PMC

(A) The inter-co-localization intervals for super- and sub-diffusion for 5 transcription factors and 10 sec binding time. (B) Co-localization regions for different factors and binding times for sub-diffusion case (H = 0.1) and super-diffusion case (H = 0.9) in 2D. (C) Co-localization ratio versus the Hurst index H in 3D.

(A) Illustration of transcription factor translocation into and out of the nucleus. (B) The burst of factor nuclear localization under the frequency of 0.15(min⁻¹) The red dash line represents the average factor number over each period. (C) The normalized expression level of X₁ gene and X₂ gene versus factor number (amplitude modulation). (D) The normalized expression level of X₁ gene and X₂ gene versus burst frequency of factor (frequency modulation). (E) X gene and Y gene co-localization ratio versus burst frequency of X transcription factor.

(A) One example of co-localization ratio convergence. The ratio rapidly becomes stable as time increases. Yellow triangle is obtained from Eq. (15), showing the consistency of calculation of co-localization ratio using different variables. (B) The inter-co-localization (ICI) interval distribution for T_c,xx and T_c,xy. The parameters are the same as displayed in (A). The inter-co-localization interval distribution can be fitted with gamma distributions. (C) The mean values of ICI distribution by varying the factor number and binding time between genes and factors. The mean ICI increases as the binding time increases, and decreases as the factor number increases. (D) The co-localization probability for X-X gene and X-Y gene. (E) Co-localization ratio for various combinations of factor number and binding time. The ratio threshold is set to be 0.52. The red dashed curve is the threshold boundary to distinguish whether the co-localization is significant or tends to be random.

(A) Demonstration of the 3D framework in a cubic, where the big green dots represent the location of the transcription factories, asteroids the transcription factors, and small dots the genes. Different families are represented by either red or blue. (B) The inter-co-localization interval distribution for T_{c, xx} and T_c,xy. The parameters are the same as in (D). (C) The mean values of ICI distributions by varying the factor number and binding time between genes and factors. The mean ICI increases as the binding time increases, and decreases as the factor number increases. (D) The distribution of the co-localization ratio for X-X genes and X-Y genes. (E) The colicalization ratio for various combinations of factor numbers and binding times. The critical ratio is 0.52. The red dashed curve is the threshold boundary to distinguish whether the co-localization is significant or tends to be random.

(A) The co-localization ratio increases as the degree of flatness of the nucleus increases when there are 5 factors and 10 sec binding time, and is independent of the number of transcription factories, at least in the flattened nucleus, such as E10 (embryonic blood), E14 (fetal liver erythroid), AS (adult anemic spleen erythroid), Sp (normal adult spleen), Th (adult thymus) , Br (fetal brain), mouse embryonic fibroblasts (MEFs) in experiments. Scale bar = 10 µm. The cubic of spherical nucleus and rectangular block of flatten nucleus demonstrate the positioning of transcription factories, factors and genes (refer to ). Note that the volumes of the cubic spherical nucleus and the rectangular flattened block are either the same (solid line), or the volume of the flattened nucleus is 5 times bigger than that of the spherical nucleus (dash line). (B) The co-localization ratio is a decreasing function of the number of transcription factors for both flattened nucleus of the same volume as the spherical one, and the nucleus of 5 times larger volume. No matter the volume, the co-localization ratio is independent of the transcription factories.

(A) Schematic representation of chromatin loops (black) extruding from a chromosome territory (gray). Transcribed genes (white) in RNAP II factories (black circles). Potentiated genes (free loops) that are not associated with RNAP II factories are temporarily not transcribed. Potentiated genes can migrate to a limited number of preassembled RNAP II factories to be transcribed (dotted arrows). Both cis and trans associations are possible. If a piece of gene is being transcribed at a particular time t, we define the state of that piece of gene as 1 (e.g., u _1,1(t) = 1, where the first subscript indicates the specific gene, and the second subscript indicates the specific factory that the gene is associated with), otherwise, the state of the gene is defined to be 0. Hence, u _1,1(t) = 1 means X gene 1 is being transcribed at factory 1, v _1,1(t) means Y gene 1 is being transcribed at factory 1, and u _3,2(t) = 1 means X gene 3 is being transcribed at factory 2. Note that in our simulation we fixed the transcription time for each gene to be 5 minutes. However a gene might be transcribed for longer than 5 minutes if another factor binding event occurs, so that the whole transcription process starts again (i.e. re-initiation). This is illustrated in the figure why genes might have various transcription time. (B) Illustration of the co-localization events (red lines) among the same family of genes (X gene) within time window [0 T] (in dashed line window) in factory 1. Once there is a gene start being transcribe inside factory 1 while there are one or more than one genes already being transcribed inside the same factory at time t, we say there is a co-localization event happened at time t. Therefore, we have 4 co-localization events among X genes in total over time T, i.e. N_xx(1,[0,T]) = 4. The inter-co-localization interval T_c,x,x tells the timing of X-X gene co-localization. (C) Illustration of co-localization events between different families of genes (X gene and Y gene) within time window [0 T] in factory 1. The X-Y gene co-localization events are similarly defined as X-X gene, as shown by the figure, where we have 4 co-localization events between X gene and Y gene over time T, i.e. N_yx(1, [0 T]) = 4, and T_c,yx is the inter-co-localization interval between Y gene and X gene.

(A) Experimental results with immuno-RNA FISH reveal co-localization between transcription factors and factories (RNAPII), genes and transcription factors, and gene pairs and transcription factors. This is the contour plot from original experimental data. (Aa) Immune-fluorescence detection of Klf1 (red) and RNAPII-S5P (green) in definitive erythroid cells, with a scale bar of 2 µm. This shows the co-localization between Klf1 and transcription factories RNAPII. This Klf1 background association rate (level) is estimated to be 20%. (Ab) The co-localization between transcription factor Klf1 and Hbb gene. (Ac) The co-localization between factor Klf1 and genes pairs (Hbb and Hist1). (B) Simulation results when we hold the Klf1 background association level as 20%, while the translocation of transcription units (genes, factors and factories) are following sub-diffusion process (H = 0.4). (Ba) Gene-factor association level (numbers indicated behind the stars) with various factors and binding time, both for 2D and 3D cases. The number of genes (for each family) is fixed to be 5 and the Klf1 (X factor) background association level is fixed to be 20% (the stars indicate the parameter values when this condition is satisfied). The detailed Klf1 association level for each X gene and Y gene are presented in the figure below, revealing the fact that the simulation results for 3D case (gene-factor association level = 0.6) match the experimental result (gene-factor association level = 0.64) quite well for a specific set of parameters (5 genes, 2 factors and 130 sec binding time). (Bb) Gene-factor association level (numbers besides the stars) with various genes and binding time for 2D and 3D cases. The number of factors is fixed to be 2 and the Klf1 background association level is fixed to be 20%. (Bc) The association rate of Klf1 (X factor) with Z gene and paired X-Z genes when there is a negative correlation between X and Z gene, or X and Z factors, under 3D case. The parameter used here are 5 genes, 2 factors, and 130 sec binding time. Preventing probability means the chance for stopping another gene (factor) to enter the factory when there is already one gene (factor) in that factory. When p = 0, it represents the independent situation of X factor (gene) and Y factor (gene).