Journal of Statistical Mechanics: Theory and Experiment
Volume 2005
October 2005
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P W Messer et al J. Stat. Mech. (2005) P10004 doi:10.1088/1742-5468/2005/10/P10004
Universality of long-range correlations in expansion–randomization systems
P W Messer1, M Lässig2 and P F Arndt1
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Figure 1. Illustration of the different mechanisms contributing to  . |
Figure 2. Average composition bias  . (a) Decay of in time for k = 1, 2, 5, 10. Rates
of the processes are: μ = 1.0, δ1 = 4.0, γ5 + = 0.2, γ2 - = 0.5. The red line is the analytic lower bound on the rate of convergence (13). (b) Stationary
 with fixed  at different rates of the elementary processes:
(1) μ = 1.0, δ3 = 15.0, γ2 + = 1.0, γ7 - = 1.0;
(2) μ = 1.0, δ1 = 16.0, γ2 + = 1.0, γ1 - = 2.0;
(3) μ = 1.0, δ2 = 6.0, γ3 + = 2.0, γ4 - = 0.5;
(4) μ = 1.0, δ1 = 4.0, γ2 + = 1.0, γ4 - = 0.5. Red
lines denote the corresponding analytic asymptotics (14). All ensemble averages were obtained by averaging
over 106 simulated sequences. |
| Figure 3. Illustration of the different mechanisms contributing to the dynamics of Peq(k, r, t). Effectively mutational events are those that randomize either sk, or sk + r. `Expansion' or `contraction' transport of joint probability from Peq(k, r ± 1) to Peq(k, r) occurs due to duplication, insertion, or deletion events at sequence positions between sk and sk + r. `Horizontal' shift from Peq(k ± 1, r) to Peq(k, r) takes place if a duplication, insertion, or deletion occurs at sequence positions prior to sk. |
Figure 4. Single-site duplication-mutation model. (a) Stationary composition correlation
C(r) at
different rates of the elementary processes; numerical results (circles) and the analytic form (23) (lines)
for μ = 1.0, δ1 varying. C(r) is averaged along the sequence. (b) Power spectra of simulated sequences for
μ = 1.0 and δ1 varying: numerical results (circles) with the analytically predicted  in those cases where δ1 ≥ 5 (lines). The dynamics of the sequences was simulated until they reached a length of
N = 227 ≈ 108. All data sets were obtained by averaging over 100 runs. |
| Figure 5. Stationary C(r) at different rates of the elementary processes for the general model with various segmental processes present: numerical results (circles) with the analytic asymptotics (27) (lines) for μ = 1.0 and varying rates of the other processes (rates not specified in the plot are zero). |
Figure 6. Numerically measured distribution functions
P(m, L) and the corresponding scaling functions  for L = 102, 103, 104. ((a), (b))
Regime (i) with χ = 0.1 and  . ((c), (d)) Regime (ii) with
χ = 1.0 and the Gaussian scaling function  . The deviations for L = 102 for both regimes are due to the fact that the analytic asymptotics is only valid for large
L. The ensemble averages were obtained by averaging over
107 sequence realizations for each parameter setting with random initial conditions,
resulting in symmetric distributions (only positive values shown). |
| Figure 7. A single sequence of length N = 400 generated by the expansion-randomization process from an initial letter + 1. (a) Strong-correlation regime (μ = 0.5, δ1 = 10.0, i.e. χ = 0.1 < 1/2). The sequence retains a net composition bias towards + 1 in its entire length, i.e., the initial composition bias is detectable. Minority islands of - 1 are found on all scales. (b) Weak-correlation regime (μ = 0.5, δ1 = 1.0, i.e. χ = 1.0 > 1/2). The sequence consists of strongly correlated islands of length ξ ≈ 5 but looks random on larger scales. The initial composition bias is not detectable. |
| Figure 8. Time-dependent correlations C(r, t). (a) Build-up of long-range correlations by stationary growth. Measured C(r, t) at various intermediate lengths N(t) = 102, 104, 106 (symbols) together with the stationary form (23) for μ = 1.0, δ1 = 8.0 (line). (b) Correlation build-up from a random sequence of length N0 = 104. At t = 0 the processes started acting on the sequence with rates μ = 1.0, δ1 = 10.0. Measured C(r, t) (symbols) of the simulated sequences after various times t (averages over 100 realizations). Black crosses denote the corresponding mean sizes r*(t) = exp(λt). Correlations have been established in the sequences according to their analytic stationary form (red line) in the regime r < r*(t), while they vanish for r > r*(t). |
| Figure 9. (a) Decay of correlations during sequence evolution at stationary length N0 = 106. Measured C(r, t) at various times Δt (symbols) together with the analytic decay of the long-range tail given by equation (54). In the previous growth phase for t < t0, correlations have been established by a single-letter duplication-mutation dynamics with μ = 1.0 and δ1 = 8.0 until the sequences reached the length N0 = 106. For Δt = t - t1 > 0, a single-letter deletion process with γ1 - = 8.0 was introduced. Note that the correlations on short scales are preserved during the second phase. (b) C(r) with two scaling regimes 1 and 2 (symbols). Process rates are: μ(1) = 1.0, δ1(1) = 10.0 and μ(2) = 1.0, δ1(2) = 2.0. The dashed red line is the analytical C(r, t) for the parameters of phase 1. The second phase lasted over a period of time that on average allowed the sequences to increase their length by a factor of 100. For each scaling regime (n = 1, 2), C(r) obeys the predicted algebraic decay with exponent α(n) = 4μeff(n)/λ(n). The transition between both regimes is sharp and its position agrees with the value predicted by (52). |
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