Positive feedback

Positive feedback closes a causal loop whose net effect reinforces an initiating change. A loop with two inhibitory links can therefore have positive feedback. In signal transduction, a product that increases the activity of the enzyme making it illustrates reinforcement; the consequences depend on the full mathematical model, not just the loop sign. See signaling feedback.

A biological example is cell-cycle activation feedback. Cyclin B binds Cdc2, whose activity promotes activation of Cdc25; Cdc25 removes inhibitory phosphorylation from Cdc2, permitting further activation. Cdc2 also inhibits Wee1, an enzyme that inhibits Cdc2 through phosphorylation. That second return contains two inhibitions and therefore also has positive feedback. These are regulatory paths, not a claim that every connection is a single direct molecular encounter. ^[ref]

Reinforcement can preserve a decision after its initiating stimulus weakens: returning from the high state may require a lower input than entering it. The cell-cycle hysteresis experiment below tests this history dependence at matched cyclin B concentrations. A steep response alone is insufficient evidence of bistability, and finite observation times can make slow relaxation look like memory. A proposed switch must specify its input, readout and observation interval.

Saturating positive feedback limits the reinforcing rate as activity increases. Here is an explicitly synthetic ordinary differential equation for normalized nonnegative input $u$, response $x$, and return strength $\beta$:The return is an increasing, nonproportional function of $x$, while removal is proportional to $x$. Time and activities are dimensionless variables. At $x=0$ the vector field points into the nonnegative domain, and $dx/dt\le u+\beta -x$ prevents unbounded growth there. This makes the toy useful for asking about a bounded switch; it supplies no molecular mechanism for the chosen rate law.

At an equilibrium point, $f=0$. For a simple equilibrium point, the sign of the derivativedetermines local asymptotic stability. If $\beta =1$, the maximum reinforcing slope is $3\sqrt{3}/8<1$, so $f$ decreases strictly: every $u\ge 0$ has one asymptotically stable equilibrium point despite positive feedback.

Three equilibrium points become possible only if the reinforcing slope exceeds removal's slope somewhere. Its maximum is $3\sqrt{3}\beta /8$, at $x=1/\sqrt{3}$. Consequently this toy requiresfor two distinct folds. This is a necessary threshold for a bistability interval in this toy, not a universal biochemical number. At a fold, both $f=0$ and $f_x=0$, givingFor $\beta =1.8$, the folds are at $(x,u)\simeq (0.350003,0.153563)$ and $(0.887240,0.094405)$. Between these input values there are two asymptotically stable equilibrium points separated by an unstable one. At $u=0.12$, the three activities are approximately $0.171331$, $0.621230$ and $1.127439$; the low and high ones are stable.

A sufficiently slow increasing sweep follows the low branch to $u_{\mathrm{on}}\simeq 0.153563$ and then jumps upward. A decreasing sweep follows the high branch to $u_{\mathrm{off}}\simeq 0.094405$ and then jumps downward: hysteresis. These are limiting quasistatic thresholds, with relaxation between input changes and negligible perturbations that could cause earlier switching. Finite sweep rates and fluctuations can alter observed thresholds. A switchlike response with only one stable equilibrium point does not have this equilibrium hysteresis mechanism.

Cell-cycle activation feedback is the reinforcement of Cdc2 activation through Cdc25 and suppression of Wee1. Its role is to help make transitions in the cell cycle decisive; the cell-cycle hysteresis experiment tests the resulting states rather than inferring them solely from circuit arrows.

A 2003 study split a Xenopus laevis egg extract into portions with different histories. One received calcium to destroy endogenous cyclin B before receiving nondegradable cyclin B; the other received nondegradable cyclin B first, then calcium. The same sustained cyclin B dose was approached from low or high Cdc2 activity. A phosphorimager quantified histone H1 phosphorylation, sampling every $15$–$30$ minutes until an apparent stationary response. At $45$–$60\mathrm{nM}$, the dose maintained high activity but failed to induce it from the low state.^[ref]

Controls tracked labeled endogenous cyclin B disappearance after calcium treatment and persistence of nondegradable cyclin B, addressing unequal remaining input. The comparison supports hysteresis under extract conditions; finite observations cannot establish indefinite persistence. It does not isolate each reinforcing path or establish thresholds in intact biological cells.^[ref]

A Xenopus laevis egg extract is a preparation of egg contents retaining biochemical machinery of the cell cycle outside intact biological cells.

Histone H1 is the protein used as a substrate for the Cdc2 activity readout in the cell-cycle hysteresis experiment.

Cyclin-dependent kinase 1, abbreviated CDK1 and historically called Cdc2, is the catalytic protein kinase in the CDK1–cyclin B assembly. CDK1 transfers a phosphate group from ATP to a protein substrate; cyclin B supplies a regulatory binding partner. Binding and phosphorylation regulate catalytic competence, so detecting CDK1 protein alone does not establish activity. The distinction makes a durable catalytic component reusable as the cell cycle changes its regulatory inputs.^[ref]

The cell-cycle activation feedback uses CDK1 activity as both an output and a regulatory input: activity promotes Cdc25 activation and suppresses Wee1, reinforcing release from inhibitory phosphorylation. The already described cell-cycle hysteresis experiment tests how that output depends on history at matched cyclin B input; the histone H1 kinase assay below explains what its output measures. A binding diagram alone cannot establish hysteresis.

A CDK1 phosphorylation state identifies modified amino acid residues. Different sites have opposite effects: in studied Xenopus laevis Cdc2, Thr14/Tyr15 are inhibitory, while Thr161 phosphorylation is activating. T denotes a threonine residue, Y a tyrosine residue, and numbers are protein primary structure positions. Do not assume this numbering for every species.^[ref]

The 1992 study supplies a useful separation of mechanisms: replacing Thr14, Tyr15 and Thr161 still permitted cyclin B binding, but replacing Thr161 prevented activation of histone H1 phosphorylation. Binding therefore did not suffice for catalytic activity in that preparation. Activating phosphorylation also occurred with catalytically inactive Cdc2, supporting a separate CDK-activating kinase rather than a requirement for intramolecular autophosphorylation.^[ref]

In this deliberately coarse classification, bound cyclin B, activating phosphorylation, and cleared inhibitory sites must coincide for the activity-permissive state. Removing inhibitory phosphate groups does not supply missing cyclin B or activating phosphorylation. Conversely, adding activating phosphorylation does not erase inhibitory modification. Actual preparations contain mixtures of states and additional regulators; this classification gives no numerical reaction rate.

A CDK-activating kinase is a protein kinase supplying activating phosphorylation to a cyclin-dependent kinase 1 preparation. The Xenopus laevis experiment above distinguished this activity from the catalytic activity of Cdc2 itself. CAK names that function here, without asserting that every species uses the same protein assembly.

Myt1 is a protein kinase distinct from Wee1. The cloned Xenopus laevis Myt1 phosphorylated Cdc2 at both inhibitory sites Thr14 and Tyr15; depletion experiments implicated it as the predominant Thr14-directed activity in Xenopus laevis egg extracts.^[ref]

A histone H1 kinase assay measures protein kinase action on supplied histone H1, a protein substrate. When the transferred phosphate carries a radioactive isotope label, label incorporated into histone H1 reports catalytic action over the assay interval. It does not count CDK1 molecules, and a crude preparation can contain other protein kinases acting on histone H1. Attribution therefore needs controls.

A 2010 study prepared human HeLa S3 extracts after a double thymidine block, using swelling, a Dounce homogenizer, centrifugation, dialysis and concentration. Added recombinant cyclin B1 incubated for $20$ minutes.^[ref]

Aliquots received histone H1 and terminal-${}^{32}\mathrm{P}$-labeled ATP for a $15$-minute room-temperature reaction, stopped with Laemmli buffer. Dried SDS-PAGE gels exposed a phosphorimager screen; a BioRad Molecular Imager FX quantified labeling. No-added-cyclin B activity set the normalization baseline; RO-3306 reduced H1 activity approximately $85\%$. Companion Western blots monitored Tyr15 phosphorylation, with actin Western blot loading control.^[ref]

Mixed extracts lose compartment boundaries; inhibitor sensitivity supports a major CDK1 contribution, with residual signal.^[ref] Substrate choice and inhibitor specificity limit attribution. The cell-cycle hysteresis experiment supplies a separate history comparison with this kind of histone H1 readout; its dose thresholds should not be transferred to human biological cells.

CDK1 activity and protein abundance are different observables. A total-CDK1 Western blot estimates protein amount under its calibration and loading assumptions. A phosphosite-specific Western blot asks about a particular CDK1 phosphorylation state. A histone H1 kinase assay asks how much substrate modification occurs under specified assay conditions. Any conversion between them needs a model and suitable controls.

An original counting example stipulates $100$ CDK1 molecules, with $60$ bound to cyclin B in each of two hypothetical samples. In sample A, $10$ bound molecules are activity-permissive and $50$ are inhibited; in sample B, $40$ are activity-permissive and $20$ are inhibited. The other $40$ are unbound in both samples. Total CDK1 and bound CDK1 agree, but activity-permissive counts differ fourfold. These are invented counts for bookkeeping, not measured biological populations.

If each activity-permissive molecule contributes the same reaction rate, substrate and ATP remain available, and the assay stays linear in time, a simple illustrative readout isHere $R$ is the recorded signal, $R_0$ is a matched blank signal, $N_{\mathrm{active}}$ is the activity-permissive count, $t$ the reaction interval, and $\alpha$ a common signal contribution per active molecule per unit time. The corrected signals in the stipulated samples have ratio $40/10=4$, although their total protein amounts agree. This is an assay accounting assumption, not an ordinary differential equation for the cell cycle. The no-added-cyclin B baseline in the inspected experiment can itself contain catalytic activity; it need not equal an enzyme-free blank $R_0$.

Substrate depletion, differing inhibitory conditions or a saturated detector can break the simple proportionality. Comparing samples therefore requires matched reaction intervals, substrate supply, loading and detector response, rather than assuming that brighter histone H1 labeling automatically means more total CDK1.

Cyclin B is a regulatory protein partner of CDK1. The protein kinase catalytic component is CDK1; bound cyclin B helps make that component competent to act. Changes in cyclin B synthesis and destruction regulate how much assembly can form, while CDK1 phosphorylation states determine whether bound assemblies are inhibited. This separates the supply of regulatory partners from release of an already assembled brake.^[ref]

More cyclin B therefore need not give a proportional increase in CDK1 activity. Compare the matched-input cell-cycle hysteresis experiment, where different histories produced different activities, and the distinct measurements under CDK1 activity and protein abundance. Neither comparison makes cyclin B itself the catalytic protein kinase.

Cdc25 is a protein phosphatase: it removes inhibitory phosphate groups from CDK1, relieving inhibition of an appropriately prepared cyclin B–CDK1 assembly. Its reaction is removal, whereas Wee1 is a protein kinase adding inhibitory modification. Removing the inhibitory sites is different from removing the activating phosphorylation supplied by CAK.^[ref]

Cdc25 action on CDK1 must also be distinguished from phosphorylation of Cdc25 itself. In cell-cycle activation feedback, CDK1-dependent regulation promotes Cdc25 activity, which can then activate additional inhibited CDK1. The complete regulatory path can involve intermediates. A histone H1 kinase assay of a mixture reports downstream protein kinase action; assigning its change to Cdc25 requires a perturbation or an independent protein phosphatase assay, rather than interpreting histone H1 labeling as phosphate removal.

Wee1 is a protein kinase that inhibits CDK1 by adding a phosphate group at an inhibitory site. Purified human Wee1 phosphorylated Cdc2 at Tyr15 in the reported in vitro assay, with no Thr14 modification detected. A catalytic-site mutation abolished both the measured protein kinase activity and the ability to inhibit division. This identifies a catalytic brake rather than merely a correlated protein amount.^[ref]

Do not attribute all inhibitory CDK1 phosphorylation to Wee1: the Xenopus laevis Myt1 experiment separately identified activity toward both Thr14 and Tyr15. Nor does every phosphorylation inhibit its target: activating Thr161 modification in the studied Xenopus laevis Cdc2 has the opposite role. In cell-cycle activation feedback, suppression of Wee1 removes a brake on CDK1, reinforcing the transition without implying that the total amount of either protein instantly disappears.