We propose that illumination with PWM reduces noise by alternating the cell between 'high' and 'low' states with smaller heterogeneity. We used. D PLAGL1, a zinc finger protein known to suppress cell proliferation Reads were aligned to the hg19 genome build using the Ion Torrent.
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The cell population exhibited a bell-shaped noise profile across both the range of light intensities blue line in Figure 1D and the range of mean mRuby expression levels blue line in Figure 1E. A similar phenomenon was observed in the dox-inducible Tet-On system solid line in Figure 2—figure supplement 1B. A Schematic representation of the LightOn expression system. C The response curves of mean mRuby expression relative to mean light intensity for AM blue and PWM with a period of min red , min yellow , and min purple.
The error bars represent standard deviations from 2—4 independent experiments. F Analysis of AM-induced mRuby expression distribution solid lines and PWM-induced mRuby expression distribution dashed lines, with a period of min for various mean light intensities, as marked on the right axis. A hypothetical highly cooperative curve is illustrated by the black dashed line. The predicted mRuby distributions based on the experimentally determined and hypothetical response curves are shown as the red and black dashed lines, respectively, in the right panel.
All measurements were performed 48 hr after light induction. All other symbols are described in detail in the 'Materials and methods' section. B Frequency distribution of mRuby expression calculated from experimental data in A. Cells from A were divided into 20 equal populations in ascending order of GFP intensities, and graphs of mRuby distribution were calculated from subpopulations 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, and 20 from bottom to top.
The light intensities AM or duty on-fraction are labeled to the right of the plots. H Surface response plot of mean mRuby expression relative to period and mean light intensity of PWM induction. I Surface response plot of mRuby distribution spreading ratio of of the 90th percentile to the 10th percentile of mRuby intensity relative to the period and mean light intensity of PWM induction. H—I Data were fitted using a cubic smoothing spline function, and plotted as common logarithms. Each sample contains 10,—50, cells.
Error bars represent standard deviations from at least two independent experiments. The mean light intensities are shown on the right axis. Benzinger and Khammash proposed that in Saccharomyces cerevisiae , noise in light-induced gene expression is caused by noise transmission from transcription factor concentration to gene expression Benzinger and Khammash, This noise can be modeled as a combination of a deterministic single-valued propagation function and an added phenomenological noise contribution, described as Equation 1 in the Materials and methods section.
The detailed calculations are described in the Materials and methods section and in Appendix 2. We then plotted a dose—response curve of the mean mRuby versus mean GFP-GAVPO expression for these 20 subpopulations red circles and fit these data to an empirical function red dashed line in the main plot in Figure 1H. We further looked at the mRuby distributions of the 20 subpopulations, assessing both the experimental and the predicted data. The CVs for experimentally measured mRuby expressions exhibited a bell-shaped function and ranged between 1.
The distributions for subpopulations transitioned from unimodal to bimodal and back to unimodal as mean GFP increased Figure 1—figure supplement 2C. On the other hand, Benzinger and Khammash's model predictions with two single-valued propagation functions can only exhibit unimodal distributions Figure 1—figure supplement 2D—I , which cannot explain our observations.
The concept of epigenetic bistability was theorized by Sneppen et al. Dodd et al. Still, experimental studies have mainly been concerned with long-term epigenetic memory Hathaway et al. Bintu et al. In dynamic system theory, this type of resonance might reduce stochasticity Collins et al. The period of min results in minimal changes in mean gene expression purple line in Figure 1C , but produces a significant reduction in gene expression noise across the intermediate mean light intensities and intermediate mean gene expression levels purple lines in Figure 1D—E.
Therefore, we were able to modulate gene expression level and noise independently by altering the mean light intensity and the period of PWM within the range sandwiched between the blue and purple lines in Figure 1D-E. A preliminary lookup table is computed by fitting expression level and distribution spreading quantified as the ratio of mRuby intensities at 90th and 10th percentiles to mean light intensity and period of PWM Figure 1—figure supplement 3H,I.
Mean light intensity and period of PWM do not act as orthogonal parameters to control gene expression mean and noise. To maintain the mean expression while adjusting noise, we needed to simultaneously adjust both mean light intensity and period of PWM Figure 1—figure supplement 3H. This observation suggested that bimodality is a robust phenomenon. To validate whether similar phenomena occur with chemical-induced p65 transactivator binding, we replaced the VP16 domain of rtTA on the classic tetON system with p65AD.
We found that this system exhibited phenomena similar to those in LightOn systems Figure 1—figure supplement 5D. We further reduced the tetO number from seven to one. As singe-cell analysis of histone modifications is able to provide greater sensitivity than protein analysis, further developments in single-cell ChIP-seq assays could help to resolve this technical problem. As shown in Figure 1I , the mean expression level of mRuby started to increase at 2— min and reached a plateau at around min for PWM and min for both AM intensities.
The delayed onset of mRuby fluorescence is partially due to its maturation time of approximately min. The noise levels remain relatively constant from min onwards Figure 1J. These observations suggest that PWM with a long period is needed to maintain a low level of noise. Previous studies of histone modification kinetics in CHO cells suggested that histone deacetylase 4 HDAC4 could be involved in our observed reduction in gene expression noise , as its rate constants are closer to the min of PWM than those of other histone modifying enzyme Bintu et al.
The concentration of the three inhibitors was 0. Data represent four independent experiments. D Proposed mechanism underlying transcription and epigenetic regulation events involved in the expression of the LightOn system. E Schematic view of the induction of bistability by a positive feedback loop, illustrating the induction of high noise levels by AM and the reduction of noise by PWM.
The black dashed line represents the boundary that separates the high and low states, also known as the unstable steady state. The cyan dashed lines represent the thresholds between the low monostable state blue and bistable states blue and red and between the bistable states blue and red and the high monostable state red. The blue arrow indicates an AM of intermediate intensity located at the bistability region.
The PWM cycle represented by the dashed rectangle alternates cells between high and low monostable states. The doxycycline concentrations are shown in the legend. A Plot of mean mRuby expression against mean light intensities. B CV of mRuby emission plotted against mean light intensity. D Plot of mean mRuby expression against mean light intensity.
A Raw traces of single-cell mRuby dynamics. The blue and red traces represent cells designated as being in a low or high state, respectively, after around 24 hr. C A raw trace of GFP fluorescence dotted green line was used to calculate a denominator curve dotted blue line that shows the cell cycle of each cell.
The sharp drop in single-cell GFP indicates a cell division event, with the denominator values immediately before and after it set to 2 and 1, and connected with linear functions. Raw GFP dotted green line or mRuby dotted red line signals were converted to normalized GFP solid green line or normalized mRuby solid red line by dividing by the denominator dotted blue line. Cells in the low mRuby expression state are shown in green.
Cells in the high mRuby expression state are shown in orange. The video shows that the mRuby expression state is mostly inherent in both daughter cells after each division. An example in which two daughter cells took different states is shown near the top at around 30 hr. This hypothetical positive feedback loop could generate bistability, probably with intermediate GAVPO dimer concentration or light intensity between the dashed cyan lines in Figure 2E.
HDACs deacetylate histones and contribute to setting the boundaries between bistability and monostability. For some cells, the local promoter-GAVPO dimer interaction is sufficiently high as to initiate the positive feedback loop and to elevate local histone acetylation and induce high gene expression red line.
Other cells fail to start the positive feedback loop, which leads to low histone acetylation and transcription blue line. Under this condition, the isogenic cells in a homogenous environment would exhibit large noise due to the history-dependent occupation of each state and stochastic switches between the two states Isaacs et al. In each state, the noise is low, and cells pass through the bimodal region rather quickly.
For PWM with a shorter period, a cell doesn't have enough time to settle at either of two unimodal states, which leads to high noise. The dashed and dotted circles in Figure 2F indicate that two cells remained in a low mRuby state through cell divisions, whereas two other cells had an elevated mRuby state which was maintained through cell division.
The solid circle indicates that two offspring cells of one cell had different mRuby states after cell division. The single-cell mRuby trajectories for cells each from low blue and high states red , normalized by cell-cycle stage, are shown in Figure 2G.
The normalization process is illustrated in Figure 2—figure supplement 3. Most of the cells at low mRuby expression state at 24 hr remain low through 40 hr, with the mRuby levels of a few cells being elevated toward the high state. Most of the cells at high mRuby state at 24 hr were elevated before 12 hr and remained in the high state through 40 hr.
A small fraction of these cells show a drop in mRuby expression, moving them towards the low state. These data indicate that the cells mostly maintain their mRuby expression states, with only a small fraction of them stochastically switching mRuby expression states. These single-cell dynamics are consistent with the bistability hypothesis, which assumes a low probability of random transitions. To validate bistability, longer live-cell imaging is required to observe sufficient occurrences of transition between the two states, partially because of the long half-life of the mRuby protein.
At present, it is a challenge to maintain longer cell tracking without interruptions for periodic replacements of the culture medium. To further assess the possibility of bistability, we examined its main characteristic, history-dependence. A cell in a bistable regime remains in the high or low state in a deterministic system, which is dependent only on the previous state of that cell.
Stochastic fluctuations induce random transitions between the two states. However, this priming would produce mRuby expression that partially remains after 48 hr. We propose that a high level of noise in gene expression is the consequence of epigenetic bimodality induced by the interplay among promoter-bound GAVPO, HATs, and histone acetylation. The mean levels of mRuby expression were also reduced Figure 3A , possible because HDACs now tilt the balance towards lower histone acetylation, leading to lower chromatin accessibility and less transcription.
However, their effect on either synthetic or endogenous gene expression has never been examined. The light intensities are specified to the right of each line. The dark control was measured in cells kept in the dark. The inserted sequences are between PB5 and PB3 not shown in the diagram.
Instead, this treatment resulted in an intermediate state with narrow GFP expression dispersion Figure 3G. We then sorted the mRuby-low and mRuby-high populations Figure 3I. We used the two cell populations and dark control cells to perform ChIP-seq assays for H3K27ac and mapped the reads to the whole genome sequence assembly of this HeLa-AB1 clone containing LightOn expression cassettes at nine loci Figure 3—figure supplement 1A—B.
The signals for the mRuby-low and the dark control populations are essentially the same, whereas the signal for the mRuby-high population is about four times higher. These signals demonstrate that the mRuby-high population from intermediate light induction is in a more epigenetically active state than the mRuby-low and dark control populations.
We also performed an ATAC-seq assay on the same cell populations. The hierarchy of chromatin accessibility is mRuby-high, mRuby-low, and dark control, with less separation between the high and low populations than H3K27ac and a clear difference between mRuby-low and dark control. We collected cells at six timepoints corresponding to different light cycle stages as illustrated in Figure 4A , and sorted tagged nuclei of each population for sequencing and analysis.
The single-cell reads for the six populations are plotted as heatmap images in Figure 4C. As expected, the fraction of cells with open chromatin was lowest for the dark control. To obtain quantitative measurements, we took the mean reads over bp for each cell and calculated the means and CVs for all populations, which are plotted against time in Figure 4D—E.
The mean number of reads started at around 0. The CV for the mean reads was initially at a high level of 3. The periodic fluctuations in chromatin accessibility did not directly translate into single-cell protein mRuby dynamics, which did not exhibit such periodic fluctuations Figure 5—figure supplement 1. The protein dynamics data indicated that the mean expression level reached a plateau at around min for PWM Figure 1l.
As indicated with blue arrows, six cell populations were collected: first at dark control 0 min , and then after one light-on cycle min , after one light-on-off cycle min , after one light-on-off cycle and one light-on cycle min , after two light-on-off cycles min , and after two light-on-off cycles and one light-on cycle min. Reads are displayed in common logarithms. The counts were added to 0. The x-axis represents the distance from the TSS. The y-axis represents the cell index.
Color represents the common logarithm of the reads. There are cells for each time point. D—E For each cell, a read was further defined as the average over the bp sequence. The mean reads D and CV of reads E over cells plotted against time. Red dashed lines represent the dynamics of the PWM light induction.
We used live-cell spinning disk confocal microscopy to image the single mRNAs. These observations are consistent with the time course of flow cytometry analysis of mRuby protein Figure 1I-J. A period of min, rather than min, was chosen for PWM in the mRNA imaging experiment so that the periodicity of noisy nuclear mRNA dynamics could be recognized with more temporal repeats while noticeable noise reduction was still exhibited.
A A representative z-projection image at a specific time. The small brighter puncta approximately nm in diameter in the nuclei represent nuclear mRNA molecules. The darker fluorescent signals are cytosolic fluorescent signals that are neglected in this analysis. The x-axis represents time with a unit of 10 min. The filled dark blue plots on top of the heatmap images represent the light induction schemes. Reads are displayed in natural logarithms. Each plot was normalized to its maximum.
For each cell, a moving average of 60 min was applied before computing the average over subpopulations. Tracking and quantification of the single-cell dynamics of the mRuby expression of representative cells plotted on a linear scale. The nucleus was segmented, tracked, isolated, and re-centered as described in the Materials and methods section.
This suggests that the cells were likely to be alternating between high and low epigenetic states and not trapped in the bistable region Figure 2E. The mean count exhibited a periodic oscillation. We did not observe periodic oscillation for the single-cell mRuby protein dynamics under PWM with min duty cycle, shown in Figure 5—figure supplement 1 , probably due in part to the longer half-life of protein.
Technical challenges that limited this type of long-term statistical analysis of single-cell mRNA dynamics remain. When compared to spinning disk confocal microscopy, light-sheet microscopy could increase the signal-to-noise ratio of these images, improve the fidelity of single mRNA detection, and ensure that an adequate number of repeating images are captured before significant photobleaching. Cell motility prevents the tracking of a sufficient number of cells over a long period, as most of the cells moved out of the imaging field at some time points.
Further development of light-sheet microscopy technology is needed to provide the capability to image a large number of large optical fields while maintaining sufficient optical and temporal resolutions over a long period. The red dashed line represents the 3-sigma cutoff for noisy outlier genes. Blue asterisks and black circles represent the CVs of the same gene sets from two biological replicates of control cells.
The boxes show the lower and upper quartiles; the whiskers show the minimum and maximal values excluding outliers; the line inside the box indicates the median; outliers red dots were calculated as values greater or lower than 1. The p-values shown in the figure were calculated using paired student t -tests. F Similar statistical analysis of CV between two biological control replicates for the same sets of genes.
C Blue asterisks and black circles represent the CVs of the same gene sets from two biological replicates of control cells. E Similar statistical analysis of CV for the same sets of genes comparing two biological control replicates. The library construction, sequencing, and data acquisition processes are described in detail in the Materials and methods section.
Outlier genes that significantly deviated from this band blue dots above the dashed red line were further selected to assess the effects of A and LMK For the filtered gene set with lower CV values less than two in control samples , there is a slight but statistically significant increase in mean CV from 1. For these genes, the mean CV exhibits a large 3.
There is much less deviation between the two biological repeats of control samples, especially for the high CV gene sets Figure 6D and F. Similar phenomena were observed with F9-AB2 cells Figure 6—figure supplement 1. It could also contribute to the heterogeneous expression states in embryonic stem cells. Synthetic inducible gene expression systems have been developed to study the function of genes in various cellular and physiological processes.
The widely used systems allow researchers to induce signals such as by doxycycline or light, and have a high dynamic range of expression. They are essentially based on the simplest form of gene regulation involving one transcriptional activator. Nevertheless, they often exhibit huge noise in gene expression, especially in mammalian cells CV as high as 5—10, shown in Figure 1—figure supplement 4 and Figure 2—figure supplement 1C. The primary molecular mechanism that contributes to such a high level of noise has not been identified.
In this study, we performed a quantitative characterization of gene expression noise in a light-induced expression circuit Wang et al. We found that PWM light controls noise in a period-dependent manner, enabling a phenomenological approach for independent modulation of expression level and gene expression dispersion by manipulation of mean light intensity and PWM period.
This approach would support more precise design of cell fate control studies. A recent study involved a more complex synthetic circuit with two orthogonal inducible systems arranged in series. These two inducible systems had unmatched noise properties, and were able to control the mean expression and noise independently by using different concentration combinations of two inducers Bonny et al.
This postulated positive feedback loop is predicted to generate bimodal expression of genes under intermediate light induction. Nevertheless, there was temporal fluctuation of the chromatin accessibility within each period. Similar phenomena were observed at the protein level. In summary, PWM noise reduction by switching between the low and high states is established at the epigenetic level. The pulsatile chromatin accessibility and nuclear mRNA levels measured under PWM light induction were not detectable at the protein level, probably because the half-life of proteins are longer than those of mRNAs.
Slow protein degradation might serve as a low-pass filter for mRNA oscillation and noise Wang et al. The mechanism that helps reduce temporal fluctuations might also limit the direct validation of bistability. The direct quantification of transitions between the high and low states is hindered by the long half-life and slow maturation time of mRuby proteins, and the limitation of the dynamic tracking length.
We observed only minor hysteresis effects, probably for similar reasons Figure 2I,J. Future experiments with bright fluorescent proteins or luciferases Suter et al. It is also possible that additional epigenetic modifiers and transcription-related factors might contribute to more complex dynamic properties. In nonlinear dynamics, chaotic oscillations can be stabilized at their unstable steady states using either the OGY feedback control theory Ott et al. Recently, feedback and periodic control of aTc and IPTG were used to stabilize the unstable steady state in a genetic toggle switch in Escherichia coli Lugagne et al.
This particular scenario would lead to a decrease in mean gene expression with an increase in mean light intensity, illustrated as the black dashed line in Figure 2E , a phenomenon never observed in our experiments. Benzinger and Khammash, have demonstrated noise reduction using PWM modulation of an ELbased light-inducible gene expression system in Saccharomyces cerevisiae.
They found that noise reduction is achieved by the PWM light working at the plateau regions of a single-valued sigmoidal active transcription factor to transcription response curve. In this scenario, transcriptional noise is mainly transferred from input noises, with bimodal expression being observed only with artificially inflated transcription factor heterogeneity in a non-isogenic cell population.
Our PWM noise reduction mechanism is different. This might lead to different levels of noise. A recent study found that the intrinsic variations between two 2A peptide-linked fluorescent proteins are minimal in mammalian cells Quarton et al. Our model suggests that the 5xUAS-mRuby at each locus has high and low expression states, and that the states at each locus are independent of each other. There are nine genomic insertion sites in HeLa-AB1 cells.
These emulated distributions are not so different from those of the experimental data Figure 1F. Further studies with newly developed single-cell epigenetic measurement techniques could help to assess this issue directly. We tested the consequences of potential disruption of this interaction on endogenous gene expression using both A and LMK In a mES-D3 Rex1-GFP reporter cell line treated with both inhibitors, we found that the Rex1-high and Rex1-low subpopulations disappeared, resulting in the formation of a single population with intermediate Rex1 expression.
Abolishing this interaction could provide a general mechanism for the modulation of noise in gene expression in mammalian cells. We found that pulsed light induction with an extended period could reduce noise, presumably because it helps the cell to avoid staying in the bimodal regime. A picture of the dynamic process underlying this noise modulation, from chromatin accessibility to mRNA and protein levels, has been established using single-cell analysis. The noise reduction exhibits a dose-response to the period of PWM, which will enable independent modulation of mean gene expression and noise in future quantitative studies of cell-fate control.
Yi Yang. The plasmid for the tetR negative feedback circuit similar to that described by Nevozhay et al. Jiandong Huang. The plasmids containing mRuby3 and mCardinal were gifts from Prof. Jun Chu. To reduce the half-life of the reporter fluorescent protein and to increase its temporal response, we fused amino acids — of the degradation domain of mouse ornithine decarboxylase MODC a PEST sequence Li et al. The nuc tag consists of three tandem repeats of the SV40 nuclear localization sequence, which facilitate the localization of the protein in the nucleus.
Puro r , Zeo r , and Bla r are resistance genes for the antibiotics puromycin, zeocin, and blasticidin, respectively. HS4 is the Chicken hypersensitive site 4 cHS4 insulator. The first six tetO sites were further removed to construct plasmid B6. All other components, unless otherwise specified, were adapted from Lu and Huang, All of the plasmid constructions were confirmed by Sanger sequencing.
The sequences of the plasmids and oligos used for the construction of plasmids will be provided upon request. L; Invitrogen following the manufacturer's protocol. Each clone was verified using flow cytometry two weeks later. One clone, named HeLa-AB1, was chosen for further analysis. The cells were illuminated with blue light for two days before sorting single cells that expressed GFP. Single clones of F9 cells were analyzed ten days after sorting.
One F9 clone, named F9-AB2, was chosen for further analysis. A; Gibco for three days. One clone, named HeLa-Tet-On, was chosen for further analysis. RRA; Takara. A standard curve was generated by serial dilution of plasmid B3 using the same concentration of wild-type HeLa genomic DNA. The total genome size of HeLa cells was estimated to be approximately 1. Jianbo Yue was cultured on a 0. Single cells expressing mCherry were sorted, expanded, and genotyped by PCR and sequencing.
A CSV file containing the designed illumination intensities and dynamics for the 24 wells was loaded into the custom Python code to control the apparatus for 48 hr unless otherwise specified. We then fitted the data using a cubic smoothing spline function csaps with MATLAB b to generate response-surface plots for the common logarithms of mean value and distribution spreading against period and mean light intensity Figure 1—figure supplement 3H—I.
We chose the distribution spreading ratio as read-out instead of the coefficient of variation, because this ratio provides a more intuitive picture of how diverse the gene expressions were and is not affected by instrumentation settings. On day 0, the medium was supplemented with 0. S; SelleckChem , 0. S; SelleckChem. Cells were then treated with 0. The remaining samples were kept on ice. To ensure the reproducibility of the measurements, we used fluorescent calibration beads Sphero Rainbow calibration beads six peaks; Catalog no.
RCP—5; Spherotech to adjust the instrumentation parameters. The relevant fluorescent channel was normalized to the mean value of the fifth peak of the Rainbow calibration beads. The expression of two fluorescent proteins linked by a 2A peptide are well-correlated at the single-cell level Quarton et al. The total population of cells Figure 1G was divided into 20 subpopulations with incremental mean GFP values and narrow ranges.
An empirical function red dashed line in the center panel of Figure 1H was fitted to the mean mRuby values vs. A hypothetical propagation function with high cooperativity was formulated with a Hill function with a Hill coefficient of 8 black dashed line in the center panel of Figure 1H. We added a lognormal noise with an amplitude of 0. HeLa-ABC2 cells were plated on 3.
Multiple XY positions within a 3 mm region were chosen, so that every cell in the optical field was always within the 7 mm blue illumination circle and the positions were separated enough to avoid overlapping in excitation and photobleaching. At each XY-position, 20 z-slices with a spacing of 0. A Prior piezo-z stage was used to ensure z precision. A nm laser was chosen for excitation to ensure the detection of single mRNA punctae while minimizing photobleaching over 3-D imaging loops.
The period of min was chosen over min to enable multiple cycles of light induction without significant photobleaching. MCP-tdTomato-nuc fluorescence is mostly localized inside the nucleus, which enabled us to segment and track individual nuclei, and to detect mRNA in a single channel Figure 5A.
With a camera pixel size of 6. The cells moved significantly during this min period, and most of the nuclei moved outside of the imaging boundary at one time or another. To find all nuclei that stayed entirely inside the boundary, we performed nuclear segmentation and tracking before using Imaris 9.
The first step in this process is the generation of a z-axis maximum projection. Connected components were searched in these binary images over time to track the nuclei that remained inside the imaging boundary. The nucleus was finally segmented using Otsu's algorithm in the cropped image. These XYZT images were normalized to the mean nuclear fluorescent intensity to compensate for MCP-tdTomato-nuc expression heterogeneity and possible photobleaching, as Imaris doesn't incorporate adaptive thresholding.
The single mRNAs were identified using Imaris's spot model with an estimated x-y diameter of 0. All of the code scripts were written in Julia 1. A customized 6-channel LED illumination apparatus was built to implement dynamic blue-light induction without interference from live-cell fluorescent imaging Appendix 1—figure 3.
The illumination nonuniformity was corrected by normalization against images of a well with media only. The nuclei of cells were segmented and binarized using the GFP images by searching for and filtering local maxima in the Gaussian blurred image as seeds, then applying the watershed algorithm to split individual cells. Each nucleus was extracted by thresholding with Otsu's algorithm in each split region. The tracking was carried out in a reverse time sequence.
To represent continuous expression dynamics, we normalized single-cell expression dynamics with the cell-cycle state for that cell Figure 2—figure supplement 3C. All the code scripts were written either in Julia 1. The reads were trimmed with cutadapt 1. The trimmed reads were filtered with cutadapt 1. The filtered reads were mapped to the reference genome hg19 and plasmid insertion sequences using bwa 0.
The alignments were sorted by leftmost coordinates using samtools 1. The duplicated reads in sorted alignments were marked with the MarkDuplicates function of Picard 2. The break sites between the hg19 and plasmid were called from the alignments using VIFI 0. These sites were potential plasmid insertion sites in hg19 coordinates. PCR and Sanger sequencing were used to confirm the plasmid insertion sites. We confirmed nine insertion sites for plasmid B1 and five for plasmid A1 Figure 3—figure supplement 1B.
The plasmid insertion sequences were inserted into the human reference genome hg19 according to the validated insertion sites to build the single cloned cell line reference genome HeLa-AB1 using a custom Python3 script. Every insertion site was updated after the front sites in the same chromosome had been inserted.
HeLa-AB1 cells were plated in a 10 cm dish at a density of 1. Cells were monodispersed and filtered through a strainer to remove clumps. Cells were kept on chilled water before and during sorting. Single cells were selected on the basis of their side and forward scattering properties. Samples were lysed, and chromatins was sonicated to obtain soluble sheared chromatin average DNA length of — bp. The antibody used for H3K27me3 was ab Abcam.
TD; Vazyme , and The period of min was chosen for PWM light induction as it reduces noise and was better suited for covering chromatin dynamics studies from 0 to 1 day Figure 4A. Finally, all reactions were pooled together and purified with a PCR minElute purification column Qiagen. At day 0, two wells A2, A3, control and one well of HeLa-AB1 cells C2, inhibitors were replenished with regular medium or with regular medium containing 0. The resulting cell pellets were washed with 1 mL PBS containing 0.
The cells were filtered through the filter cap of the flow tube Catalog no. In the third step, after the scRNA-seq libraries were constructed, each library was split into two sets. Reads were preprocessed using the fastp 0.
The preprocessed reads were mapped to the single cloned cell line reference genome HeLa-AB1 using bowite2 2. The alignments were preprocessed with samtools 1. The deduplicated alignments were filtered using samtools 1. These reads were filtered using custom bash code to remove the reads that were mapped to mitochondria. For population sequencing data, all sample filtered reads were processed using the multiBamSummary function of deeptools 3. The scale factor and filtered reads for each sample were processed using the bamCoverage function of deeptools 3.
For single-cell sequencing data, the filtered reads were processed using the bamCoverage function of deeptools 3. The coverage tracks were processed using custom python3 scripts to calculate the coverage around the TSS sites. The coding region of vector CEG was added to the reference file, and a new reference was made using the mkref function of cellranger 3. The human and mouse mixed reads were processed by cellranger 3.
The matrix files were processed by R package scater 1. The cells were identified as human cells if the total UMI counts mapped to the mouse genome were less than 0. The cells were identified as mouse cells if the total UMI counts mapped to the human genome were less than 0. The UMI counts were normalized by the library size. Genes with mean counts under 1 CPM counts per million in any sample were neglected. Coefficients of variance CV and means for the remaining genes were computed for all three samples.
The normalized count matrices from two control samples were pooled together as the control. Genome assembly mm10 was selected as mouse reference in ChIP-Atlas. Genome assembly hg19 was chosen as a human reference in ChIP-Atlas.
The UMI counts of filtered genes were processed by the R package saver 1. Data are described by the sample mean and standard deviation SD. Sample sizes and details of statistical tests are provided in the corresponding figure legends. The metal frame was computer numerical control CNC -machined from aluminum alloy, sandblasted, and darkened. A computer programmable channel constant currents source 0—10 mA in steps provided precise and independent control of currents for each LED.
The control code was written in Python3 and modulated the currents at 1 Hz or slower. B View of a fully assembled illumination apparatus. C The channel programable constant current board and connected power supply. D The channel blue LED board. F The channel blue lights were visualized with a frosted glass plate. A Light intensity — digital input curves for the channel LED board. The circles represent measurements made using a laser power meter.
The lines are cubic smooth spline fitted curves. Each line represents a channel. B The lookup curves used to find digital input values y-axis for target light intensities x-axis for any of the 24 LEDs. The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Competing interests No competing interests declared. Author contributions Validation, Investigation, Methodology, Writing - review and editing. Data curation, Software, Formal analysis, Methodology, Writing - review and editing. Conceptualization, Formal analysis, Supervision, Funding acquisition, Investigation, Methodology, Writing - original draft, Project administration, Writing - review and editing. Our editorial process produces two outputs: i public reviews designed to be posted alongside the preprint for the benefit of readers; ii feedback on the manuscript for the authors, including requests for revisions, shown below.
We also include an acceptance summary that explains what the editors found interesting or important about the work. This paper uses a light-induced, synthetic gene expression system in mammalian cells to show that mean gene expression and variability 'noise' of expression can be independently tuned by modulating the light input. This expands this general strategy from yeast to mammalian cells and provides a tool to study the functional consequences of expression variability in mammalian cells.
The paper also reports an impressive amount of single-cell data on gene expression and chromatin state which suggest that variations in histone acetylation state contribute to the expression variability. Your article has been reviewed by 3 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Kevin Struhl as the Senior Editor.
The reviewers have opted to remain anonymous. The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission. Result 2 is new for the mammalian system but has previously been shown in yeast.
Result 1 requires additional data to be shown conclusively. Positive feedback via histone modifications could be consistent with either scenario, and depends on the experimental evidence for alternate, deterministically stable steady states. The experimental data in Figures 1D and 3D only weakly support the notion of 'low' and 'high' alternate states especially because protein abundances are bounded from below and potentially also from above due to general expression capacity or simply applied input ranges , compared to the simulated data in Figure 2F.
More direct evidence for bistability would be given by experiments that demonstrate hysteresis, which should be considered-see iv. The authors should either 1 de-emphasize the mathematical model, acknowledging its limitations, or 2 include a calibrated mathematical model-for example, using a stochastic model as in Benzinger and Khammash.
The inclusion of multiple reporters see iii would be a benefit. The current model represents cell-to-cell variability by cell-specific parametrization, and hence only contains extrinsic noise components. Thus, at intermediate AM light intensity, the mRuby gene expression should be deca-modal 10 modes , not bimodal as shown in Figures 1 — 2.
Something seems inconsistent between the local chromatin positive feedback model and the observed data. The mathematical model used to validate the observations does not model the total expression from 9 independent promoters, which is a critical omission given the cis-nature of the positive feedback loop. The fact that these 9 promoters generate 2 peaks at intermediate light intensity suggests that the GAVPO bistability likely originates from a trans-effect, i.
Is it possible that the bistability simply comes from the dimerization event caused by light? To unambiguously show bistability, the authors should 1 measure and show dynamic signatures of bistability e. However, this is a crude perturbation because A likely affects the expression of many genes, including any cis- and trans-factors that affect mRuby gene expression.
If you see bistability, then an alternative should be considered more seriously. Use a monomeric transcription factor regulated by a chemical e. Zinc-finger-ER-p65 and engineer a mRuby promoter with a single binding site. For scatterers with geometries that do not conform to separable coordinates, an integral equation method can be used, such as the direct numerical solution of integral equations, or the null field method also known as T-matrix approach [, , 98].
These analytical approaches allow us to reduce an MS problem for an arbitrary configuration of a finite number of scatterers to a family of single scattering problems. Hence, the analytical tools used to find the response of configurations of multiple objects follow from the analysis of scattering by single scatterers [, , , 28]. Below we will give a literature study of single scattering problems, followed by a literature review of investigations conducted on MS problems for obstacles of different geometries.
Both theoretical and experimental aspects are considered. Gaunaurd [63] gives an extensive and comprehensive bibliog- raphy and a review of topic on scattering from cylindrical and spherical shells and solids. More recent studies address cylinder scattering with relation to fiber scattering in composites, piezoelectrics, anisotropic solids, and inhomogeneous solids and take into account multiple scattering in a scatterer.
Elastic and acoustic wave scattering from a solid cylinder An investigation of acoustic scattering of normal incident plane acoustic waves by isotropic elastic cylinders and spheres was initiated by Faran in [55]. Faran ex- tended the theory of the scattering of plane sound waves to include SH waves. The scattering field was given as a function of specular reflection and circumferentially trav- eling geometric waves and elastic circumferential waves.
Computed scattering patterns were in a good agreement with experimental measurements. The normal scattering of incident plane waves by cylinders and plates immersed in water was studied by Maze et al. Backscattered spectrum was calculated using the Resonance Isolation and Identification Method; supplementary resonances were obtained for aluminum. Experimentally de- tected resonances were considered as normal modes of the target. The solution of the problem of scattering of an obliquely incident plane wave from an isotropic elastic cylin- der was obtained and compared with experimental measurements by Li et al.
The solution was given in terms of phase angles. The excitation of supplementary reso- nances was noticed for small incidence angles. A theoretical model of the scattering of an obliquely incident plane wave from an infinite aluminum cylinder was developed by Flax et al. In the limiting case of taking the angle of incidence as zero, the solutions given in [92] and [57] correspond to the previously given scattering description of Faran [55]. Honanvar et al. The displacement field was decomposed into scalar and vector potentials using the displacement decomposition proposed by Morse and Feshbach [] that will be discussed in Section 2.
Potentials were expanded in series of normal modes. In this model the influence of each element of the stiff- ness matrix on the various vibration resonant modes was shown. In the case of weak anisotropy the solution reduces to a simple model for an isotropic cylinder. Numerical calculations were performed first for an isotropic aluminum cylinder. Then the effect of perturbation of the elastic constants of an aluminum cylinder on the form function was evaluated.
Ahmad et al. The existence of two distinct types of transversely isotropic materials, namely type I and type II, was shown. The expected behavior of the form functions of such materials was described. Critical angles of the incidence were shown, away from these angles an angular pattern of scattering field alters significantly.
Honarvar et al. Rahman et al. They also noted that these representations lead to identical characteristic equations and the same final result. Scattering from cylindrical shells consisting of one layer Scattering from an infinite isotropic elastic hollow cylindrical shell by an obliquely in- cident plane acoustic wave was investigated by Maze et al. The far-field form function was calculated for an aluminum hollow cylindrical shell in water by the direct summation of the Rayleigh series.
The theoretical results are compared with the experi- mental results showing a good agreement between them. Veksler et al. The resonances were computed from the dispersion relations. The behavior of different scattered waves were studied, and the resonance contribution of these waves were shown for a thick walled shell. When the inner radius of the shell b tended to the outer radius, the scattering by an empty cavity was considered.
When the inner radius b tended to 0 another limiting situation was obtained: the scattering by an elastic cylinder in an elastic matrix. In [] an asymptotic solution of the problem of acoustic wave scattering from heavily fluid loaded thin isotropic cylindrical and spherical shells is derived by Norris et al. The proposed method is developed on the basis of thin shell theory and is effective in the midfrequency range, and it describes the total acoustic response as the sum of background response and resonant contribution.
Most scat- terers have simple geometry, such as a cylindrical or spherical shape. Below we will consider these works in detail. The effect of MS between the scatterers will be considered in Section 1. Acoustic scattering from multi-layered cylindrical shells and solids The scattering field of an oblique plane wave of incidence from a circular clad rod of infinite length was calculated by Honarvar et al.
Using Resonance Scattering Theory RST the effect of various resonance frequencies on the variation of cladding thickness was evaluated, where high frequency resonances were shown to be more sensi- tive than low frequency resonances.
Form functions of a copper-clad aluminum rod were evaluated for different incident angles of a plane wave. The comparison of numerical calculations with experimental measurements show good agreement. Orthotropic cylin- drical shells submerged in and filled with compressible ideal fluids were considered by Hasheminejad et al.
The method of wavefunction expansion is used to study the effects of inner fluid loading along with the shell thickness on the frequency response of the shell. The correlation between the perturbation in elastic constants of loaded shell material and the sensitivity of resonances associated with various modes appearing in the backscattered field is obtained.
The methods developed in [72] are generalized in [73] for an axially polarized piezo- electric material. Hasheminejad et al. Shells are submerged in and filled with compressible ideal fluids. Numerical calcu- lations for the total form function amplitude including the associated global scattering, the far-field inherent background and the resonance scattering coefficients of the n th normal mode are performed based on acoustical RST for different electrical boundary conditions.
Jamali et al. Numerical results are presented for a form function and radiation force. Elastic scattering from multi-layered cylindrical shells and solids Beattie et al. Scattering from a fiber-matrix interphase in a four-phase composed system, consisting of a matrix and a three-phase fiber, was studied by Huang et al. Different simplified approximate models were considered to show the effect of the fiber-matrix interphase on scattering. Sinclair et al. A normal mode technique was used to derive the scattering spectrum.
Numerical calculations were compared with experimental data for a form function. The original formulation of scattering from a transversely isotropic cylinder sub- merged in an acoustic medium given in [78] was generalized by Fan et al. The formulation of scatter- ing problem was provided for incident plane longitudinal, axially polarized shear and transversely polarized modes. The solution was obtained using normal mode expan- sions. Obtained results point to the sensitivity of several resonances to the perturbation of elastic constants of the cylinder for each type of scattered waves.
Niklasson et al. It was noted that in the case when there is a cavity in a transversely isotropic medium, the reflected shear waves are much stronger than in the case with a solid cylinder. Fan et al. The circumferential resonance modes of a submerged cylinder were studied over a large range of incidence angles using RST.
The shift of resonance frequencies to higher frequencies was noticed with the growth of incidence angle. Qian et al. The solution was obtained using the method of wavefunction expansion. The stress distribution, mechanical displacements and electrical potential around the piezoelectric cylinder were calculated.
The effects of incident angles over a range of normalized frequency along with the change in radius of the cylinder on the mechanical stress field and electric field concentrations were studied in detail, and the physical explanations were given for such effects. The results of this paper can be used in modeling piezoelectric composites, particularly, when piezoelectric cylinders are aligned sparsely in a matrix medium.
Cai [30] presented an analytical solution for the scattering of antiplane elastic waves by a layered circular elastic cylinder embedded in an elastic medium of infinite extent. Numerical calculations are performed for a ceramic-fiber reinforced metal-matrix com- posite system. The effects of the geometrical and physical properties of the interphase were investigated. When the outer cylindrical layer is more compliant and the inner core undergoes a rigid body motion, a resonance mode was noticed.
This approach was generalized by Cai, in [27], to study scattering by a multilayered scatterer using MS pro- cess, which is based on the observation that elastic MS occurs in a single scatterer which has an inner structure. The proposed approach can be extended for studying multi-core structures, i. Early work was conducted almost one and a half centuries ago by Rayleigh [] on the scattering of sound-waves by small elastic discontinuities, e.
Herzfeld studied the scattering of longitudinal waves by an elastic sphere submerged in a viscous fluid, but he did not include the scattered shear wave. The scattering field was obtained in the form of a progressing series of zonal harmonics where each harmonic depends on the radius vector and is multiplied by an unknown coefficient. Therefore, the unknowns were determined from the surface conditions at the interface by taking the factors of each zonal harmonic separately.
Epstein et al. Explicit expressions for the attenuation to water particles in air were derived from general results, considering the particular case of liquid droplets suspended in gases. The comparison of calculated and experimental data was shown for an attenuation. Hickling [76] both theoretically and experimentally stud- ied the scattering of shear and compressional waves from a homogeneous solid sphere submerged in an acoustic medium; numerical calculations of backscattering spectrum and pulse forms of echoes were presented.
An approach that describes the acoustical background of a submerged elastic isotropic spherical shell for a suitable thickness over an entire frequency range was developed by Werby []. It was shown that a background at a higher frequencies and thicker shells is equivalent to a rigid background, whereas low frequencies with a thin shell approxi- mation it tended toward a soft background. Dacol et al. This method was further generalized to the case of an arbitrary two-point correlation function in the positions of any two scatterers.
Martin [97] studied the problem of acoustic scattering from a sphere, specifically scattering by an inhomogeneous sphere submerged in a homogeneous fluid, and scattering by a ho- mogeneous sphere with a concentric inhomogeneous coating.
The ra- dial parts of the solutions were given in terms of Coulomb wave functions or Whittaker functions. Elastic scattering from spherical shells and solids The idea of acoustic scattering from a sphere was generalized to model elastic scattering problems from a sphere embedded in an elastic matrix [], [86], [50], [56], [].
Ying et al. The scattered field was obtained using continuity conditions for displacements and stresses at the interface. Calculations were performed for the problem of scattering of a longitudinal wave by an isotropic elastic sphere and a rigid sphere embedded in an elastic medium, and a spherical cavity surrounded by an elastic medium.
Mechanical properties of scatterers differed from those of the surrounding matrix material. Rayleigh scattering and some other limiting cases were discussed in detail. Expressions for the scattering cross section were derived for an elastic sphere, a fluid sphere and a spherical cavity. Knopoff [86] studied the scattering of incident plane shear waves by a perfectly rigid, infinitely dense sphere situated in an elastic medium.
The scattered field includes both longitudinal P and transverse S modes. Numerical calculations were shown for a group of obstacles with radii very small compared with wave length and radii equivalent to the wave length. Einspruch et al. For incident longitudinal waves, the scatterer was considered as a fluid-filled cavity embedded in an isotropic elastic matrix and the scattering coefficients were obtained from the 3 by 3 system of linear equations.
For incident transverse waves the scattering problem was considered as 3D and the obstacle was taken as a fluid- filled cavity, an empty cavity, a rigid sphere, and an elastic sphere embedded in an elastic medium with different physical parameters. The scattering cross section for a transverse wave was calculated and the Rayleigh limit was considered. Flax et al. Numerical results are presented for resonances and a backscattered form function of the solid inclusions surrounded by a Lucite or iron sphere embedded in an elastic matrix.
Sessarego [] studied the scattering by an elastic sphere embedded in sediment. A physical interpretation of scattered circum- ferential waves was given in terms of monostatic and bistatic scattering cross sections. The resonance behavior of the target was determined numerically in the individual nor- mal mode amplitudes. Numerical computations were in agreement with experimental measurements for an aluminum sphere embedded in Plexiglas. The extended bibliographical overview and concept of MS from obstacles are given by Martin [98].
There are several analytical methods to solve MS problems. An MS solution can follow from the analysis of wave scattering by single obstacles. For obstacles with variable-separable geometries, the solution of an MS problem can be obtained by the multipole expansion method [, , 98, ].
The multipole expansion method for variable-separable geometries can be applied in two ways. The method is exact, and leads to an infinite system of algebraic equations; in the system, the infinite sums are truncated to use in practice. This approach will be used in the analysis to follow.
This iterative procedure can be described as follows. The first-order scattering results from the excitation of scatterers by primary incident wave. The field scattered by each scatterer is found by translating the primary incident wave to the origin of local coordinate system and using single scattering analysis.
The second order scattering from one of the scatterers results from the excitation by the first order of scattering from the remaining scatterers, and so on to higher order of scattering. The calculation of higher orders of scattering is continued until the convergence of results is reached.
For scatterers with geometries that do not conform to separable coordinates, MS solutions can be calculated using the integral equation methods [58], [], []. The method was generalized in [] to investigate an acoustic MS from elastic multilayered scatterers, and further by Peterson et al.
Review of acoustic MS from solids and shells Radlinski and Meyers [] have investigated the scattering of waves radiated by an oscillating cylinder surrounded by a circular cage of rigid cylinders. They have shown a good agreement between a 2D analysis and experimental measurements of a farfield radiation for cages consisting of 6 and 12 cylinders.
Klyukin [85] has investigated both theoretically and experimentally the problem of an acoustic MS of plane waves by a 2D grating of rigid cylinders; a comparison of results for reflection characteristics shows a good agreement with the experimental data. Three different solutions for an MS problem for 2 cylinders are presented and com- pared with experimental data by Gustafson and Stepanishen [69]. An acoustic MS of plane waves by an array of rigid cylinders moving in an ideal inviscid fluid has been examined by Lin and Raptis [93].
Their technique involves formulating analytical expressions of scattering functions, and determining the effects of MS and vibration of cylinders on the scattering pattern; calculations are performed for a cluster of carbon and brass cylinders submerged in water, having different geometrical configu- rations of one, two, three, and seven circular cylinders. They have also reported a comparison of results for the semiclassical and quantum dynamics of the point particle-three hard disc system giving hints for more complex dynamical systems.
Kubenko, V. Scharstein [] researched the effects of coupling between two parallel soft cylinders of different radii via a comparison of far scattered fields computed by his proposed method and a superposition of non-interacting far fields of the isolated cylinders. The effect of incident waves on circular arrays of identical circular cylinders has been considered by Evans and Porter [52]; resultant forces versus wavenumber are calculated for symmetric and asymmetric arrangement of four, five and six cylinders; real parts and magnitudes of velocity potentials have been shown for a ring of four, five and six cylinders.
Wirzba [] has studied 2D convergence problems of periodic orbit expansions of the non-overlapping disconnected n-disk repellers, and derived the T- matrix of the n-disk scattering systems by the methods of stationary scattering theory. Decanini et al. In the second part of paper [44], Decanini et al.
Grote and Kirsch [67] have derived a Dirichlet-to-Neumann DtN boundary condition for the numerical solution of MS problems for the obstacle consisting of several disjointed components; being a natural boundary condition, the DtN condition fits into a variational formulation of BVP allowing easy use of FEM, and yields an exact formula for the far-field form function.
Hasheminejad and Alibakhshi [70] have studied MS effects of 2-D acoustic scattering in fiber suspensions considering the interactions of a plane compressional sound wave with a cluster of two flexible fibers submerged in a boundless viscous fluid medium; comparisons of angular distribution of the form function are presented for polymeric, elastic, and rigid cylinders at selected distances and frequencies. Sherer [] studied acoustic MS generated by two types of axisymmetric sources by a grating of arbitrary parallel multiple rigid circular cylinders of varying radii; the incident field is determined from a cylindrical line source and a spatially distributed acoustic source.
Lethuillier et al. Analytical solutions for MS problem were obtained using the multipole method in conjunction with the concept of T-matrix. Nu- merical and experimental results are given for grating of 2 to 5 shells to study the resonant interaction between close shells.
This approach also was employed by Bas et al. The S scattering matrix of an N-shell cluster was defined and a resonance spectrum was investigated. Numerical results for aligned cylinders were in agreement with previously published results [91]. The multi- pole method via T-matrix formulation was further employed by Cai et al. An isovelocity examples were given for waveguide with uniform physical properties and constant waveguide depth.
MS in the waveguide is compared with the corresponding 2D case for the cluster of 20 by 40 identical cylinders. Along this line Antoine et al. Iterative Krylov space methods were applied taking the advantage of the structure of algebraic linear system. Sodagar et al. For shells, the general solutions of governing equations of 3D elasticity are found using the Helmholtz decomposition.
A detailed study of resonances of the shells and the effect of the center-to-center distance of the shells on these resonances is conducted. Numerical results are compared with experimental data for a grating of aligned cylinders. Schwartz et al. It was shown that for primitive ordered cubic suspensions, governing equations reduced to a system of coupled equations which had a solution of the form predicted by Biot; for disordered suspensions the Biot formula did not hold.
Illustrative numerical calculations for the case of densely packed composites were presented. Review of elastic MS from solids and shells An extensive bibliography and a wide variety of problems that have been solved using the T-matrix approach was given in [, ]. The T- matrix approach was further elaborated by Varadan and Varadan [, ] to study MS of P and SV waves from an elastic scatterer in an elastic medium.
Cheng [37] provided a formal solution of MS of incident elastic P waves from a grating of rigid cylinders embedded in an elastic medium. Distributions of normalized stress and maximum normalized stress were shown for 2 identical circular cylinders. Sancar and Pao [] derived a power spectral function of pulses backscattered from two cylindrical cavities in a solid. The MS of waves from two cylindrical cavities in a solid was investigated. Luppe et al. The dispersion relations of coherent fast longitudinal and slow longitudinal waves as well as shear waves were calculated.
The shear wave decouples and propagates independently. The coupling effects of the longitudinal waves are noticed when forward scattering by a single cylinder of the slow wave into the fast is larger than forward scattering with no conversion. This approach was employed by Lonne [95] to predict attenuation of a unidirectional layer of Carbon fibers in an epoxy matrix, including the coupling between MS by fibers and viscoelastic losses phenomena.
Sato and Shindo [] examined MS of elastic P and SV waves from randomly distributed parallel fibers with graded interfacial layers embedded in a metal matrix composite containing randomly distributed parallel fibers with graded interfacial lay- ers. They obtained an analytical solution based on the multipole expansion method.
The effect of an imperfect layer on phase velocity versus frequency was shown for both P and SV waves. Biwa et al. The scattering coefficients are found numerically using a collocation method. This idea was generalized by Sumiya et al. Sheikhhassani and Dravinski [] investigated MS of SH wave by an arbitrary num- ber of multilayered inclusions in half space using a direct boundary integral equation method.
They analyzed the effects of MS, geometry, and impedance contrast of the layers on the surface motion. The effective wave speed and attenuation were given in explicit form when inclusions were voids. The method allows them to reduce the number of actual scatterers to a lesser number of abstract scatterers. Within transfor- mation acoustics, scholars such as Cummer et al. A perfect cloak has either an infinite mass IC or zero stiffness PM which is unreal- istic.
Norris and Nagy [] adopted a discrete layered approach to achieve IC made from three acoustic fluids; meanwhile, Urzhumov et al. Torrent and Sanchez-Dehesa [, ] used a homogenization technique on an acoustic cloak based upon multilayered structures.
They showed a trade-off between cloaking performance and construction difficulty. To overcome this drawback, Norris [] showed that perfect cloaking can be achieved with finite mass through the use of PM. However, the manufacturing difficulty remains because the cell size of the PM material has to be the same order as the wave length.
Within impedance acoustic cloak- ing, scholars such as Chen et al. On the other hand, active cloaking has the advantage of being broadband [, , ]. Despite the main interest in modeling passive cloaking devices, active exterior cloak- ing has been investigated broadly, and interest has focused on the Helmholtz equation in two dimensions [, , , , , ].
Miller [] created a cloak based on wave measurement and showed how the necessary surface sources should be calculated. He provided a formula for source amplitudes which depends on the measurements at all sensing points in the near-field, but could not derive a unique relationship between the source amplitudes and the incident field.
Vasquez et al. Importantly, this integral equation yields a linear relation between the source amplitudes and the incident wave field. In the pro- ceeding work [], the linear relation was given in more explicit form using multipole sources and numerical results were compared with SVD solutions of the linearized sys- tem [, ].
The approach was further generalized and extended to a 3D case in [] to handle the three-dimensional Helmholtz equation and seek non-resonant frequencies of the cloaked object. Paul Lueg first formulated the basic ideas of anti-sound in his U. Nelson and Elliott [] described the idea of completely suppressing the sound field in a finite volume inside an unbounded domain using the Kirchhoff- Helmholtz integral formula and continuous distribution of monopoles and dipoles.
The principles and practical application of anti-vibration techniques have been developed in [, 61]. The main function of anti-sound is to reduce the sound radiated from a sound source or to create a silence zone using sources but this active field is not required to be non-radiating.
Chapter 2 formulates the mathematical model of the problem of scattering of incident waves from solids, and gives definitions of the impedance and matricant matrices and preliminaries on acoustics and elasticity the- ory. Impedance matrices are defined for both spherically and cylindrically anisotropic media.
Chapter 3 analyzes the mathematical models described in Chapter 2 in detail, and presents a global impedance matrix method for a multilaminate general anisotropic medium and an explicit method for finding the impedance in piecewise uniform, trans- versely isotropic materials. It also describes the Global Matrix Methods for scattering from a multilaminate isotropic solid.
Acoustic and elastic scattering from a single scatterer is considered and a T-matrix is developed for the scatterer. Numerical results represent the form function, total scattering cross section, and total fields. Chapter 4 generalizes ideas given in Chapter 3 to study an MS and radiation from cylindrical structures, and includes a full interaction between the scatterers in both acoustic and elastic media. Nu- merical results for MS represent total fields and form functions in near and far-radiated fields for different configurations such as waveguides, Helmholtz resonators, slabs, and rings of cylinders.
Chapter 5 details the iterative methods suitable with parallel com- puting for solving the MS problems considered in Chapter 4. Iterative approaches by means of Neumann series expansions are given for a fixed value of frequency and gener- alized for a band of frequencies. Taking advantage of the Block Toeplitz structure of the linear system, another iterative technique is presented. Chapters 6 and 7 describe the modeling of active cloaking devices generated by active multipole sources that render an object invisible to incident waves in the acoustic and elastic media.
Finally, Chapter 8 gives conclusions and discusses future work. In Section 2. The impedance and matricant matrices are defined in Section 2. Scattering from homogeneous isotropic cylinders is considered in Section 2. Because of the symmetry, the relations between matrices 2.
The medium consists of n anisotropic layers with different densities and elasticity tensors in general. It is indicated in [] that eq. A transversely isotropic material is characterized by five non-trivial stiffnesses Cij. The stress-strain law for the transversely isotropic material is a particular case of eq. General solutions We consider time harmonic wave motion in a radially inhomogeneous spherically anisotropic medium. The density and the elasticity tensor of a radially inhomogeneous anisotropic cylinder depend on the coordinate r only.
The dynamic equilibrium equation 2. We seek solutions of eq. It was shown in [] that the separable solution of the form 2. Introducing the assumed form of solution 2. Elastic waves in a radially inhomogeneous multilayered sphere Let us investigate a spherically anisotropic medium as a radially inhomogeneous multi- layered sphere consisting of J isotropic spherical layers. Based on vector spherical harmonic functions 2. Thus, incorporating equations 2. The relations derived hereinafter are valid for both cylindrically and spherically anisotropic media.
In this section, we study the relation between vectors U r and V r. For the moment we may consider m as general. The relations between the matricant of 2. Comparing Eqs. Therefore, the dis- placement decomposition 2. Ahmad and Rahman [5],[4], [] showed that the above mentioned two representations lead to identical characteristic equations and the same final result. However, in [80], the response to [], authors noted that their solution has a stronger physical basis.
Nonetheless, in this dissertation, we use the Buchwald potentials approach for the purpose of mathematical simplicity. A cylindrical layer is an elastic solid; the medium inside the cylinder is either fluid, gas, or elastic core; the outer region is either acoustic medium or elastic matrix. Note that the equations given in Section 2. In this section, we formulate the wave scattering problem, find the general solutions of governing equations and derive the formulas for pressure, displacements and stresses in acoustic and elastic media.
The scattering coefficients will be calculated using the Impedance method and Global matrix method and verified with COMSOL results in the next chapter. Figure 2. Incident field: plane wave, cylindrical line source Figure 2. In eq. Response of cylinder The response of cylinder S1 to the incident waves pinc can be defined by the transition matrix T. In general, for obstacles with no rotational symmetry, the T matrix is non- diagonal.
For cylinders with a rotational symmetry, T is diagonal. The equation of motion of the elasticity theory is given by eq. Incorporating eqs. For a planar motion, in the absence of forcing f introducing the Helmholtz decomposition of a vector field u, eq. Introducing eqs. Introducing 2. Inner region Acoustic medium. Rigid core and hollow region. For an acoustic medium allowing all shear terms to tend to zero, only the first columns of the matrices Xln and Ynl remain in eq.
Here An is the incident wave coefficient and defined in Section 2. The scattering coefficient Bn is to be determined. Elastic matrix. In a solid elastic matrix, we consider a longitudinal P and transversal SV incidence separately.
Here, the scattering coefficients Bp,n and Bs,n can be determined from the boundary conditions. It begins with the impedance matrix method proposed in Section 3. Section 3. The method described in this section also serves as a tool to compare with more general solution methods based on the Riccati matrix differential equation for the impedance matrix [] for a cylindrically anisotropic medium.
In section 3. Transformation from elas- tic to acoustic matricant is shown in Section 3. A Global matrix method is described in Section 3. These include the use of scalar and vector potentials [], the transfer matrix method [23, 81, , 79], and the delta matrix method [, 48].
Alternatively, computationally stable methods have also been developed, e. The goal of this Section is to provide a methodology for modeling such materials. Consider a time harmonic wave motion in radially inhomogeneous anisotropic solids. The method proposed in this section is applicable for both cylindrically anisotropic and spherically anisotropic media.
The associated equilibrium equations for linear elasto- dynamics in cylindrical coordinates are summarized in Section 2. We seek solutions of equilibrium equations 2. We develop an approach suitable for radially inhomogeneous piecewise uniform anisotropic medium by explicit calculation of the global impedance matrix Z of 2. In particular, this method can be applied for piecewise uniform transversely isotropic TI cylinders for which the explicit formulas are available for the conditional impedance z [] and the two point impedance matrix Z of a given TI layer that will be derived in Section 3.
The approach is also suitable in spherical coordinates for a radially inhomogeneous sphere for which the material is piecewise uniform TI about er vector. However, the explicit formulas for impedance matrices z and Z for TI material in spherical coordinates are not available; we consider spherically anisotropic medium as radially inhomogeneous multilayered medium consisting of N isotropic layers, and derive the explicit formular for Z of a given isotropic layer in Section 3.
The approach is based on a recursive algorithm proposed by Rokhlin et al. The analysis in [] was restricted to multilay- ered media in Cartesian coordinates, whereas the present method is applicable to both cylindrically and spherically layered anisotropic medium. We will refer to Rokhlin and Wang [] several times in this section to note the similarities and differences of the approaches.
Results of this section are published in [] for a cylindrically anisotropic medium. From the second row of eq. Note that eqs. Employing 3. The global impedance matrix for the N -layered medium is obtained by using eq. The main differences between the present results and those of [] are, first that by construction the local Zk and global ZK two point impedance matrices are Hermitian matrices. Secondly, the present results are valid for cylindrically and spherically layered structures, as compared with those of [] which are for multilayered structures in Cartesian coordinates.
Despite the differences, we note that the two point impedance matrix Zk of eq. Incorporating Cauchy relations 2. Inserting eq. Introducing solutions 3. The formula for Xl follows by substituting the potentials 3. The derivation of the matrix zl r can be found in []. The explicit form of the two point impedance matrix see eq. Assume that the cylinder consists of J layers.
Consider a perpendicular wave incidence, i. The total pressure p is defined as a sum of incident pinc and scattered psc pressure fields, and satisfies the Helmholtz equation 2. The incident field pinc is given by 2.
The total radial stress and displacement fields in the surrounding fluid are defined by 2. The medium inside the multilaminate is either fluid, gas, or elastic core; the outer region is either acoustic medium or elastic matrix. We assume a perfect interface between each layer of a multilayered cylinder which requires continuity of stresses and displacements at the interface. Considering acoustic fluid in the interior, we write eq.
The stresses and displacements for the outer acoustic medium are given by 2. We will compare the result for Bn obtained in this section with the one that follows from the application of the Global matrix method to the solution of scattering of a multilayered cylindrical structure in the succeeding section. The stresses and displacements for the inner acoustic region are given by 2.
We derive an explicit formula for the impedance Z of a given isotropic layer, and find scattering coefficients using an acoustic impedance. We assume that each spherical layer satisfies the eqs. Assume a perfect interface between each layer in the multilayered sphere which requires continuity of stresses and displacements at the interface.
Outer acoustic region. The total pressure p x is a sum of incident pinc and scattered psc pressure fields, and satisfies the acoustic Helmholtz wave equation 2. The incident pressure is taken as an obliquely traveling plane wave of unit amplitude and written in terms of regular solutions of eq. The outgoing scattered wave pressure psc is taken in terms of irregular solutions of 3.
The displacement and the traction eq. Inner acoustic region. Continuity of displacements and traction on the interface implies the boundary conditions 3. Equation 2. Using 3. Their original formulation was for a cylinder submerged in an acoustic, compressible, inviscid fluid, and was was general- ized in [54] to the case of a cylinder embedded in an elastic matrix.
Orthotropic cylindrical shells submerged in and filled with compressible ideal fluids were considered by [73] using a a state space formulation for the sequentially laminated piecewise homogeneous config- uration. Piezoelectric hollow cylinders have been considered recently in two separate papers [, 73]. In this section, the mathematical model of acoustic and elastic wave scattering from a submerged TI cylinder is developed.
Both solid and hollow shell configurations are considered by combining an integral solution based on the Shuvalov formulation for a shell of non-zero interior radius with the impedance operator of a uniform core region. The impedance is expressed using the exact solution for a solid cylinder.
The impedance is zero for a hollow cylinder. The equilibrium equations written in terms of displacements are given by eqs. We seek an in-plane time harmonic solution of on equation of motion in the form 2. We decom- pose the displacement vector uinc p using Helmholtz potentials 2.
Rigid inner surface. If the inner surface of cylinder is rigid, i. If the inner surface of cylinder is traction free, i. From eq. Let us rearrange eq. The total pressure field p x is defined by eq. Subindex p is dropped hereunder for acoustic medium to simplify a notation. The outgoing scattered wave pressure psc at point P has form 2. The scattering coefficients Bn are derived next. The radial velocity components of the incident and scattered fields in the fluid, which follow from 2.
Such a relation follows from the continuity condi- tions. Let v0n b and p0n b be the radial velocity and pressure at the inner surface of the cylinder for circumferential mode n. Consider a two point impedance matrix Z r, r0 defined by eq. Our goal is to find a field pressure, velocity, displacement, stress at an arbitrary point P x , and investigate a far field response of cylinder to an excitation.
First, we study the scattering of acoustic waves from an isotropic multilayered cylin- der by considering a clad-rod consisting of an elastic isotropic core cladded with other isotropic materials, and a fluid filled cylinder submerged in water. Then, we investigate an elastic wave scattering from an infinite multilayered cylinder embedded in an elastic medium, considering the incidence of longitudinal and transverse waves seperately. A fluid-filled multilayered cylinder embedded in an elastic matrix will be studied as a par- ticular case.
We obtain the scattering coefficients for both acoustic and elastic waves, and relate them to the transition matrix T m that is also called the T-matrix. We will use T-matrix T m of cylinder Sm in isolation to study MS of acoustic and elastic waves from the cluster of cylinders in the proceeding chapters.
In this section, we only investigate a scattering from a single scatterer, and drop super-index m for simplicity of notation hereinafter. We will validate our approach by comparing our results with finding available in literature and with COMSOL simulations. For perpen- dicular acoustic incidence, i. The displace- ments and stress are defined by eq. The total outer acoustic pressure p x is a sum of incident pinc and scattered psc pressure fields defined by 2.
The outer pressure fields satisfies the acoustic Helmholtz equation 2. For a solid elastic cylinder with no cladding, the scalar potentials, eq. For a solid elastic cylinder, the T-matrix components have simplified form, and eqs. Consequently, both the impedance z1 defined by 2.
Hence, for a rigid cylinder, eq. Acoustic scattering from a fluid filled cylinder Consider scattering of acoustic incident waves from an anisotropic layered elastic cylin- der immersed in an ideal fluid illustrated in Figure 2. For perpendicular acoustic incidence, i. The inner and outer pressure fields satisfy the acoustic Helmholtz equation 2. The displacements and stress are de- fined by eq. The determinants in eqs.
We evaluate eq. Numerical results a Calculation performed by Flax et al. Figure 3. Properties of materials considered in this section are given Table 3. In Figure 3. The graphs in Figure a were obtained by Leon et al. The properties of shell taken from [89] are given in Table 3. COMSOL results are shown for a lesser number of nf req because of prohibitively long computation time.
Variation of modulus of backscattering form function and TSCS with normalized frequency ka for air-filled multilayers submerged in water is depicted in Figures 3. Graphs in Figure b show a full agreement with findings in [] depicted in Figure a. Figure a shows calculations performed in []. Graphs in Figures a - b are consistent. Figure displays calculations performed in []. Comparison of Figures a and 3. Figure a shows calculations performed in Figure 8 [].
Figures b - e display our computations for comparison. In [], the coefficients fn were obtained via Fourier sine and cosine series expansion, whereas here the form function coefficients fn are defined by eq. Figures on the left display the total field zoomed out. Figures on the right show enlarged view of total field zoomed in. Figures on the left display a total field zoomed out. Figures on the right show an enlarged view, a total field zoomed in.
Left color bars correspond to total displacement of the shell and right color bars correspond to total outer acoustic pressure field. Polar plots coincide with findings in [55]: Figures 5, 9, and 13 respectively. A scattering from a fluid-filled multilayered cylinder can be analyzed analogously and will be discussed in parallel. The displacements and stress are defined by eq. An anisotropic clad-rod consists of an elastic isotropic core cladded with anisotropic multilayered metamaterial.
We assume that metamaterial cladding is comprised of J isotropic layers, and is perfectly attached to the rod core , see Figure 3. The far scattered field amplitude function follows the asymptotic form of the Hankel function for large values of argument see eq. The right-hand side of eq. The results are presented in Figures 3. The Figure on the left displays calculations performed in [17], both theoretically and experimentally.
The graphs are in good agreement. Figure a on the left displays calculations performed in [17]. Figures 3. Figure a on the right displays calculations performed in []. In Section 4. In general, each obstacle may have no rotational symme- try. An arbitrary planar configuration of shells is given in Fig. Figure 4. The total pressure field p x is defined by 2. For a m source at point S depicted in Figure 4.
Considering a scattering from a single cylinder in Section 2. Below, we will consider MS from a grating of cylinders by taking into account the full interaction between the obstacles. We shall consider a grating of M obstacles, and in a particular case, MS from two cylinders. The total incident field impinging on the cylinder Sj is a sum of the last two terms on the right hand side of eq. Thus, the system of equations 4. The scattered field psc is defined by eq.
To find far-field behavior of psc , we will write it in terms of position vector x. The computations are performed on Matlab for different configurations depicted in Figures 4. In Figure 4. Figure b is our Matlab result based on the MST and given for comparison with findings in [].
Plots in Figures a and b show a full correspondence and confirm our theoretical prediction. The left-hand side and the right- hand side pictures illustrate configurations considered in [] and [71] correspondingly. Here d is the distance between the centers of the two cylinders. The polar plots show a good agreement between our theoretical predictions depicted in Figure 4.
Now let us consider some applications of MST. Specifically, we design waveguides and resonators as an arrangement of empty thin aluminum shells and active tuned shells [, ]. Shells have mechanical properties depicted in Table 4. It can be tuned by select- ing the shell thickness, spring stiffness and added mass, and matching its impedance to the impedance of water.
The T matrix for an active shell is derived in []. Fig- ures 4. Analyzing the total field around the configuration, we can notice that active shells act as water. Pictures for a total field around a waveguide and a configuration with active shells are almost identical.
Consider now another planar configuration of shells arranged in a closed rectangular frame shown in Figures b-c. The individual shells of these arrangements can be hidden in water by optimizing the spring-mass parameters. Figures 4. Figures a and b show identical behavior, similarly Figures a and b are alike and have a full correspondence.
An arrangement of active and passive empty shells produces equal total field amplitudes. When all shells are active, i. Figures show that at resonance frequencies, the frame acts as a resonator resembling the dynamical behavior of two interacting dipoles. The dipole behavior of the structure is noticed again. In general, each obstacle may have no rotational symmetry. We will refer to obstacles simply as cylinders, but this may include hollow, rigid, and elastic solid cylinders of outer radii am , as well as thin and thick cylindrical shells of outer am and inner bm radii with and without attachments inside the shells.
The Helmholtz decomposition 2. We decompose the total displacement field u as a sum of incident field uinc and scattered field usc such that eq. The response of each cylinder Sm to incident elastic waves can be defined by its transition matrix T m. The total incident field impinging on cylinder Sj is a sum of the last two terms on the right hand side of eq. The cylinders are positioned at points O1 0, 1.
Properties of considered material are given in Table 4. The complexity of the system grows with the number of scatterers and frequency. A direct solution of the system leads to an excessive simulation run time. Therefore, in this chapter, we develop and provide iter- ative approaches suitable for a parallel computation of linear systems and applicable in many different areas of science and engineering.
For example, these iterative techniques can be used in acoustics, elastodynamics, electrodynamics, probability and mathemat- ical statistics, algebra, numerical solution of integral equations, random walk, theory of stationary time series and signals, etc. We start in Section 5. Specifically, we study the structure of multilevel matrices and consider some examples.
In Section 5. The iterative methods for a solution of linear systems are described in Section 5. For an arbitrary planar configuration of scatterers, one of the iterative approaches uses a Neumann series expansion to invert the matrix X and solve a linear system.
This approach is presented in Section 5. The generalization of this iterative approach for a band of frequencies is provided in Section 5. Section 5. Particularly, the application of the matrix vector product method and Fast Fourier Transform is presented. Another iterative approach is proposed for an evenly distributed cluster of scatterers using the Block Toeplitz structure of the system.
Numerical results are illustrated in Section 5. They include the Neumann expansion study given in Section 5. The spectral radius of the matrix is investigated as a function of wavenumber and the validity of the Neumann series solution is shown by modifying the number of scatterers M and the distance between the cylinders d.
Numerical results are also presented for the CPU time taken to solve the linear system by varying values of M and d. Toeplitz [], honoring his early work from on bilinear L-forms in relation to Laurent series. A Toeplitz matrix has a specific structure such that its each descending diagonal from the left to the right is constant. As we will see further, the special features of the Toeplitz matrix will allow us to apply the Krylov subspace methods [] that are nowadays considered among the most powerful iterative techniques available for large scale linear systems.
Here we dropped upper indices j and m of matrix Pj,m for simplicity of notation. P0 The Circulant matrices that have upper triangular or lower triangular form are called semicirculant matrices. These block matrices can also consist of blocks of Toeplitz structure leading to the construction of multilevel block matrices that will be addressed further.
The inverse of an infinite Toeplitz matrix has also infinite Toeplitz structure. Con- sidering different classes of matrices, it is assumed that one class contains matrices of the same size that form a linear manifold in the space of matrices of the same q size. We will denote a general type of matrices by symbol G, Toeplitz matrices by T, and Circulant matrices by C.
The general type of matrices G can be defined by eq. Let matrices of K type be defined by eq. Let K type satisfy the system 5. Let K1 ,. We can construct new composite types based on these types, e. The matrices of composite types are called multilevel matrices and characterized by block partitioning of different levels.
We will denote blocks as follows. The matrix A itself is a single block of level 0. If the matrix A has a composite type K1. The last level s is formed by the elements that cannot be partitioned, and assumed to contain only complex numbers. Multilevel partitioning is of interest if the level blocks have some structure. In the proceeding section, we will consider the matrices of composite types Tm1 Gm2 and Tm1 , m2 Gm3. The former defines a two-level block matrix, i.
Let each cylinder have the same physical properties and radius and satisfy the condition of continuity of normal stresses and displacements at the interfaces. Then for a configuration with My rows and Mx columns of i. Figure 5. On the right picture, d is the distance between the centers of two cylinders.
Example of BT matrix of level 1. Example of BT matrix of level 2. At the same time, as a BT matrix, the matrix X has order My see eq. Biorthogonal polynomials are a generalization of or- thogonal polynomials and defined as polynomials that are orthogonal to several different measures. Biorthogonal polynomials share some properties of orthogonal polynomials. The basics of the theory of orthogonal polynomials were established by Chebyshev [], and further developed by A.
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