双稳性

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文件:Bistability graph.svg
A graph of the potential energy of a bistable system; it has two local minima [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ 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]\displaystyle{ 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.


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 Hysteresis有关。


数学模型

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。

In biological and chemical systems

文件:Stimuli.pdf
Three-dimensional invariant measure for cellular-differentiation featuring a two-stable mode. The axes denote cell counts for three types of cells: progenitor ([math]\displaystyle{ z }[/math]), osteoblast ([math]\displaystyle{ y }[/math]), and chondrocyte ([math]\displaystyle{ x }[/math]). Pro-osteoblast stimulus promotes P→O transition.[4]

thumb|Three-dimensional invariant measure for cellular-differentiation featuring a two-stable mode. The axes denote cell counts for three types of cells: progenitor (z), osteoblast (y), and chondrocyte (x). Pro-osteoblast stimulus promotes P→O transition.

= = 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.

Bistability is key for understanding basic phenomena of cellular functioning, such as decision-making processes in cell cycle progression, cellular differentiation,[5] and apoptosis. It is also involved in loss of cellular homeostasis associated with early events in cancer onset and in prion diseases as well as in the origin of new species (speciation).[6]

Bistability is key for understanding basic phenomena of cellular functioning, such as decision-making processes in cell cycle progression, cellular differentiation, and apoptosis. It is also involved in loss of cellular homeostasis associated with early events in cancer onset and in prion diseases as well as in the origin of new species (speciation).

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.[7] Studies have shown that numerous biological systems, such as Xenopus oocyte maturation,[8] 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.[7] 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.[7]

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.O. Brandman, J. E. Ferrell Jr., R. Li, T. Meyer, Science 310, 496 (2005) Studies have shown that numerous biological systems, such as Xenopus oocyte maturation, 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. 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.

通过正反馈环路和超灵敏的调节步骤可以产生双稳态。正反馈回路，比如简单的 x 激活 y 和 y 激活 x 基序，本质上将输出信号与输入信号联系起来，并且已经被认为是细胞信号转导中一个重要的调节基序，因为正反馈回路可以创造出一个全有或全无的决定的开关。研究表明，许多生物系统，如非洲爪蟾卵母细胞成熟、哺乳动物的钙信号转导和芽殖酵母的极性，都包含时间(慢和快)正反馈回路，或者不止一个在不同时间发生的反馈回路。具有两个不同的时间正反馈回路或”双时间开关”允许(a)增加调节: 两个开关具有独立的可变激活和失活时间; (b)多时间尺度上的链接反馈回路可以过滤噪声。

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

在生物化学系统中，只有在特定的参数值范围内才会出现双稳态，在这种情况下，参数往往可以被解释为反馈的强度。在几个典型例子中，系统只有一个稳定不动点，且参数值很低。在参数的临界值处，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。然后，不稳定解与初始稳定解在参数的较高值形成另一个鞍结分岔，只留下较高的固定解。因此，在两个临界值之间的参数值，系统有两个稳定的解。一个展示了类似功能的动力系统的例子是

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

\frac{\mathrm{d}x}{\mathrm{d}t} = r + \frac{x^5}{1+x^5} - x


\frac{\mathrm{d}x}{\mathrm{d}t} = r + \frac{x^5}{1+x^5} - x

where [math]\displaystyle{ x }[/math] is the output, and [math]\displaystyle{ r }[/math] is the parameter, acting as the input.[9]

where x is the output, and r is the parameter, acting as the input.

其中 x 是输出，r 是参数，作为输入。

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.[10]

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.

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。

Bistability has also been described in the embryonic development of Drosophila melanogaster (the fruit fly). Examples are anterior-posterior[11] and dorso-ventral[12][13] axis formation and eye development.[14]

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.

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。

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.[15] 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.

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.

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。

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.[6]

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.

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。

Bistable chemical systems have been studied extensively to analyze relaxation kinetics, non-equilibrium thermodynamics, stochastic resonance, as well as climate change.[6] In bistable spatially extended systems the onset of local correlations and propagation of traveling waves have been analyzed.[16][17]

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.

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。

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)[18]), 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.

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.

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。

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.

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。

In mechanical systems

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.

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.

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.

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.

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。

文件:Ratchet example.gif
A ratchet in action. Each tooth in the ratchet together with the regions to either side of it constitutes a simple bistable mechanism.

thumbnail|A ratchet in action. Each tooth in the ratchet together with the regions to either side of it constitutes a simple bistable mechanism.

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。

See also

  • In psychology
  • ferroelectric, ferromagnetic, hysteresis, bistable perception
  • astable multivibrator, monostable multivibrator.
  • Schmitt trigger
  • strong Allee effect
  • Multistable perception describes the spontaneous or exogenous alternation of different percepts in face of the same physical stimulus.
  • Interferometric modulator display, a bistable reflective display technology found in mirasol displays by Qualcomm

在心理学上，铁电的，铁磁的，滞后的，双稳态知觉的，不稳定的多谐振荡器，单稳态多谐振荡器。多稳态知觉描述了面对同样的物理刺激，不同感知的自发或外源交互作用。

  • 干涉测量调制器显示器，Qualcomm 公司在 mirasol 显示器中发现的双稳态反射显示技术

References

  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. 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'.
  4. ↑ Kryven, I.; Röblitz, S.; Schütte, Ch. (2015). "Solution of the chemical master equation by radial basis functions approximation with interface tracking". BMC Systems Biology. 9 (1): 67. doi:10.1186/s12918-015-0210-y. PMC 4599742. PMID 26449665. open access
  5. ↑ 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.
  6. ↑ 6.0 6.1 6.2 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.
  7. ↑ 7.0 7.1 7.2 O. Brandman, J. E. Ferrell Jr., R. Li, T. Meyer, Science 310, 496 (2005)
  8. ↑ Ferrell JE Jr.; Machleder EM (1998). "The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes". Science. 280 (5365): 895–8. Bibcode:1998Sci...280..895F. doi:10.1126/science.280.5365.895. PMID 9572732. S2CID 34863795.
  9. ↑ 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.
  10. ↑ 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.
  11. ↑ Lopes, Francisco J. P.; Vieira, Fernando M. C.; Holloway, David M.; Bisch, Paulo M.; Spirov, Alexander V.; Ohler, Uwe (26 September 2008). "Spatial Bistability Generates hunchback Expression Sharpness in the Drosophila Embryo". PLOS Computational Biology. 4 (9): e1000184. Bibcode:2008PLSCB...4E0184L. doi:10.1371/journal.pcbi.1000184. PMC 2527687. PMID 18818726.
  12. ↑ 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. S2CID 4415152.
  13. ↑ 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.
  14. ↑ 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.
  15. ↑ 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.
  16. ↑ 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. S2CID 17770042.
  17. ↑ 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.
  18. ↑ 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.

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