# The evolutionary dynamics of extrachromosomal DNA in human cancers – Nature Genetics

Our research complies with all relevant ethical guidelines. FISH images from GBMs were obtained from patients treated at UCLA participating in a multi-institutional phase II clinical trial of lapatinib sponsored by the North American Brain Tumor Consortium NABTC 04-01, a biomarker and phase II study of lapatinib GW572016 (lapatinib) in recurrent GBM. The collection and use of patient samples was approved by the UCLA institutional review board. These samples have been described previously, including in Nathanson et al.9. FISH images from NBs were acquired as part of routine molecular tumor diagnostics. Patients were registered and treated according to the trial protocols of the Society of Paediatric Oncology European Neuroblastoma Network HR-NBL-1 trial (NCT01704716) or the German Society of Pediatric Oncology and Hematology (GPOH) NB2004 trial. This study was conducted in accordance with the World Medical Association Declaration of Helsinki (2013) and good clinical practice; informed consent was obtained from all patients or their guardians. The collection and use of patient specimens was approved by the institutional review boards of the St. Anna Kinderspital in Vienna, the Charité-Universitätsmedizin Berlin and the Medical Faculty, University of Cologne. Specimens and clinical data were archived and made available by Charité-Universitätsmedizin Berlin, the St. Anna Kinderspital or the National Neuroblastoma Biobank and Neuroblastoma Trial Registry (University Children’s Hospital Cologne) of the GPOH.

### Cell culture

Cell lines were purchased from ATCC or the DSMZ-German Collection of Microorganisms and Cell Cultures (Leibniz Institute) or were a kind gift from J.H. Schulte. GBM39-HSR and GBM39-EC were derived from a patient with GBM as described previously9. Hap1 cells (Horizon Discovery) were maintained in IMDM supplemented with GlutaMAX and 10% FCS (Gibco).

PC3 cells were cultured in DMEM with 10% FCS. COLO320-HSR and COLO320-DM were cultured in DMEM/F12 50:50% with 10% FCS. SNU16 were grown in Roswell Park Memorial Institute (RPMI) 1640 with 10% FCS. GBM39-HSR and GBM39-EC neurospheres were grown in DMEM/F12 with B27, GlutaMAX, heparin (5 μg ml−1), EGF (20 ng ml−1), and fibroblast growth factor (20 ng ml−1). TR14 cells were grown in RPMI 1640 with 20% FCS. Cell numbers were counted with a TC20 automated cell counter (Bio-Rad Laboratories). For drug treatments, the drug was replaced every 3–4 d.

Cells were concentrated in metaphase by treatment with KaryoMAX Colcemid (Gibco) at 100 ng ml−1 for between 3 h and overnight (depending on the cell cycle speed). Cells were washed once with PBS and a single-cell suspension was incubated in 75 mM KCl for 15 min at 37 °C. Cells were then fixed with Carnoy’s fixative (3:1 methanol:glacial acetic acid) and spun down. Cells were washed with fixative three additional times. Cells were then dropped onto humidified glass slides.

### FISH

Fixed samples on coverslips or slides were equilibrated briefly in 2× SSC buffer. They were then dehydrated in ascending ethanol concentrations of 70, 85 and 100% for approximately 2 min each. FISH probes were diluted in hybridization buffer (Empire Genomics) and added to the sample with the addition of a coverslip or slide. Samples were denatured at 72 °C for 2 min and then hybridized at 37 °C overnight in a humid and dark chamber. Samples were then washed with 0.4× SSC then 2× SSC 0.1% Tween 20 (all washes lasting approximately 2 min). 4,6-Diamidino-2-phenylindole (DAPI) (100 ng ml−1) was applied to samples for 10 min. Samples were then washed again with 2× SSC 0.1% Tween 20 then 2× SSC. Samples were briefly washed in double-distilled H2O and mounted with ProLong Gold. Slides were sealed with nail polish.

### Dual immunofluorescence–FISH

Asynchronous cells were grown on poly-L-lysine-coated coverslips (laminin for GBM39-EC). Cells were washed once with PBS and fixed with cold 4% paraformaldehyde (PFA) at room temperature for 10–15 min. Samples were permeabilized with 0.5% Triton X-100 in PBS for 10 min at room temperature and then washed with PBS. Samples were then blocked with 3% BSA in PBS 0.05% Triton X-100 for 30 min at room temperature. Samples were incubated in primary antibody, diluted in blocking buffer (1:100–1:200) for either 1 h at room temperature or overnight at 4 °C. Samples were washed three times in PBS 0.05% Triton X-100. Samples were incubated in secondary antibody, diluted in blocking buffer for 1 h at room temperature (all subsequent steps in the dark) and then washed three times in PBS 0.05% Triton X-100. Cells were washed once with PBS and refixed with cold 4% PFA for 20 min at room temperature. Cells were washed once with PBS then once with 2× SSC buffer. FISH proceeded as described above with the following difference: denaturation was performed at 80 °C for 20 min.

### Microscopy

Conventional fluorescence microscopy was performed using an Olympus BX43 microscope; images were acquired with a QIClick cooled camera. Confocal microscopy was performed using a Leica SP8 microscope with lightning deconvolution (University of California San Diego School of Medicine Microscopy Core). NB cell lines were imaged with a Leica TCS SP5 microscope, HCC PL APO lambda blue ×63 1.4 oil lens or with DeltaVision Elite Cell Imaging System (Applied Precision) and microscope (model IX-71; Olympus) controlled by the SoftWoRx software v.6.5.2 (Applied Precision) and a 60x objective lens with a CoolSNAP HQ2 camera (Photometrics).

### NB patient tissue FISH

FISH analysis was performed on 4-µm sections of formalin-fixed, paraffin-embedded blocks. Slides were deparaffinized, dehydrated and incubated in pretreatment solution (Dako) for 10 min at 95–99 °C. Samples were treated with pepsin solution for 2 min at 37 °C. For hybridization, the ZytoLight SPEC MYCN/2q11 Dual Color Probe (ZytoVision) was used. Incubation took place overnight at 37 °C, followed by counterstaining with DAPI. For each case, signals were counted in 50 nonoverlapping tumor cells using a fluorescence microscope (BX63 Automated Fluorescence Microscope; Olympus). Computer-based documentation and image analysis was performed with the SoloWeb Imaging System (BioView Ltd) MYCN amplification (MYCN FISH+) was defined as an MYCN/2q11.2 ratio >4.0, as described in the INRG report32.

### Quantification of FISH foci

Quantification of FISH foci was performed using the ImageJ Find plugin maxima function in a supervised fashion. To quantify pixel intensity, the ImageJ Pixel intensity function was used. The FISH images of the tissue of these two patients with GBM were obtained as part of a phase II lapatinib GBM clinical trial described previously. In brief, patients were administered 750 mg of lapatinib orally twice a day for 7–10 days (depending on whether the treatment interval fell over a weekend) before surgery, the time to steady state. Blood and tissue samples were obtained at the time of resection9.

### Construction of PC3-TetO cell line

The insertion of TetO repeats was conducted through CRISPR–cas9-mediated approaches. The plasmids pSP2-96-mer TetO-EFS-BlaR and F9-TetR-EGFP-IRES-PuroR used in this were kind gifts from H. Zhao21. Briefly, the intergenic region between MYC and PVT1 was selected as the insertion region on the basis that it is amplified in PC3 cells on ecDNA with high frequency. DNA sequences were retrieved from the UCSC Genome Brower; repetitive and low complexity DNA sequences were annotated and masked by RepeatMasker in the UCSC Genome Browser. The guide sequences of sgRNAs were designed by the CRISPRdirect web tool33 and their amplification was confirmed with whole-genome sequencing data. The guide sequence selected was constructed into pSpCas9(BB)-2A-Puro (PX459). pSpCas9(BB)-2A-Puro(PX459) was a gift from F. Zhang (Addgene plasmid no. 62988; http://n2t.net/addgene:62988; research resource identifier: Addgene_62988). The repair donor was obtained through PCR amplification, using the pSP2-96-merTetO-EFS-BlaR plasmid as template, as well as primers containing the 50-nucleotide homology arm upstream and downstream of the predicted cutting site.

The transfection of the CRISPR–Cas9 plasmid and 96-mer TetO EGFP-BlastR donor into PC3 cells was conducted with the X-tremeGENE HP transfection reagent according to the manufacturer’s instructions with the CRISPR–Cas9 plasmid only or the 96-mer TetO EGFP-BlastR only used as negative control. Two days after transfection, blasticidin was added to the culture medium for 3 d, at a time point when most of the cells in the negative control groups had died while more cells survived in the group with transfection of the CRISPR–Cas9 plasmid and donor. The surviving cells were subjected to limited dilution in a 96-well plate, with blasticidin being added all the time. Surviving clones were expanded and their genomic DNA (gDNA) was extracted and subjected to genotyping with a pair of primers flanking the inserted region. The PCR product of the genotyping results was subjected to Sanger sequencing to confirm the insertion at the predicted cutting site. Clones with positive genotyping bands were expanded and metaphase cells were collected. Double FISH with FISH probe against the Tet operator and against the MYC FISH probe was performed on the metaphase spread. PC3 cells with TetO repeats were infected with lentivirus containing F9-TetR-EGFP-IRES-PuroR; 2 d after infection, puromycin was added to the culture medium to establish a stable cell line that could be used to image ecDNA with the aid of EGFP visualization.

### Live-cell imaging of ecDNA

The PC3 TetO TetR-GFP cell line was transfected with a PiggyBac vector expressing H2B-SNAPf and the super PiggyBac transposase (2:1 ratio) as described previously34. Stable transfectants were selected by 500 µg ml−1 G418 and sorted by flow cytometry. To facilitate long-term time-lapse imaging, 10 µg ml−1 human fibronectin was coated in each well of an 8-well Lab-Tek chambered cover glass. Before imaging, cells were stained with 25 nM SNAP tag ligand JF669 (ref. 22) at 37 °C for 30 min followed by 3 washes with regular medium for 30 min in total. Cells were then transferred to an imaging buffer containing 20% serum in 1× Opti-Klear live-cell imaging buffer at 37 °C. Cells were imaged on a Zeiss LSM880 microscope prestabilized at 37 °C for 2 h. We illuminated the sample with a 1.5% 488-nm laser and 0.75% 633-nm laser with the EC Plan-Neofluar ×40/1.30 oil lens, beam splitter MBS 488/561/633 and filters BP 495–550 + LP 570. Z-stacks were acquired with a 0.3-µm z step size with 4-min intervals between each volumetric imaging for a total of 16 h.

### Colony formation assay

TR14 cells were taken from 60 d of treatment with either dimethylsulfoxide (DMSO), 50 nM palbociclib or 5 nM abemaciclib, and seeded into a poly-D-lysine-coated 24-well plate at 20,000 cells per well. After 24 h, the cells from each condition were treated with either DMSO, 50 nM palbociclib or 5 nM abemaciclib over 20 d in triplicate. At 20 d, crystal violet staining was performed. Briefly, the cell culture medium was aspirated, cells were washed gently with PBS, fixed in 4% PFA in PBS for 20 min, stained with 2 ml crystal violet solution (50 mg in 50 ml 10% ethanol in Milli-Q water), washed once with PBS and dried for 30 min. The area intensity was calculated using the ColonyArea plugin in ImageJ v2 (NIH)35.

### CellTiter-Glo

TR14 cells were taken from 60 d of treatment with either DMSO, 50 nM palbociclib or 5 nM abemaciclib and seeded into white flat-bottom 96-well plates (Corning) in 100 µl medium at a density of 500 cells per well. After 24 h, cells were treated with either vehicle, 50 nM palbociclib or 5 nM abemaciclib (50 µl of drug solution per well). Cell viability was determined using the CellTiter-Glo Luminescent Cell Viability Assay (Promega Corporation) at 3, 6 and 9 d after the drug was added, according to the manufacturer’s protocol.

### Flow cytometry

Single-cell suspensions were made and passed through a cell filter to ensure single-cell suspension. Cells were suspended in flow cytometry buffer (Hanks’ Balanced Salt Solution buffer without calcium and magnesium, 1× GlutaMAX, 0.5% (v/v) FCS, 10 mM HEPES). EGFRvIII monoclonal antibody 806 (ref. 36) was added at 1 μg per million cells and incubated on ice for 1 h. Cells were washed in flow cytometry buffer and resuspended in buffer with anti-mouse Alexa Fluor 488 antibody (1:1,000, catalog no. A11017; Thermo Fisher Scientific) for 45 min on ice in the dark. Cells were washed again with flow cytometry buffer and resuspended in flow cytometry buffer at approximately 4 million cells per milliliter. Cells were sorted using a Sony SH800 FACS sorter, which was calibrated; gating was informed using a secondary-only negative control. The sorting strategy is shown in Supplementary Fig. 14.

### Quantitative PCR

DNA extraction was performed using the NucleoSpin Tissue Kit (Macherey-Nagel) according to the manufacturer’s protocol. Quantitative PCR (qPCR) was performed using 50 ng or 1.5 µl of template DNA and 0.5 µM primers with the SYBR Green PCR Master Mix (Thermo Fisher Scientific) in FrameStar 96-well PCR plates (4titude). Reactions were run and monitored on a StepOnePlus Real-Time PCR System (Thermo Fisher Scientific) and Ct values were calculated with the StepOne Plus software v.2.3 (Thermo Fisher Scientific): CDK4 forward: AAAGTTACCACCACACCCCC; CDK4 reverse: AGTGCTAAGAAAGCGGCACT.

### Guide RNA design for CRISPR-C

sgRNAs were designed to target the ends of a previously reported DHFR-containing ecDNA amplicon (clone: PD29424h)17; 1,000 base pairs (bp) of sequence flanking each end of this segment (Chr5: 79,841,431–81,655,326; hg19) were used to design guides using the Integrated DNA Technologies (IDT) Custom Alt-R CRISPR–Cas9 guide RNA software (https://www.idtdna.com/site/order/designtool/index/CRISPR_CUSTOM). These sequences were ordered as Alt-R sgRNAs (IDT).

### ecDNA induction by CRISPR-C

Hap1 cells were trypsinized, quenched with IMDM (GlutaMAX, 10% FCS), counted and centrifuged at 300g for 5 min. Cells were washed once with PBS before resuspension in Neon Resuspension Buffer to 1.1 × 107 cells ml−1. Ribonucleoprotein (RNP) complexes were formed as follows: Cas9 (IDT) was diluted to 36 μM in Neon Resuspension Buffer. Equal volumes of diluted Cas9 and sgRNA (44 μM in TE, pH 8.0) were mixed and incubated at room temperature for 10–20 min. Left (DHFR_H2_sgL) and right (DHFR_H2_sgR) sgRNA RNPs were assembled separately. Then, 5.5 μl of each RNP, 5.5 22 μl of electroporation enhancer (10.8 μM; IDT) and 99 μl of cells were mixed and electroporated according to the manufacturer’s instructions using a 100-μl Neon pipet tip and electroporated with the Neon Transfection System (Thermo Fisher Scientific) using the following settings: 1,575 V, 10-ms pulse width, 3 pulses. Single-guide controls were prepared as above except 11 μl of the appropriate sgRNA was used. Electroporated cells were dispensed into 3.2 ml of medium (± hypoxanthine and thymidine supplementation as appropriate) and split into 6 wells of a 24-well plate. Negative electroporation control cells were resuspended in Neon Resuspension Buffer and then added directly to wells containing fresh medium.

For neutral selection, cells were cultured in 24-well plates and passaged every 2 d. During passaging, 80–90% of the cells in each well were used for gDNA isolation, while the rest were transferred to a new plate containing fresh medium. For hypoxanthine- and thymidine-supplemented wells, hypoxanthine and thymidine supplement (100X, catalog no. 11067030; Gibco) was added to final concentrations of 100 μΜ hypoxanthine and 16 μΜ thymidine.

For positive selection, 3 d after electroporation, cells were passaged into 12-well plates and the day 3 time point was collected. Cells were changed to medium containing the indicated concentration of methotrexate (Calbiochem) 4 d after electroporation. Medium was changed every 2–3 days and cells were passaged when at 70–80% confluence. Cells were collected after 14 days of methotrexate incubation (18 days after electroporation). The final DMSO concentration in methotrexate-treated wells was 0.1%.

Cells were collected at the indicated time points as follows: cells were washed with 1 ml per well prewarmed PBS (Gibco), followed by the addition of 100 μl TrypLE Express (Thermo Fisher Scientific) and incubation at 37 °C for 5–10 min. TrypLE was quenched with 800 μl IMDM (GlutaMAX, 10% FCS) and the cell suspension was pelleted at 300 g for 5 min at 4 °C. The supernatant was discarded and the cell pellets were stored at −80 °C.

The CRISPR-C schematic was created with BioRender.

### ddPCR to determine ecDNA or chromosomal scar frequency

gDNA was isolated using DNeasy columns (QIAGEN) according to the manufacturer’s instructions, including a 10-min incubation at 56 °C during the proteinase K digestion step; DNA was eluted with 100 μl EB buffer.

Amplicons for the ecDNA junction, chromosomal scar junction and glyceraldehyde 3-phosphate dehydrogenase (GAPDH) were designed using the IDT PrimerQuest software (https://www.idtdna.com/PrimerQuest/Home/Index). Dual-quenched probes (IDT) were used: FAM-labeled probes were used for both the ecDNA and chromosomal scar junction amplicons to facilitate multiplexing with the GAPDH amplicon utilizing a HEX-labeled probe. All probe and primer sequences are available in Supplementary Information. Droplets were created using droplet-generating oil for probes, DG8 cartridges, DG8 gaskets and the QX200 Droplet generator (Bio-Rad Laboratories); amplification was performed using the ddPCR Supermix for Probes (Bio-Rad Laboratories). The ddPCR Supermix amplification reactions were set up according to the manufacturer’s specifications (Bio-Rad Laboratories). Approximately 60 ng of gDNA was used in a 20 μl reaction with a final primer concentration of 900 nM (225 nM for each primer), 125 nM FAM probe and 125 nM HEX probe. The reaction was partitioned into droplets for amplification according to the manufacturer’s protocol (Bio-Rad Laboratories). Droplets were transferred to a 96-well PCR plate and heat-sealed using the PX1 PCR plate sealer (Bio-Rad Laboratories). Droplets were amplified using the following cycling conditions: 95 °C for 10 min, 40 cycles (94 °C for 30 s, 56.1 °C for 60 s), 98 °C for 10 min. After thermal cycling, droplets were scanned individually using the QX200 Droplet Digital PCR system (Bio-Rad Laboratories). Positive and negative droplets in each fluorescent channel (HEX, FAM) were distinguished on the basis of fluorescence amplitude using a global threshold set by the minimal intrinsic fluorescence signal resulting from imperfect quenching of the fluorogenic probes (negative droplets) compared to the strong fluorescence signal from cleaved probes in droplets with amplified template(s). The frequency of ecDNA or chromosomal scar was calculated by dividing their measured concentration by the concentration of the GAPDH amplicon.

### Quantification of single-cell ecDNA segregation patterns

We generated the theoretically expected distribution of ecDNA copy number fractions after a single cell division under different models of ecDNA segregation by stochastic computer simulations implemented in C++. Briefly a single cell is initiated with a random number of ecDNA copies n, drawn from a uniform distribution U(20,200). EcDNA is amplified and 2n ecDNA copies are segregated between two daughter cells after a binomial trial B(2n, p) with segregation probability p. In this case, p = 1/2 corresponds to random segregation and p > 1/2 to a biased random segregation. This results in two daughter cells with ecDNA copy number n1 ≈ B(2n, p) and n2 = n − n1. The fraction of segregated ecDNA, f, is then calculated as:

$$f_1 = \frac{{n_1}}{{n_1 + n_2}}\,{{{\mathrm{and}}}}\,f_2 = \frac{{n_2}}{{n_1 + n_2}}.$$

Iterating the process 107 times generates the expected distribution of f as shown in Fig. 1c. Similarly, we generated an expected distribution of f for chromosomal patterns of inheritance. For perfect chromosomal segregation, we have f1 = f2 = 1/2. To allow for mis-segregation, we introduced a probability u = 0.05 such that n1 = n ± 1 and n2 = n − n1. We used Kolmogorov–Smirnov statistics to compare the theoretically expected and experimentally observed distributions of ecDNA copy number fractions under these different scenarios.

### Stochastic simulations of ecDNA population dynamics

We implemented individually based stochastic computer simulations of the ecDNA population dynamics in C++. For each cell, the exact number of ecDNA copies was recorded through the simulation. Cells were chosen randomly but proportional to fitness for proliferation using a Gillespie algorithm. The simulation was initiated with one cell carrying n0 copies of ecDNA. The proliferation rate of cells without ecDNA was set to r = 1 (time is measured in generations). A fitness effect for cells with ecDNA then corresponded to a proliferation rate r+ = s. In this example, s > 1 models a fitness advantage, 0 < s < 1 a fitness disadvantage and s = 1 corresponds to no fitness difference (neutral dynamics, r+ = r). During proliferation, the number of ecDNA copies in that cell are doubled and randomly distributed into both daughter cells according to a binomial trail B(n, p) with success rate p = 1/2. If a cell carries no ecDNA, no daughter cell inherits ecDNA. We terminated simulations at a specified cell population size. We output the copy number of ecDNA for each cell at the end of each simulation, which allowed us to construct other quantities of interest, such as the ecDNA copy number distribution, the time dynamics of moments, the power law scaling of tails or Shannon diversity index. We used the Kolmogorov–Smirnov statistics to test similarity between simulated and experimental ecDNA copy number distributions and Shapiro–Wilk statistics to test for deviations from normality.

### Sampling and resolution limits

We ran an in silico trial to test our ability to reconstruct the true ecDNA copy number distribution from a sampled subset of varying sizes. We constructed a simulated ecDNA copy number distribution from 2 × 106 cells using our stochastic simulations. We then performed 500 random samples of 25, 50, 100 and 500 cells, reconstructed the sampled ecDNA copy number distribution and compared similarity to the true copy number distribution using the Kolmogorov–Smirnov statistics. The distribution converges to the true distribution with increasing sampling size and a comparably small sample of 100–500 cells is sufficient to reconstruct the true underlying ecDNA copy number distribution.

### Mathematical description of ecDNA dynamics

#### Deterministic two-population model without selection

In the simplest representation of the model, we discriminated cells that did or did not carry copies of ecDNA. We denoted cells with copies of ecDNA as N+(t) and cells without copies of ecDNA with N(t). We can write for the change of these cells in time t:

$$\frac{{\partial N^ – \left( t \right)}}{{\partial t}} = N^ – \left( t \right) + \upsilon \left( {N^ + \left( t \right)} \right)N^ + \left( t \right)$$

$$\frac{{\partial N^ + (t)}}{{\partial t}} = N^ + \left( t \right) – \upsilon (N^ + \left( t \right))N^ + (t)$$

where $$\upsilon \left( {N^ + \left( t \right)} \right)$$ corresponds to the loss rate of random complete asymmetric ecDNA segregation. We found for the fraction of cells carrying ecDNA f+(t) in an exponentially growing population:

$$f^ + (t) = \frac{2}{{2 + t}}$$

The fraction of cells carrying ecDNA decreases with approximately 1/t if ecDNA is neutral. Thus, copies of neutral ecDNA are only present in a small subpopulation of tumor cells.

### Deterministic two-population model with selection

The above equations can be modified to allow for a fitness advantage s > 1 for cells carrying ecDNA:

$$\frac{{\partial N^ – \left( t \right)}}{{\partial t}} = N^ – \left( t \right) + s\upsilon \left( {N^ + \left( t \right)} \right)N^ + \left( t \right)$$

$$\frac{{\partial N^ + (t)}}{{\partial t}} = sN^ + \left( t \right) – s\upsilon (N^ + \left( t \right))N^ + (t)$$

The solution to this set of equations is:

$$N^ + \left( t \right) = \left( {1 – f^ – } \right){\mathrm{e}}^{st – (1 – s){\int}_0^t {f^ – } (\tau ){\mathrm{d}}\tau }$$

In the case of positive selection, the fraction of cells with ecDNA is $$f^ + \to 1$$. For a sufficiently long time, the tumor will be dominated by cells carrying ecDNA.

### Stochastic dynamics of neutral ecDNA

We were also interested in the stochastic properties of ecDNA dynamics in a growing population. Therefore, we moved to a more fine-grained picture and considered the number of cells Nk(t) with k copies of ecDNA at time t. The dynamic equation for neutral copies of ecDNA becomes:

$$\frac{{\partial N_k(t)}}{{\partial t}} = – N_k\left( t \right) + 2\mathop {\sum}\limits_{i = k/2}^\infty {N_i(t)} \left( {\begin{array}{*{20}{c}} {2i} \\ k \end{array}} \right)\frac{1}{{2^{2i}}}$$

It is more convenient to work with the cell density ρ instead of cell number N. Normalizing the above equation, we get for the density ρk of cells with k ecDNA copies:

$$\frac{{\partial \rho _k(t)}}{{\partial t}} = – 2\rho _k\left( t \right) + 2\mathop {\sum}\limits_{i = k/2}^\infty {\rho _i(t)} \left( {\begin{array}{*{20}{c}} {2i} \\ k \end{array}} \right)\frac{1}{{2^{2i}}}$$

### Moment dynamics for neutral ecDNA copies

With the above equation for the density of cells with k ecDNA copies, we can calculate the moments of the underlying probability density function. In general, the lth moment can by calculated via:

$$M^{\left( l \right)}\left( t \right) = \mathop {\sum}\limits_{i = 0}^\infty {i^l} \rho _i(t)$$

It can be shown that all moments scale with $$M^{\left( l \right)}(t)\approx t^{l – 1}$$ and we found explicitly for the first two moments:

$$M^{(1)} = 1\,{{{\mathrm{and}}}}\,M^{\left( 2 \right)}\left( t \right) = t.$$

The mean ecDNA copy number in an exponentially growing population is constant for neutral ecDNA copies. The variance of the ecDNA copy number increases linearly in time.

### Stochastic dynamics of ecDNA under positive selection

The above equations can be generalized to accommodate positive selection (s > 1) for ecDNA copies. The set of dynamic equations for cell densities becomes:

$$\left. {\frac{{\partial \rho _k\left( t \right)}}{{\partial t}}} \right|_{k > 0} = s\left. {\frac{{\partial \rho _k\left( t \right)}}{{\partial t}}} \right|_{s = 1} + \left( {s – 1} \right)\rho _k\rho _0$$

$$\frac{{\partial \rho _0(t)}}{{\partial t}} = s\left. {\frac{{\partial \rho _k(t)}}{{\partial t}}} \right|_{s = 1} + (s – 1)(1 – \rho _0)\rho _0$$

A general solution to these equations is challenging. Nonetheless, important quantities, for example, the moment dynamics and scaling behavior can be calculated explicitly.

### Moment dynamics for ecDNA under positive selection

A generalized equation for the dynamics of moments directly follows from the above equations. We have:

$$\frac{{\partial M^{\left( l \right)}(t)}}{{\partial t}} = s\left. {\frac{{\partial M^{\left( l \right)}(t)}}{{\partial t}}} \right|_{s = 1} + \left( {s – 1} \right)\rho _0M^{\left( l \right)}(t)$$

This implies for the first moment $$\frac{{\partial M^{\left( 1 \right)}(t)}}{{\partial t}} = (s – 1)\rho _0M^{\left( 1 \right)}(t)$$, which then can be solved for the first moment:

$$M^{\left( 1 \right)}\left( t \right) = {\mathrm{e}}^{(s – 1){\int}_0^t {\mathrm{d}\tau \rho _0(\tau )} }$$

Similarly, the dynamic equation for the second moment becomes $$\frac{{\partial M^{\left( 2 \right)}(t)}}{{\partial t}} = M^{\left( 1 \right)}\left( t \right) + (s – 1)\rho _0M^{\left( 2 \right)}(t)$$ and we find

$$M^{\left( 2 \right)}\left( t \right) = tM^{\left( 1 \right)}(t)$$

Initially, the first moment increases exponentially. However, with increasing mean copy number, the rate of cells transitioning into a state without ecDNA is decreasing and the increase of the mean ecDNA copy number slowly levels off. Note, for s = 1 we recovered the previous results for the moments of neutral ecDNA amplifications.

### Genome editing using CRISPR–Cas9 ribonucleoprotein

Genome editing in COLO320-DM cells was performed using Alt-R S.p. Cas9 Nuclease V3 (catalog no. 1081058; IDT) complexed with sgRNA (Synthego) according to the Synthego RNP transfection protocol using the Neon Transfection System (catalog no. MPK5000; Thermo Fisher Scientific). Briefly, 10 pmol Cas9 protein and 60 pmol sgRNA for each 10 μl reaction were incubated in Neon Buffer R for 10 min at room temperature. Cells were washed with 1× PBS, resuspended in Buffer R and 200,000 cells were mixed with, for the preincubated RNP complex, for each 10-μl reaction. The cell mixture was electroporated according to the manufacturer’s protocol using the following settings: 1,700 V, 20 ms, 1 pulse. Cells were cultured for 10 d afterwards; cell counts and ecDNA copy number data were collected at days 3, 6 and 10. To estimate the ecDNA copy numbers, we performed metaphase chromosome spreading followed by FISH as described above. All sgRNA sequences are in Supplementary Table 3.

### FISH probes

The following probes were used for FISH as indicated: ZytoLight SPEC CDK4/CEN 12 Dual Color Probe (ZytoVision); ZytoLight SPEC MYCN/2q11 Dual Color Probe (ZytoVision); Empire Genomics EGFR FISH Probe; Empire Genomics MYC FISH Probe; Empire Genomics FGFR2 FISH Probe; Empire Genomics CDK4 FISH Probe; Empire Genomics MYCN FISH Probe.

### Antibodies

The following antibodies were used at concentrations of 1:100–1:200 for immunofluorescence and 1:1,000 for immunoblotting (unless otherwise indicated in specific Methods sections): Aurora B Polyclonal Antibody (catalog no. A300-431A; Thermo Fisher Scientific); EGFRvIII monoclonal antibody 806 (ref. 36); anti-mouse Alexa Fluor 488.

### Statistics and reproducibility

Sample sizes for the biological experiments analyzing copy number distributions were informed by stochastic simulations. Investigators were not blinded to experimental groups.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

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