12. Dwell time decides the answer · Prof. Kwon's TCR-HLA and CD4/CD8, First Steps

Binding strength K_D is a ratio of two numbers.

K_D = k_off / k_on

k_on is how fast they meet and stick; k_off is how fast they come apart. And when people measured TCRs, k_on was much the same for all of them. Nearly all the difference lived in k_off.

The reciprocal of k_off is the dwell time — call it a half-life. A good agonist lasts seconds; a weak one lasts under one.

Why time becomes the answer — kinetic proofreading

A TCR binding pMHC does not send a signal immediately. Lck phosphorylates the first site on a CD3 tail, then the next, then ZAP-70 arrives and binds… there are several steps.

Here is the decisive part. Let go halfway and every step taken so far disappears. The phosphates come straight back off. Start again.

P(signal) = ( k_p / (k_p + k_off) )^N

N is the number of steps that must be taken. The exponent is the whole point: make the dwell time a little longer and the signal gets a great deal stronger.

Timothy McKeithan proposed this model in 1995, borrowing the idea from how a ribosome avoids inserting the wrong amino acid.

Between a 0.5 s and a 5 s half-life the signal spreads 19-fold at N = 4.

Python Runs in your browser

# How much does dwell time change the signal
import thla

print(f"{'half-life(s)':>13}{'k_off':>10}{'N=2':>10}{'N=4':>10}{'N=6':>10}")
for t in [0.2, 0.5, 1, 2, 3, 5, 10, 20]:
    row = f"{t:>13.1f}{thla.koff(t):>10.3f}"
    for n in [2, 4, 6]:
        row += f"{thla.proofread(t, steps=n):>10.4f}"
    print(row)
print()
print("The more steps there are, the further a difference in time is stretched.")

That non-linearity is what pulls 'two nearly identical fragments' apart.

Python Runs in your browser

# Clearer as a ratio
import thla

base = thla.proofread(0.5, steps=4)
print("taking a 0.5 s half-life as 1")
for t in [0.5, 1, 2, 5, 10]:
    r = thla.proofread(t, steps=4) / base
    print(f"   {t:>5.1f}s  {r:>7.1f}x  " + "█" * max(1, int(r * 2)))
print()
print("A 10-fold longer bond gives a 19-fold signal. Not linear.")

The larger N gets, the sharper the threshold.

Python Runs in your browser

# As a picture
import matplotlib.pyplot as plt
import numpy as np
import thla

t = np.linspace(0.1, 12, 300)
fig, ax = plt.subplots(figsize=(8, 4.2))
for n, col in [(1, "#B9C4D8"), (2, "#8DA5CC"), (4, "#4C7DE0"), (8, "#22357A")]:
    ax.plot(t, [thla.proofread(x, steps=n) for x in t], lw=2.2, color=col, label=f"N = {n}")
ax.axvline(1.0, ls="--", c="#E8483F", lw=1.2)
ax.set_xlabel("complex half-life (s)")
ax.set_ylabel("P(signal completes)")
ax.legend(title="proofreading steps")
plt.show()

Move the half-life and the step count. A threshold opens up between agonist and antagonist.

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One residue flips the answer

The most famous test of this model is the altered peptide ligand: take a good fragment and change a single amino acid.

Fragment	Dwell time	What the T cell does
original (agonist)	long	proliferates, makes cytokines, kills — everything
one residue changed (partial agonist)	middling	only some of it — makes cytokines but does not proliferate
one more change (antagonist)	short	does nothing, and blocks the response to the original fragment as well

Same TCR, same plate, nearly the same fragment. The only thing that changed is the dwell time — and the cell's behaviour changes qualitatively.

🤔 Check yourself

Which physical quantity matters most for a TCR telling fragments apart?

🤔 Check yourself

In the kinetic proofreading model, what happens as the exponent N grows?

🤔 Check yourself

What characterises an antagonist peptide?