Circuits: Amplification, Feedback, Adaptation and Switches

One photon, absorbed by one rhodopsin molecule in a rod cell, changes the current flowing through that cell by about one picoamp (a trillionth of an ampere). A single molecule cannot carry a picoamp, so how does the cell register it?

The answer is arithmetic, and the arithmetic comes with a bill.

We need a new vocabulary for signaling

A pathway diagram is a wiring list. Its arrows say which protein switches on which, not how strongly, for how long, or whether the response can be undone. Prof. Deppmann’s lectures open this topic with a slide titled “We need a new vocabulary for signaling,” and the words come from amplifier and control engineering:

• Gain: how many output molecules one input molecule produces.
• Negative feedback: the output acts to reduce its own production.
• Positive feedback: the output acts to increase its own production.
• Feed-forward: an input reaches the output by a direct route and an indirect one at once.
• Adaptation: the response returns toward rest while the stimulus is still present.
• Switch-like (ultrasensitive) response: the output climbs from near zero to near maximum over a narrow range of input.
• Robustness: the output holds when the amounts or rates of the parts vary.
• Insulation: keeping one pathway’s signal away from another’s targets. Cross-talk is the leak.

His lectures also borrow a picture from process control: water, malt and hops flow into a fermenter, a threshold indicator trips when a level is crossed, and the product goes to a storage tank. Two of his questions run through this lesson: “Insulation versus connectivity?” and “Does a signaling pathway have to be the most logical or efficient of all possible scenarios?”

The idea: gain is a product, and it has a price

A rod cell in the dark leaks, like a tap left running. Ion channels in the membrane of its outer segment are held open by cGMP, a small signaling molecule, and ions flow through them all the time. That flow is the dark current, about 20 picoamps. A photon does not add current. It briefly reduces one that is already running.

Here is the chain. A photon is absorbed by one molecule of rhodopsin, the light-sensitive protein of the rod, and flips it into an active shape. Active rhodopsin switches on many molecules of transducin, a G protein that can act only after it binds GTP. Each active transducin switches on phosphodiesterase (PDE), an enzyme that destroys cGMP. The cGMP level falls, some channels close, and the current dips by about one picoamp, a few percent of the dark current. Denis Baylor, Trevor Lamb and King-Wai Yau recorded that dip from a single toad rod in 1979.

Gain is multiplication. A step is catalytic when one active molecule keeps switching on new copies of the next component while it stays active, and one-to-one when it handles a single partner. In the rod, rhodopsin to transducin and PDE to cGMP are catalytic, and the step between them multiplies nothing. Leskov and colleagues measured the first as a rate: one active rhodopsin switches on transducin, and through it the phosphodiesterase, at 120 to 150 molecules per second.

gain of a cascade = gain of step 1 × gain of step 2 × gain of step 3

With invented numbers, two catalytic steps that reach 100 copies each, around a one-to-one step, give 100 × 1 × 100 = 10,000. The interactive below uses a different cascade, from epinephrine to glycogen breakdown, with a slider for each step. Change one step at a time and watch the product.

That gives two separate dials. A bigger response needs each active molecule to reach more copies. A longer one needs each component to stay active longer.

Amplification has a price. Multiplying a signal also multiplies its mistakes, and a one-picoamp dip sits on a baseline that never stops moving. Baylor’s group could trust the dip only because they counted. They dimmed the flashes until most produced nothing and sorted trials into blank and response. If each response is one photon caught at random, the fraction of blanks must follow a Poisson distribution, the signature of rare independent events. It did. A chain whose steps keep multiplying also needs a way to stop, which is the next section.

A loud messenger also travels. Amplification makes a small, soluble messenger strong enough to reach proteins its receptor was never meant to reach, so a cell that answers different receptors differently must keep each pool of messenger near where it was made. It slows the messenger, binds it near its source and destroys it nearby. Anchoring proteins such as mAKAP go further and hold a kinase, the phosphodiesterase that destroys its messenger, and the kinase’s target within nanometres of each other. The cyclic AMP that matters is then the cyclic AMP made right beside the kinase. That is insulation, and it has to be built.

Negative feedback sets how long a signal lasts

A thermostat is negative feedback: the heat it makes warms the sensor that tells it to stop. A pathway does the same when something its output switches on turns the pathway down. The brake takes time to build, so the loop’s timing sets how long the signal lasts, even while the ligand is still there.

In a figure from Prof. Deppmann’s lectures, feedback loops set the activity window of a signal: about 20 minutes after TNF (tumor necrosis factor), 30 after EGF (epidermal growth factor), 90 after NGF (nerve growth factor) and 120 after LPS (a molecule from bacterial outer membranes). The slide attributes each window to the loops that handle that signal.

Design problem

Design a pathway that turns itself off after activation. You have a growth factor, its receptor, a kinase and one protein X whose properties you may choose. Decide what X senses, what it does and when. The growth factor must stay present while the output falls.

Positive feedback turns a dial into a switch

If a population of cells gives a smooth dose-response curve, does each cell give a smooth response? At Stanford, James Ferrell and Eric Machleder loaded a gel with one frog egg per lane. Each was a Xenopus oocyte, an immature egg about 1.2 millimetres across, big enough to assay alone on an immunoblot (a gel-based protein detection). The eggs had been incubated in a range of progesterone concentrations, the hormone that starts maturation. The blot showed whether MAP kinase was on, because the active enzyme runs slightly slower through the gel. The usual version of the experiment pooled groups of eggs in one tube per dose and gave the textbook lazy S: more hormone, more active kinase.

A pooled curve cannot tell two stories apart: every egg responds a little more at each dose, or some fraction is fully on and the rest fully off. Read one egg at a time, the second was true. Ferrell and Machleder put a number on the steepness on 8 May 1998: a Hill coefficient of at least 35. The Hill coefficient n measures how abruptly a response turns on, and hemoglobin binding oxygen, with n of about 3, is the textbook case.

response = dose^n / (K^n + dose^n)

dose at 90% of maximum ÷ dose at 10% of maximum = 81^(1/n)

Going from 10 to 90 percent takes an 81-fold rise in dose when n = 1, about 4.3-fold when n = 3, and about 1.13-fold when n = 35.

Where does that steepness come from? ERK, a MAP kinase, turns on only after it receives two phosphates, a threonine and a tyrosine one residue apart, both installed by the enzyme above it, MEK. One tag leaves it off. Requiring two modifications turns a proportional response into a threshold, and stacking the three tiers of the relay multiplies the steepness. On 17 September 1996, Chi-Ying Huang and Ferrell published a calculation, confirmed by measurements in oocyte extracts, that gave the cascade a Hill coefficient of about 4 to 5.

Thirty-five is a different kind of object. Ferrell and Machleder proposed that a positive feedback loop in which the cascade is embedded adds the rest, so an egg that has begun to switch on drives itself the remaining distance. Steepness alone is still reversible, since the output follows the dose back down. Positive feedback adds memory: later work from the same laboratory (Xiong and Ferrell, 2003) showed that an oocyte that has committed to maturing stays matured after the hormone is removed.

That is why the pooled curve misled. Eggs differ in where their thresholds lie, so at a middling dose some have flipped and others have not, and a homogenate reports the fraction that flipped as though every egg had responded partway.

Adaptation: returning to rest while the signal is still on

A bacterium that smells something worth swimming toward stops tumbling and runs straight. After a while, with the attractant still there, it tumbles at its former rate again. It must, because to register a climb in concentration the cell has to forget the level it has reached. That return is adaptation.

The receptor controls CheA, which hands a phosphate to CheY, and phosphorylated CheY drives tumbling. A second loop resets the receptor: the enzyme CheR hangs methyl groups on it, the enzyme CheB takes them off, and the pair keeps adding or stripping until the receptor’s output is back where it rests. The methylation level is a running total of the gap between current output and resting output, and it stops changing only when the gap is exactly zero. Engineers call this integral feedback. A shower knob you keep turning until the water feels right works the same way: your speed sets how long you take, and “right” sets where you stop.

Ordinary enzymology says that doubling an enzyme shifts the outcome. If the stopping point is set by the zero, though, a cell with more or less enzyme should still return exactly to the tumbling rate it had before the attractant. Only that resting rate, and the time the return takes, should change with the enzyme level. Barkai and Leibler derived that on paper in 1997. On 14 January 1999, Alon, Surette, Barkai and Leibler tested it by making E. coli carry different amounts of one circuit protein at a time, then adding attractant. Adaptation time and steady-state behavior varied strongly with protein concentration. The precision of adaptation did not.

The protection is narrow. Overload the correcting enzyme and the cell still returns exactly to the tumbling rate it had before the attractant, but that resting rate is no longer the one its normal neighbors have, and the return takes a different time.

So does a pathway have to be the most logical or efficient of all possible scenarios? This one is not the simplest wiring that works. It is wired so that one property stays put when the parts are sloppy: how exactly the cell returns to its own starting rate.