Random Walk Enzymes: Information Theory, Quantum Isomorphism, and Entropy Dispersion

Abstract

Activation-induced deoxycytidine deaminase (AID) is a key enzyme in the human immune system. AID binds to and catalyzes random point mutations on the immunoglobulin (Ig) gene, leading to diversification of the Ig gene sequence by random walk motions, scanning for cytidines and turning them to uracils. The mutation patterns deposited by AID on its substrate DNA sequences can be interpreted as random binary words, and the information content of this stochastically generated library of mutated DNA sequences can be measured by its entropy. In this paper, we derive an analytical formula for this entropy and show that the stochastic scanning + catalytic dynamics of AID is controlled by a characteristic length that depends on the diffusion coefficient of AID and the catalytic rate. Experiments showed that the deamination rates have a sequence context dependence, where mutations are generated at higher intensities on DNA sequences with higher densities of mutable sites. We derive an isomorphism between this classical system and a quantum mechanical model and use this isomorphism to explain why AID appears to focus its scanning on regions with higher concentrations of deaminable sites. Using path integral Monte Carlo simulations of the quantum isomorphic system, we demonstrate how AID's scanning indeed depends on the context of the DNA sequence and how this affects the entropy of the library of generated mutant clones. Examining detailed features in the entropy of the experimentally generated clone library, we provide clear evidence that the random walk of AID on its substrate DNA is focused near hot spots. The model calculations applied to the experimental data show that the observed per-site mutation frequencies display similar contextual dependences as observed in the experiments, in which hot motifs are located adjacent to several different types of hot and cold motifs.

FIGURE 1.

Experimental setup and results. (A) Deamination assay reports AID-catalyzed deaminations on target cassette with multiple trinucleotide motifs NNC embedded in lacZα reporter gene, and examples of mutation patterns where each “T” indicates a C→U mutation. (B) Intrinsic deamination rates for each trinucleotide motif in the (hot hot’)-(hot cold) cassette consisting of a (AAC AGC)₁₅-sss-(AAC GTC)₁₅ sequence. (C) Per-site mutation frequencies observed in experiments. The hot motifs (red) are hotter when amongst hot’ motifs (orange) than amongst cold motifs (blue). (D) A hypothetical path and a set of deamination events along this path, illustrating how the convolution between the scanning dynamics and catalysis of AID generates mutant DNA.

FIGURE 2.

Results from Monte Carlo path integral simulations on a (hot hot’)-(hot-cold) mixed cassette. (A) Footprints of AID on the sequence. (B) Observed per-site mutation frequencies, showing similar contextual dependence observed in the experiments shown in Fig. 1.

FIGURE 3.

Results from Monte Carlo path integral simulations on a (hot hot’)-(hot frigid)-(hot frigid) cassette. (A) Footprints of AID on the sequence. (B) Observed per-site mutation frequencies. (C) Far left: Entropy densities of a word starting at the center of the (hot hot’) region going to the right (5’ to 3’) or the left (3’ to 5’) as a function of word length. Blue solids lines are results for a 60 s incubation time. Dashed and dotted dashed lines are for 30 s and 15 s, respectively. Middle: Orange line shows entropy densities of words staring at the center of the (hot frigid) region in the middle of the cassette for incubation times of 60 s (solid orange), 30 s (dashed) and 15 s (dotted dashed). Far right: Green line, dashed and dotted dashed line show entropy densities of words staring at the center of the (hot frigid) region on the far right of the cassette after 60 s, 30 s and 15 s, similar to the other two panels. The length of the word is shortest on the far right, indicating that AID’s diffusion is slow there. In the middle of the cassette, correlations among mutations are weak generating long words, suggesting that AID’s diffusion is fast here.

FIGURE 4.

Entropy density obtained from the simulation results for the (hot hot’)-(hot frigid)-(hot frigid) cassette, in units of bits of information divided by the length of the word read in the 5’ to 3’ direction, for words of different lengths and at different starting positions along the sequence. In the middle of the cassette where the (hot frigid) cluster is, the entropy shows almost no dependence on word length, indicating that long words are preferentially generated here due to the fast diffusion of AID.

FIGURE 5.

Experimental results corresponding to the cassette studied by the simulations shown in Fig. 3. (A) Per-site mutation frequencies. (B) Decay of entropy density as a function of word length, for words centered at the center of the (hot hot’) cluster on the far left (blue), the middle (orange) and the far right (green). Consistent with the simulations, the words are shortest on the far left, suggesting that AID diffusion is slowest here, while the words are longest in the middle of the cassette, indicating that AID diffusion is fastest there.