Effect of Interaction between Chromatin Loops on Cell-to-Cell Variability in Gene Expression

Abstract

According to recent experimental evidence, the interaction between chromatin loops, which can be characterized by three factors-connection pattern, distance between regulatory elements, and communication form, play an important role in determining the level of cell-to-cell variability in gene expression. These quantitative experiments call for a corresponding modeling effect that addresses the question of how changes in these factors affect variability at the expression level in a systematic rather than case-by-case fashion. Here we make such an effort, based on a mechanic model that maps three fundamental patterns for two interacting DNA loops into a 4-state model of stochastic transcription. We first show that in contrast to side-by-side loops, nested loops enhance mRNA expression and reduce expression noise whereas alternating loops have just opposite effects. Then, we compare effects of facilitated tracking and direct looping on gene expression. We find that the former performs better than the latter in controlling mean expression and in tuning expression noise, but this control or tuning is distance-dependent, remarkable for moderate loop lengths, and there is a limit loop length such that the difference in effect between two communication forms almost disappears. Our analysis and results justify the facilitated chromatin-looping hypothesis.

Fig 1. Schematic diagram for two representative forms that communication between enhancer and promoter takes: facilitated tracking and direct looping.

It is unclear which form is more reasonable. This paper will justify the facilitated chromatin–looping hypothesis [2].

Fig 2. Modeling interactions between a pair of DNA loops: from physical models to biological models and to theoretical models.

The blue and green loops influence gene expression in direct and indirect manners, respectively. The first column depicts fundamental biological structures for three kinds of interactions between two DNA loops, where the Su and Hw (green dock) may form a loop; the enhancer and promoter (blue dock) may form another loop. The second column depicts physical structures for respective DNA–looping interactions in the first column, which consider two different paths of looping (i.e., the Su and Hw pair or the enhancer and promoter pair is first looping). The third column represents respective theoretical models by mapping the physical models in the second column into a multistate model of gene expression, where transition rates between active and inactive states actually represent the looping rates, which depend on the loop lengths (along the DNA line), denoted by d₁ for the blue loop but by d₂ for the green loop.

Fig 3

Shown is the dependence of mean expression (A, B) and expression noise (C, D) on the blue loop length (along the DNA line) for three fundamental patterns: alternating loops, nested loops and side-by-side loops. Here the side-by-side structure is taken as a control or reference since the blue loop length does not affect the expression level nor the noise intensity (see green lines). In all cases, parameter values are set as μ = 10, δ = 1, r = 0, d₂ = 1500, λ₂₁ = λ₃₂ = λ₃₄ = λ₄₁ = 0.3, k₂ = 0.5, and $k_1=4e^{-0.5d_1}+1$ with d₁ ∈ (0,1000).

Fig 4. Comparison of effects between the cases of no tracking (r = 0) and tracking (positive r): cross–type structure.

(A,B) dependence of mean expression and noise intensity on the green loop length for a fixed tracking ratio (r = 0.1), where the dashed lines correspond to the side-by-side loops; (C,D) dependence of mean expression and noise intensity on tracking ratio for a fixed green loop length; (E,F) three–dimensional pseudo diagrams for dependence of the mean expression level/the noise intensity on both the loop length and the tracking ratio, where the color change in the bar represents the change in the mean level or in the noise. Parameter values are set as d₁ = 1500, μ = 10, δ = 1, r = 0.1, λ₂₁ = λ₃₂ = λ₃₄ = λ₄₁ = 0.3, k₁ = 0.5, and $k_2=4e^{-0.5d_2}+1$ with d₂ ∈ (0,10000) in (A) and (B) but k₂ = 0.5 in (C) and (D).

Fig 5. Comparison of effects between the cases of tracking (r > 0) and non–tracking (r = 0): inline–type structure.

(A,B) dependence of mean expression and noise intensity on the green loop length for a fixed tracking ratio (r = 0.1), where the dashed lines correspond to the side-by-side loops; (C,D) dependence of mean expression and noise intensity on tracking ratio for a fixed green loop length; (E,F) three–dimensional pseudo diagrams for dependence of the mean expression level/the noise intensity on both the loop length and the tracking ratio, where the color change in the bar represents the change in the mean level or in the noise. Parameter values are set as d₁ = 1500, λ₂₁ = λ₃₂ = λ₃₄ = λ₄₁ = 0.3, μ = 10, δ = 1, r = 0.1, $k_1=k_2=4e^{-0.5d_2}+1$ with d₂ ∈ (0,10000) in (A) and (B) but $k_1=k_2=4e^{-0.5d_1}+1$ with d₁ ∈ (0,10000) in (C) and (D).

Fig 6

Dependence of relative change ratios on the green loop length: (A) mean expression and (B) noise intensity. Here, parameter values are set as d₁ = 1500, λ₂₁ = λ₃₂ = λ₃₄ = λ₄₁ = 0.3, μ = 10, δ = 1, r = 0.15, and k₁ = k₂ = 0.5 for alternating loops but $k_1=k_2=4e^{-0.5d_2}+1$ with d₂ ∈ (40,10000) for nested loops.