Elongation dynamics shape bursty transcription and translation

Abstract

Cells in isogenic populations may differ substantially in their molecular make up because of the stochastic nature of molecular processes. Stochastic bursts in process activity have a great potential for generating molecular noise. They are characterized by (short) periods of high process activity followed by (long) periods of process silence causing different cells to experience activity periods varying in size, duration, and timing. We present an analytically solvable model of bursts in molecular networks, originally developed for the analysis of telecommunication networks. We define general measures for model-independent characterization of bursts (burst size, significance, and duration) from stochastic time series. Inspired by the discovery of bursts in mRNA and protein production by others, we use those indices to investigate the role of stochastic motion of motor proteins along biopolymer chains in determining burst properties. Collisions between neighboring motor proteins can attenuate bursts introduced at the initiation site on the chain. Pausing of motor proteins can give rise to bursts. We investigate how these effects are modulated by the length of the biopolymer chain and the kinetic properties of motion. We discuss the consequences of those results for transcription and translation.

Fig. 1.

Bursts generated by the minimal model. (A) The network consists of a switching source and a Poissonian product generator. Full arrows denote reactions. Product P is synthesized only in the ON state. k_sw⁺, k_sw⁻, k_ini, and k_deg denote the ON switching, OFF switching, production, and degradation rate constant, respectively. (B) Simulation of bursty accumulation of P. Two timescales correspond to uninterrupted and interrupted production events. Bars denote OFF (white) and ON (gray) state. (C) Waiting times for nonbursty production events (vertical lines). On average, 1 P is produced during the ON state. The resulting intervals between production events correlate weakly with OFF and ON states.

Fig. 2.

Theoretical analysis of the minimal burst model (Fig. 1). Columns correspond to different parameterizations. (Upper) The waiting time PDF for a pure Poisson process and for 2 IPPs with different burst characteristics (Eq. 1). The vertical dashed lines indicate the threshold of timescale separation τ_X. (Lower) Sequence size function (Eq. 4). The Φ evaluated at τ_X yields the burst size β (horizontal dotted lines).

Fig. 3.

Diagram of a sequence size function Φ. (Upper) Time series of production events. Horizontal bars denote intervals longer than thresholds ϑ_A–D. (Bottom) For a given ϑ, Φ is constructed by dividing the total number of intervals by the number of intervals longer than ϑ. Timescale separation introduces a regime where ϑ is longer than intervals within bursts but shorter than interruptions between them; a plateau appears. The point of timescale separation τ_X lies in the middle of 2 inflection points τ_1,2 determined from the second derivative of Φ. The value of Φ at τ_X is the burst size β.

Fig. 4.

Canonical model of macromolecular trafficking along a biopolymer. In the ON state of the switch, proteins initiate elongation with a rate constant k_ini. Elongation occurs with a rate constant k_el. “O” and “U” denote occupied and unoccupied state of the site, respectively. Motors leave the chain with a rate constant k_ter and accumulate a product P. The product is degraded with a rate constant k_deg.

Fig. 5.

Mean site occupancy. (A) At the beginning (solid line) and at the end (dashed line) of a 100-site polymer (k_sw⁺ = k_sw⁻ = 1 [1/T], k_ini = 100 k_sw⁻); error bars, standard deviation; (B) along 1- to 100-site polymers (dotted), numbers indicate the length, circles mark mean occupancy at the polymer's end; parameters as in A, and k_el = k_ini.

Fig. 6.

Analysis of the canonical model of macromolecular trafficking. Gillespie simulations of at least 1E-05 events, k_sw⁺ = k_sw⁻ = 1 [1/T], k_el = k_ini = 100 k_sw⁻. (A–D) The dashed line marks the minimal burst model (no elongation). (A) The waiting time PDF for 10- and 1,000-site polymers. Circles, the mean waiting time. (B–D) The mean waiting time and its standard deviation (B), burst size (C), and burst significance (D) for different chain lengths.

Fig. 7.

Model of macromolecular trafficking with pausing motor proteins. Proteins initiate motion with a fixed rate constant k_ini. A site can be unoccupied (U), occupied (O) by a motor, or occupied by a motor in a paused state (S). Motors can pause at a rate k_p⁺ at every site. The lifetime of the paused state is 1/k_p⁻. Other parameters are the same as in Fig. 4.

Fig. 8.

The effect of chain length and pausing parameters on waiting time statistics for the model with pausing. Data obtained from Gillespie simulations of 1E-06 events. Common parameters, k_ini = 100 [1/T], k_el = k_ter = k_ini. (A) The appearance of the timescale separation due to the pausing of proteins. Chain length, 100 sites. Bursts arise when (i) k_p⁺ allows for only few pauses during the elongation and (ii) 1/k_p⁻ is long enough for many initiations to occur (solid line). Circles, the mean waiting time. (B) Waiting time PDF for chain lengths 50 and 500 sites. The lifetime of the paused state is fixed: 1/k_p⁻ = 100 [T]. At a pausing rate constant k_p⁺ = 0.01 [1/T], the increase in the chain length increases the probability of multiple pauses during the progression: bursts disappear (dotted line). Reduction of k_p⁺ to 0.001 [1/T] recovers bursts (dashed line). Circles, 〈t〉. (C) The mean waiting time and its standard deviation as function of chain length. Lines, k_p⁺ = k_p⁻ = 0.01 [1/T]. Symbols at L = 500 correspond to the dashed line in B: k_p⁺ = 0.001 [1/T]. (D) Sequence size function for polymers as in B. Circles, β.

Fig. 9.

Superposition of independent bursters (k_sw⁺ = 0.1 [1/T], k_sw⁻ = 1 [1/T], k_ini = k_sw⁻). (A) Analytical waiting time PDF as a function of the number of burst sources. The curve becomes almost exponential for 8 sources. (B) Analytical sequence size function for 1–20 sources. The 2 roots of its second derivative (filled circles and triangles) vanish for >8 sources. (Inset) Burst significance as function of the number of sources.

Fig. 10.

Detailed model of elongation: mean waiting time 〈t〉 for mRNA production as function of initiation intervals τ_ini = 1/k_ini. The gene consists of 1,000 nt, RNA polymerase occupies 50 nt (35). Parameters of the initiation switch are τ_on = 6 and τ_off = 37 min (3). Elongation occurs at 50 nt/s (28). Means were obtained from Gillespie simulations of at least 1E-04 events. Solid line without symbols denotes the mean, and the dashed line denotes one standard deviation plus the mean for the minimal burst model (no elongation). The vertical line indicates the mean waiting time of mRNA in the experiment of Golding et al. (3). Deviation from the solid line results from collisions of RNA polymerases.