265100. PMC 357011. PMID 14766974.
  7. ↑ Ferrell JE Jr. (2008). "Feedback regulation of opposing enzymes generates robust, all-or-none bistable responses". Current Biology. 18 (6): R244–R245. doi:10.1016/j.cub.2008.02.035. PMC 2832910. PMID 18364225.
  8. ↑ Wang, Yu-Chiun; Ferguson, Edwin L. (10 March 2005). "Spatial bistability of Dpp–receptor interactions during Drosophila dorsal–ventral patterning". Nature. 434 (7030): 229–234. Bibcode:2005Natur.434..229W. doi:10.1038/nature03318. PMID 15759004.
  9. ↑ Umulis, D. M.; Mihaela Serpe; Michael B. O’Connor; Hans G. Othmer (1 August 2006). "Robust, bistable patterning of the dorsal surface of the Drosophila embryo". Proceedings of the National Academy of Sciences. 103 (31): 11613–11618. Bibcode:2006PNAS..10311613U. doi:10.1073/pnas.0510398103. PMC 1544218. PMID 16864795.
  10. ↑ Graham, T. G. W.; Tabei, S. M. A.; Dinner, A. R.; Rebay, I. (22 June 2010). "Modeling bistable cell-fate choices in the Drosophila eye: qualitative and quantitative perspectives". Development. 137 (14): 2265–2278. doi:10.1242/dev.044826. PMC 2889600. PMID 20570936.
  11. ↑ 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.
  12. ↑ Elf, J.; Ehrenberg, M. (2004). "Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases". Systems Biology. 1 (2): 230–236. doi:10.1049/sb:20045021. PMID 17051695.
  13. ↑ Kochanczyk, M.; Jaruszewicz, J.; Lipniacki, T. (Jul 2013). "Stochastic transitions in a bistable reaction system on the membrane". Journal of the Royal Society Interface. 10 (84): 20130151. doi:10.1098/rsif.2013.0151. PMC 3673150. PMID 23635492.
  14. ↑ Nielsen; Dolganov, Nadia A.; Rasmussen, Thomas; Otto, Glen; Miller, Michael C.; Felt, Stephen A.; Torreilles, Stéphanie; Schoolnik, Gary K.; et al. (2010). Isberg, Ralph R. (ed.). "A Bistable Switch and Anatomical Site Control Vibrio cholerae Virulence Gene Expression in the Intestine". PLOS Pathogens. 6 (9): 1. doi:10.1371/journal.ppat.1001102. PMC 2940755. PMID 20862321.
  15. ↑ Ghaffarizadeh A, Flann NS, Podgorski GJ (2014). "Multistable switches and their role in cellular differentiation networks". BMC Bioinformatics. 15: S7+. doi:10.1186/1471-2105-15-s7-s7. PMC 4110729. PMID 25078021.

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