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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。

==In biological and chemical systems==

==生物化学模型==

[[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>]]

= = 在生物和化学系统中用于细胞分化的三维不变测度具有双稳态模式。轴表示三种类型细胞的细胞计数: 祖细胞(z)、成骨细胞(y)和软骨细胞(x)。Pro-osteoblast stimulus promotes P→O transition.

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>。

超灵敏的正反馈调节可以产生双稳态。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"/>。

 

双稳态是理解细胞功能基本现象的关键，例如细胞周期进程中的决策过程、细胞分化和凋亡。它还参与了与癌症发病早期事件、朊病毒疾病以及新物种起源(物种形成)有关的细胞内稳态的丧失。

 

Bistability can be generated by a positive feedback loop with an ultrasensitive regulatory step.  Positive feedback loops, such as the simple X activates Y and Y activates X motif, essentially links output signals to their input signals and have been noted to be an important regulatory motif in cellular signal transduction because positive feedback loops can create switches with an all-or-nothing decision.<ref name="O. Brandman, J. E 2005">O. Brandman, J. E. Ferrell Jr., R. Li, T. Meyer, Science 310, 496 (2005)</ref> Studies have shown that numerous biological systems, such as ''Xenopus'' oocyte maturation,<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> mammalian calcium signal transduction, and polarity in budding yeast, incorporate temporal (slow and fast) positive feedback loops, or more than one feedback loop that occurs at different times.<ref name="O. Brandman, J. E 2005"/> Having two different temporal positive feedback loops or "dual-time switches" allows for (a) increased regulation: two switches that have independent changeable activation and deactivation times; and (b) linked feedback loops on multiple timescales can filter noise.<ref name="O. Brandman, J. E 2005"/>

 

 

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

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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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>

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>