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

Perfect or near-perfect adaptation

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.

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. This is a fundamental property of biological systems, allowing organisms to maintain long term homeostasis and carry out processes in a changing environment. Examples of perfect or near-perfect adaptation can be found in cell signaling, chemotaxis, photoreceptor responses, enzyme kinetics, and more.

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

Mathematical Framework

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: some stimulus $X(t)$, corresponding output $Z(t)$, and an additional internal variable $Y(t)$. Adaptation occurs as $Y$ and $Z$ control each other's activities response to $X$.

Let $Z=Z_{s}(X)$ be the steady state response to a constant input $X$.

Now suppose the input increases by $\Delta X$. Then the output given by Z changes from $Z_{s}(X)$ to a peak value of $Z_{p}(X+\Delta X,X)$, before adapting to a new steady state value of $Z_{s}(X+\Delta X)$. For perfect adaptation, $Z_{s}(X+\Delta X)=Z_{s}(X)$.

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

Separation of time scales

Using the notations above, define $\tau _{Z}$, the time scale of the output $Z$. Similarly, define $\tau _{Y}$, the time scale of the subsequent adaptation that occurs through $Y$.

The first key requirement for adaptation to occur is $\tau _{Z}<<\tau _{Y}$, a separation of time scales.

Conceptually, this indicates that given some new stimulus, the system first responds quickly to the new input, reflecting high sensitivity towards that stimulus. This is followed by a slower adaptation back to a steady state output. 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.

Response sensitivity and adaptation accuracy

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:

$\varepsilon :={|Z_{s}(X+\Delta X)-Z_{s}(X)| \over |Z_{p}(X+\Delta X,X)-Z_{s}(X)|}$.

The second key requirement for adaptation is $\varepsilon <1$: the new steady state should not be larger than the peak value of the output. The lowest value of $\varepsilon =0$ corresponds to perfect adaptation.

Network requirement for adaptation

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

• Two sub-pathways, which start with a stimulus and result in a response, are necessary for adaptation within a system: one direct pathway ($X$ to $Z$), and one indirect pathway through an internal variable ($X$ to $Y$ to $Z$).
• 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.

Network motifs

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

In the following sections, let $X$ be an input, let $Z$ be some variable producing the output, and let $Y$ 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: ${dZ \over dt}=k_{1}X\cdot (1-Z)-k_{2}Z\cdot Y$, ${dY \over dt}=k_{3}Z\cdot {(1-Y) \over K_{3}+1-Y}-k_{4}{Y \over K_{4}+Y}$. 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:$k_{1}=200,k_{2}=200,k_{3}=10,K_{3}=0.01,k_{4}=4,K_{4}=0.01$.

Negative feedback

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

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

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. Firstly, $Y$ must respond with high sensitivity to the activity of the output-producing variable. Then, Michaelis-Menten constants for the activation of $Y$ by $Z$ and the inhibition of $Y$ by external factors must be small. Secondly, the rates of the activation of $Z$ by $X$ and the activation of $Y$ by $Z$ should be relatively large. When these parameter constraints are met, adaptation is seen to be fairly robust with respect to the remaining parameters.

Schematics of a simple incoherent feedforward loop

This was plotted on MATLAB2018 using the following rate equations for a typical IFFL: ${dZ \over dt}=k_{1}X\cdot (1-Z)-k_{2}Z\cdot Y$, ${dY \over dt}=k_{3}X\cdot {(1-Y) \over K_{3}+1-Y}-k_{4}Y$. Delay in feedback activity relative to the output can be observed, which portrays the time separation necessity for adaptation. Parameters used are: $k_{1}=10,k_{2}=100,k_{3}=0.1,K_{3}=0.001,k_{4}=1$.

Incoherent feedforward

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

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

Adaptation itself is not hard to obtain from IFFL; all that is needed is a time separation between response time and adaptation time. However, some of the parameters must be carefully controlled to achieve perfect or near-perfect adaptation. Specifically, the activation of $Y$ should be operating close to saturation, while the inactivation of $Y$ should be operating far from saturation. Mathematically, a smaller Michaelis-Menten constant and slower kinetics for the activation of $Y$ 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.

Schematics of a simple state-dependent inactivation adaptation pathway

State-dependent inactivation

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

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

This network is robust with respect to changes in the parameters $k_{1},k_{2}$.

The ratio of $k_{1}/k_{2}$ determines the height of the peak and the timescale of the adaptation.

The toilet flush phenomenon

In the state-dependent inactivation network shown above, adaptation is brought about by the depletion of "off" state $Z$. Once the system has adapted, there exist no more available "off" state $Z$ that can receive the input, so no additional responses can be produced, even if the stimulus strength is increased. To be able to respond to a new stimulus, the current incoming input must first cease, so that the "inactive" state $Z$ can return to their "off" states. 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.

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

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

• State-dependent inactivation dynamics with different parameters
• 
  This was plotted on MATLAB2018 using the following rate equations for a typical state-dependent inactivation pathway: ${dZ_{on} \over dt}=k_{1}X\cdot (1-Z_{on}-Z_{in})-k_{2}Z_{on}$, ${dZ_{in} \over dt}=k_{2}Z_{on}$. The separation of timescales is shown by the steep initial increase in output, followed by a slower return to steady state. Parameters used are: $k_{1}=1,k_{2}=1$. A smaller second peak is observed for the second surge in input; subsequent increases in input results in no response because all of the $Z$ have been inactivated to $Z_{in}$.
• 
  Also plotted on MATLAB2018. The same rate equations were used as the previous plot. Parameters used for this plot are rate constants $k_{1}=1,k_{2}=1$. Though the same parameters were used as before, because the input is much greater, the peak of the output reaches a higher value. The first peak corresponds to the first surge in input. Unlike before, no response is observed for subsequent increases in input.
• 
  Also plotted on MATLAB2018. The same rate equations were used as the previous plot. Parameters used for this plot are rate constants $k_{1}=10,k_{2}=10$. The ratio $k_{1}/k_{2}$ is the same as the previous plot, so the peak of the output approaches the same value as before. However, because the kinetics of the system is faster, a "pointier" output peak is observed: the output reaches its peak faster, and adapts to its unresponsive steady state faster.
• 
  Also plotted on MATLAB2018. The same rate equations were used as the previous plot. Parameters used for this plot are rate constants $k_{1}=10,k_{2}=1$. The ratio $k_{1}/k_{2}$ is higher than the previous plot, so the peak of the output approaches a higher value than before.
• 
  This was plotted on MATLAB2018. The following rate equations for a state-dependent inactivation network in which the input is in the form of a molecule were used: ${d(XZ)_{on} \over dt}=k_{1}(X-(XZ)_{on}-(XZ)_{in})\cdot (1-(XZ)_{on}-(XZ)_{in})-k_{2}(XZ)_{on}$, ${d(XZ)_{in} \over dt}=k_{2}(XZ)_{on}$. Time scale separation can be observed from the steep upwards motion of the output peaks compared to the relatively slower returns back to steady state. Parameters used are: $k_{1}=4,k_{2}=4$. With each increase in input, an output is produced with decreasing strength. The strength decreases because with each output produced, more and more $XZ_{on}$ are inactivated. On the other hand, with each decrease in input, $XZ_{in}$ is degraded and more $Z_{off}$ is available.

Biological applications

Bacterial chemotaxis

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

Cell signaling: receptor regulation

Adaptation occurs via methylation of the receptor. 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. To achieve perfect adaptation, the methylation rate function depends only on the output and not the input. 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.

Flagellar regulation

The bacterial flagellar motor achieves adaptation through the flagellar switch protein FliM. 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. 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.

Eukaryotic chemotaxis

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

Olfactory sensing in mammalian neurons

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). This causes an influx of calcium ions, consequently inducing the activation of calmodulin kinase II, which brings about the deactivation of AC. This is a classic example of a negative feedback loop.

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

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. G-protein coupled receptor photon sensors are activated by light, thereby decreasing the level of cGMP. This inhibits the influx of calcium ions, which effectively activates ORK, the compound which phosphorylates and deactivates the photon sensor. This is a negative feedback loop with light being the input, the photon sensor producing the output, and ORK providing negative feedback.

Voltage-gated sodium channels

Voltage-gated sodium channels play an important role in neuron signals, muscle contractions, any processes induced by action potentials, and more. 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. This is an example of adaptation via state-dependent inactivation.

EGF receptor binding

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. This pathway allows for adaptation via state-dependent inactivation where subsequent stimuli can still produce output until the receptors are depleted.

Related

• Weber-Fechner law
• Hill coefficient
• Michaelis-Menten
• Mass action kinetics