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This is a draft for an unofficial Wikipedia article describing perfect adaptation.

Perfect or near-perfect adaptation

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At the cellular level, adaptation is a characteristic of some pathway in which the system initially responds to an environmental stressor, but eventually returns to a steady state.[1][2]

A system that is able to return precisely to or near its initial state after a response is said to have perfect or near-perfect adaptation.[3] This is a fundamental property of biological systems, allowing organisms to maintain long term homeostasis and carry out processes in a changing environment.[1] Examples of perfect or near-perfect adaptation can be found in cell signaling, chemotaxis, photoreceptor responses, enzyme kinetics, and more.[2]

A system has robust perfect adaptation when the adaptation is independent of other parameters and is intrinsic to the system.[2] On the other hand, a system has nonrobust perfect adaptation if a careful selection of parameters must be made to achieve perfect adaptation.[2]

Mathematical Framework

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Depicts a generic system with perfect adaptation that is given some additional stimulus at the position indicated by the red line. The peak output is reached immediately after the additional input is given, reflecting the fast time scale of the response. On the other hand, a slower curve can be observed for when the output is returning to its steady state, indicating the relatively slow time scale of the response.

The simplest adaptation pathway requires a minimum of 3 variables controlling the pathway:[4] some stimulus , corresponding output , and an additional internal variable .[3] Adaptation occurs as and control each other's activities response to .[3]

Let be the steady state response to a constant input .[3]

Now suppose the input increases by .[3] Then the output given by Z changes from to a peak value of , before adapting to a new steady state value of .[3] For perfect adaptation, .[3]

In the following sections, fundamental requirements for adaptation to occur will be described using the same notations as above.

Separation of time scales

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Using the notations above, define , the time scale of the output . Similarly, define , the time scale of the subsequent adaptation that occurs through .[3]

The first key requirement for adaptation to occur is , a separation of time scales.[3]

Conceptually, this indicates that given some new stimulus, the system first responds quickly to the new input, reflecting high sensitivity towards that stimulus.[3] This is followed by a slower adaptation back to a steady state output.[3] The delay in adaptation gives the system enough time to respond to the stimuli, which gives the system high sensitivity towards a larger range of stimuli strength.[3]

Response sensitivity and adaptation accuracy

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The long time scale for adaptation is characterized by the adaptation error, which is given by how much the steady state has changed relative to the peak response following an increase in input:[3]

.

The second key requirement for adaptation is : the new steady state should not be larger than the peak value of the output.[3] The lowest value of corresponds to perfect adaptation.[3]

Network requirement for adaptation

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The third key requirement for adaptation is a network requirement of the pathway:[3]

  • Two sub-pathways, which start with a stimulus and result in a response, are necessary for adaptation within a system: one direct pathway ( to ), and one indirect pathway through an internal variable ( to to ).
  • These two pathways must have opposite signs, meaning one is excitatory, and the other is inhibitory.

The full derivation can be found in Adaptation in Living Systems by Tu and Rappel.[3]

Network motifs

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Several common pathway motifs allowing for perfect or near-perfect adaptation can be found in biological systems.[1] Below, a few of these motifs are described briefly.

In the following sections, let be an input, let be some variable producing the output, and let be an internal variable that provides feedback as the system's response progresses.

Schematics of a simple negative feedback loop
This was plotted on MATLAB2018 using the following rate equations for a typical NFL:[1][4] , . Notice the slight delay of the feedback dynamics relative to the output peaks, displaying the separation of time scales required for adaptation. Parameter values are:.

Negative feedback

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Negative feedback loops (NFL) are a common motif for adaptation in biological systems.[1]

In a simple NFL, the direct pathway from input to output is excitatory, whereas the indirect pathway that goes through is inhibitory.[3] Given some input, produces a response to the stimulus and begins to activate .[3] However, upon activation, inhibits the activity of , thus inhibiting output production.[3]


Negative feedback is not robust with respect to a few of the parameters: the appropriate parameters are needed for perfect or near-perfect adaption to be achieved.[1][4][5] Firstly, must respond with high sensitivity to the activity of the output-producing variable.[1] Then, Michaelis-Menten constants for the activation of by and the inhibition of by external factors must be small.[4] Secondly, the rates of the activation of by and the activation of by should be relatively large.[1] When these parameter constraints are met, adaptation is seen to be fairly robust with respect to the remaining parameters.[1][4]

Schematics of a simple incoherent feedforward loop
This was plotted on MATLAB2018 using the following rate equations for a typical IFFL:[1][4] , . Delay in feedback activity relative to the output can be observed, which portrays the time separation necessity for adaptation. Parameters used are: .

Incoherent feedforward

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The incoherent feedforward loop (IFFL) is named as such because it consists of two feedforward pathways of opposite (incoherent) signs.[3] Typically, the direct pathway from input to output is positive, whereas the indirect pathway through is negative.[3]

Unlike NFL, in IFFL, the input directly controls .[3] Given an input, works to produce an output, while works to inhibit the activity of .[1] There is no feedback in this model; the flow of the pathway is strictly downstream.[1]

Adaptation itself is not hard to obtain from IFFL; all that is needed is a time separation between response time and adaptation time.[1] However, some of the parameters must be carefully controlled to achieve perfect or near-perfect adaptation.[1] Specifically, the activation of should be operating close to saturation, while the inactivation of should be operating far from saturation.[1] Mathematically, a smaller Michaelis-Menten constant and slower kinetics for the activation of combined with slower mass action kinetics for the inhibition of Y will produce perfect or near-perfect adaptation that is robust with respect to other parameters.[4]

Schematics of a simple state-dependent inactivation adaptation pathway

State-dependent inactivation

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The most prominent example of state-dependent inactivation is the dynamics of the voltage-dependent sodium channel.[6] In a state-dependent inactivation network, the variable producing the output is also the variable that enables adaptation.[1]

Initially, is in an "off" state, during which it is ready to receive an input.[1] This input will trigger into an "on" state, in which it can produce an output.[1] The "on" state is then converted into an "inactivated" state, in which it cannot produce an output nor receive an input.[1] An "inactivated" state will return to an "off" state once the stimulus ends.[7]

This network is robust with respect to changes in the parameters .[1]

The ratio of determines the height of the peak and the timescale of the adaptation.[7]

The toilet flush phenomenon

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In the state-dependent inactivation network shown above, adaptation is brought about by the depletion of "off" state .[1] Once the system has adapted, there exist no more available "off" state that can receive the input, so no additional responses can be produced, even if the stimulus strength is increased.[1] To be able to respond to a new stimulus, the current incoming input must first cease, so that the "inactive" state can return to their "off" states.[1] This phenomenon resembles the flushing of a toilet: a toilet that has been flushed cannot be flushed again until the handle has been released and the tank has refilled itself with water.[1]

Now, consider a system similar to the one above, but now, the input () itself is a molecule that binds to the output-producing molecule () to activate it;[1] this complex then produces the response.[1] Then, the complex is inactivated, much like the previous system.[1] A sub-maximal level of molecules will not deplete the molecules, so after the first round of adaptation has ben achieved, subsequent increases in input may cause additional outputs until the "off" state molecules are depleted.[1][8]

Like the state-dependent inactivation model before, the performance of perfect or near-perfect adaptation does not depend on parameter values.[1]

Biological applications

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Bacterial chemotaxis

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Bacteria such as E. coli have membrane bound chemo-receptors which can sense external chemical signals, allowing bacteria to undergo chemotaxis.[3] In this process, there are two observable adaptation pathways: one is observed for the bacterial flagellar switch[9][10], and another is observed for the cell signaling process.[11]

Cell signaling: receptor regulation

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Adaptation occurs via methylation of the receptor.[3] This methylation process is much slower than the rate at which ligands bind to receptors, as well as kinase activity dynamics, which allows the response time scale to be much faster than the adaptation time scale.[3] To achieve perfect adaptation, the methylation rate function depends only on the output and not the input.[3] It has been found that the adaptation process positively enhances the sensitivity of the output, whereas the output negatively affects the rate of methylation, proving to be a form of negative feedback.[12][13]

Flagellar regulation

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The bacterial flagellar motor achieves adaptation through the flagellar switch protein FliM.[9] Both CheY-P and FliM positively enhance the rate of output, which in this case is the tumbling behavior of the bacteria, whereas the output inhibits FliM at a slower timescale.[9] This pathway is much like that of the receptor regulation adaptation and is also a form of negative feedback; however, the adaptation here is only partial and is slower than the receptor methylation adaptation pathway.[10]

Eukaryotic chemotaxis

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Eukaryotic chemotaxis is crucial in many biological processes such as wound healing[14] and cancer cell migration[15]. It has been found that eukaryotic chemotaxis relies on the incoherent feedforward network to achieve adaptation.[16] The biochemical components allowing for adaptation are: chemoattractant concentration as stimuli, activated Ras, which produces the appropriate response, and RasGAP, which inactivates Ras.[3][17]

Olfactory sensing in mammalian neurons

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Olfactory fatigue is a common phenomenon and a well-studied example of neural adaptation. In the olfactory sensing pathway, an odorant (the stimulus) binds to olfactory receptors, which induces the activation of adenylyl cyclase (AC).[18] This causes an influx of calcium ions, consequently inducing the activation of calmodulin kinase II, which brings about the deactivation of AC.[18] This is a classic example of a negative feedback loop.[3]

In this example, the spots in the "lilac chaser" illusion fade away after several seconds when the black cross is stared at long enough. This leaves a grey background and the cross. Some viewers may notice that the moving space has faded into a moving blue-green spot, possibly with a short trail following it. Furthermore, moving one's eyes away from the image after a period of time may result in a brief, strong afterimage of a circle of green spots.

Light sensing in mammalian neurons

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A prolonged stimulation to the cone cells of mammalian eyes can also cause exhaustion, another implication of neural adaptation, and taken advantage of in many optical illusions.[19] G-protein coupled receptor photon sensors are activated by light, thereby decreasing the level of cGMP.[20][21] This inhibits the influx of calcium ions, which effectively activates ORK, the compound which phosphorylates and deactivates the photon sensor.[3] This is a negative feedback loop with light being the input, the photon sensor producing the output, and ORK providing negative feedback.[3]

Voltage-gated sodium channels

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Voltage-gated sodium channels play an important role in neuron signals, muscle contractions, any processes induced by action potentials, and more.[6] The sodium channel activates in response to depolarization, followed by auto-inactivation, then a slow return to its original "off" state, in which it awaits the next depolarization.[1] This is an example of adaptation via state-dependent inactivation.[22]

EGF receptor binding

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When the appropriate ligand binds to an EGF receptor, the receptor is activated, and the ligand-receptor complex is internalized into the cell, where the receptor is either recycled, or degradation of both the receptor and ligand occurs.[1][8] This pathway allows for adaptation via state-dependent inactivation where subsequent stimuli can still produce output until the receptors are depleted.[1]

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References

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  1. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj Ferrell, James E. (2016-02-XX). "Perfect and Near-Perfect Adaptation in Cell Signaling". Cell Systems. 2 (2): 62–67. doi:10.1016/j.cels.2016.02.006. ISSN 2405-4712. {{cite journal}}: Check date values in: |date= (help)
  2. ^ a b c d Drengstig, Tormod; Ueda, Hiroki R.; Ruoff, Peter (2008-12-02). "Predicting Perfect Adaptation Motifs in Reaction Kinetic Networks". The Journal of Physical Chemistry B. 112 (51): 16752–16758. doi:10.1021/jp806818c. ISSN 1520-6106.
  3. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae Tu, Yuhai; Rappel, Wouter-Jan (2018-03-10). "Adaptation in Living Systems". Annual Review of Condensed Matter Physics. 9 (1): 183–205. doi:10.1146/annurev-conmatphys-033117-054046. ISSN 1947-5454. PMC 6060625. PMID 30057689.{{cite journal}}: CS1 maint: PMC format (link)
  4. ^ a b c d e f g Ma, Wenzhe; Trusina, Ala; El-Samad, Hana; Lim, Wendell A.; Tang, Chao (2009-08-21). "Defining Network Topologies that Can Achieve Biochemical Adaptation". Cell. 138 (4): 760–773. doi:10.1016/j.cell.2009.06.013. ISSN 0092-8674. PMC 3068210. PMID 19703401.
  5. ^ Xiao, Fangzhou; Doyle, John C. (2018-04-11). "Robust Perfect Adaptation in Biomolecular Reaction Networks". bioRxiv: 299057. doi:10.1101/299057.
  6. ^ a b Catterall, William A. (2014-01-XX). "Structure and function of voltage-gated sodium channels at atomic resolution". Experimental Physiology. 99 (1): 35–51. doi:10.1113/expphysiol.2013.071969. ISSN 1469-445X. PMC 3885250. PMID 24097157. {{cite journal}}: Check date values in: |date= (help)
  7. ^ a b Friedlander, Tamar; Brenner, Naama (2009-12-29). "Adaptive response by state-dependent inactivation". Proceedings of the National Academy of Sciences. 106 (52): 22558–22563. doi:10.1073/pnas.0902146106. ISSN 0027-8424. PMC 2799740. PMID 20018770.{{cite journal}}: CS1 maint: PMC format (link)
  8. ^ a b Goh, Lai Kuan; Sorkin, Alexander (2013-05-XX). "Endocytosis of Receptor Tyrosine Kinases". Cold Spring Harbor Perspectives in Biology. 5 (5). doi:10.1101/cshperspect.a017459. ISSN 1943-0264. PMC 3632065. PMID 23637288. {{cite journal}}: Check date values in: |date= (help)
  9. ^ a b c Tu, Yuhai; Berg, Howard C. (2012-11-09). "Tandem adaptation with a common design in Escherichia coli chemotaxis". Journal of Molecular Biology. 423 (5): 782–788. doi:10.1016/j.jmb.2012.08.012. ISSN 1089-8638. PMC 3472109. PMID 22922485.
  10. ^ a b Yuan, Junhua; Branch, Richard W.; Hosu, Basarab G.; Berg, Howard C. (2012-04-11). "Adaptation at the output of the chemotaxis signalling pathway". Nature. 484 (7393): 233–236. doi:10.1038/nature10964. ISSN 1476-4687. PMC 3335734. PMID 22498629.
  11. ^ Wadhams, George H.; Armitage, Judith P. (2004-12-XX). "Making sense of it all: bacterial chemotaxis". Nature Reviews Molecular Cell Biology. 5 (12): 1024–1037. doi:10.1038/nrm1524. ISSN 1471-0080. {{cite journal}}: Check date values in: |date= (help)
  12. ^ Tu, Yuhai; Shimizu, Thomas S.; Berg, Howard C. (2008-09-30). "Modeling the chemotactic response of Escherichia coli to time-varying stimuli". Proceedings of the National Academy of Sciences of the United States of America. 105 (39): 14855–14860. doi:10.1073/pnas.0807569105. ISSN 1091-6490. PMC 2551628. PMID 18812513.
  13. ^ Block, S. M.; Segall, J. E.; Berg, H. C. (1983-04-XX). "Adaptation kinetics in bacterial chemotaxis". Journal of Bacteriology. 154 (1): 312–323. doi:10.1128/JB.154.1.312-323.1983. ISSN 0021-9193. PMC 217461. PMID 6339475. {{cite journal}}: Check date values in: |date= (help)CS1 maint: PMC format (link)
  14. ^ Clark, Richard A. F. (1988), "Wound Repair", The Molecular and Cellular Biology of Wound Repair, Boston, MA: Springer US, pp. 3–50, ISBN 978-1-4899-0187-3, retrieved 2021-05-07
  15. ^ Condeelis, John; Singer, Robert H.; Segall, Jeffrey E. (2005). "The great escape: when cancer cells hijack the genes for chemotaxis and motility". Annual Review of Cell and Developmental Biology. 21: 695–718. doi:10.1146/annurev.cellbio.21.122303.120306. ISSN 1081-0706. PMID 16212512.
  16. ^ Takeda, Kosuke; Shao, Danying; Adler, Micha; Charest, Pascale G.; Loomis, William F.; Levine, Herbert; Groisman, Alex; Rappel, Wouter-Jan; Firtel, Richard A. (2012-01-03). "Incoherent feedforward control governs adaptation of activated ras in a eukaryotic chemotaxis pathway". Science Signaling. 5 (205): ra2. doi:10.1126/scisignal.2002413. ISSN 1937-9145. PMC 3928814. PMID 22215733.
  17. ^ Rappel, Wouter-Jan; Firtel, Richard A. (2012-03-15). "Adaptation in a eukaryotic pathway: combining experiments with modeling". Cell Cycle (Georgetown, Tex.). 11 (6): 1051–1052. doi:10.4161/cc.11.6.19715. ISSN 1551-4005. PMC 3335914. PMID 22391206.
  18. ^ a b Kurahashi, T.; Menini, A. (1997-02-20). "Mechanism of odorant adaptation in the olfactory receptor cell". Nature. 385 (6618): 725–729. doi:10.1038/385725a0. ISSN 0028-0836. PMID 9034189.
  19. ^ Nakatani, K.; Tamura, T.; Yau, K. W. (1991-03-XX). "Light adaptation in retinal rods of the rabbit and two other nonprimate mammals". The Journal of General Physiology. 97 (3): 413–435. doi:10.1085/jgp.97.3.413. ISSN 0022-1295. PMC 2216483. PMID 2037836. {{cite journal}}: Check date values in: |date= (help)
  20. ^ Fain, G. L.; Matthews, H. R.; Cornwall, M. C.; Koutalos, Y. (2001-01-XX). "Adaptation in vertebrate photoreceptors". Physiological Reviews. 81 (1): 117–151. doi:10.1152/physrev.2001.81.1.117. ISSN 0031-9333. PMID 11152756. {{cite journal}}: Check date values in: |date= (help)
  21. ^ Burns, M. E.; Baylor, D. A. (2001). "Activation, deactivation, and adaptation in vertebrate photoreceptor cells". Annual Review of Neuroscience. 24: 779–805. doi:10.1146/annurev.neuro.24.1.779. ISSN 0147-006X. PMID 11520918.
  22. ^ Milescu, Lorin S.; Yamanishi, Tadashi; Ptak, Krzysztof; Smith, Jeffrey C. (2010-09-08). "Kinetic properties and functional dynamics of sodium channels during repetitive spiking in a slow pacemaker neuron". The Journal of Neuroscience: The Official Journal of the Society for Neuroscience. 30 (36): 12113–12127. doi:10.1523/JNEUROSCI.0445-10.2010. ISSN 1529-2401. PMC 2945634. PMID 20826674.