Abstract
Single molecule FRET (smFRET) is a useful tool for studying biomolecular sub-populations and their dynamics. Advanced smFRET-based techniques often track multiple parameters simultaneously, increasing the information content of the measurement. Photon-by-photon hidden Markov modelling (H2MM) is a smFRET analysis tool that quantifies FRET dynamics of single biomolecules, even if they occur in sub-milliseconds. However, sub-populations can be characterized by additional experimentally-derived parameters other than the FRET efficiency. We introduce multi-parameter H2MM (mpH2MM) that identifies sub-populations and their transition dynamics based on multiple experimentally-derived parameters, simultaneously. We show the use of this tool in deciphering the number of underlying sub-populations, their mean characteristics and the rate constants of their transitions for a DNA hairpin exhibiting milliseconds FRET dynamics, and for the RNA polymerase promoter open complex exhibiting sub-millisecond FRET dynamics of the transcription bubble. Overall, we show that using mpH2MM facilitates the identification and quantification of biomolecular sub-populations in smFRET measurements that are otherwise difficult to identify. Finally we provide the means to use mpH2MM in analyzing FRET dynamics in advanced multi-color smFRET-based measurements.
1 Introduction
The role of structural dynamics in biomolecular function has come to the forefront of biophysical research[1, 2]. Biomolecules in solution exhibit structural dynamics at a hierarchy of timescales and modes, from bond rotations to movements of entire globular domains, occurring at times from picoseconds to seconds and slower[3]. In many cases, the stages in the functions that biomolecules carry are promoted by different sub-populations of closely-related structures, better known as conformations. This is indeed the case in the coupling of catalysis with substrate binding domain dynamics in some enzymes[4, 5], the dynamics of the DNA bubble in transcription initiation to support transcription start site selection[6, 7], involved in DNA mismatch repair[8], to facilitate protein translocation across membranes[9], assisting chaperone action[10], involved in the allosteric regulation of the AAA+ disaggregase[11], influences the selectivity in ABC im-porters[12], and many other important bio-processes, in which structural dynamics is coupled to biological function or influences it[1, 2]. Thus, methods capable of identifying and characterizing distinctly time-separated structural sub-populations of biomolecules are of great interest in biomolecular sciences and in dynamic structural biology.
NMR- and EPR-based methods[13–16] and single-molecule methods[17–21] have come to the forefront, each with their own advantages and disadvantages. Singlemolecule methods allow measuring one biomolecule at a time and tracking different experimental parameters simultaneously. One of the most popular single-molecule approaches is single-molecule FRET (smFRET)[22], where the bio-molecule of interest is site-specifically labeled at two strategic points with two fluorescent dyes, which can exhibit transfer of the excitation energy from the donor dye to the acceptor dye with a probability (or efficiency; E) that is inversely proportional to the sixth power of the distance between them according to the phenomenon of Fluorescence-(or Förster) resonance energy transfer (FRET)[23–25]. In the absence of factors correcting for differences in donor and acceptor fluorescence quantum yields and detection efficiencies (better known as the γ factor), acceptor direct excitation, and leakage of donor photons into the acceptor channel, the raw ratio of acceptor photons to donor and acceptor photons, better known as the proximity ratio (PR), serves as the apparent or uncorrected FRET efficiency[26].
smFRET has proven a powerful tool to disentangle sub-populations of biomacromolecules undergoing dynamic transitions over a range of time scales, but remains limited by the time resolution of the apparatus, which in the case of freely diffusing molecules, is often on the order of a few milliseconds, and down to the sub-milliseconds using advanced analyses of photon statis-tics within single-molecule photon bursts, summarized in recent reviews of the field[1, 2]. Photon-by-photon hidden Markov modelling (H2MM)[27] can extract the number of sub-populations involved in the dynamics, their proximity ratio values and transition rate constants. Nevertheless, while advanced smFRET setups often detect multiple parameters beyond the simple proximity ratio, such as in alternating-laser excitation (ALEX; also known as pulsed-interleaved excitation, PIE)[28, 29] or in multi-color smFRET-based measurements[30–37], H2MM in its current iteration only reports on the proximity ratio of a single donor-acceptor pair of dyes.
Here,we introduce multi-parameter H2MM (mpH2MM), which enables incorporation of multiple parameters, through additional photon streams in nanosecond ALEX (nsALEX)(a.k.a. PIE)[28, 29] experiments. We apply this approach to 1) a DNA hairpin exhibiting well-defined two-state opening and closing dynamics[38], and 2) with the transcription bubble in the RNA polymerase (RNAP) promoter open (RPo) complex, exhibiting sub-millisecond dynamics (figure 1)[6, 7]. We show that mpH2MM is able to extract sub-populations based on both the nsALEX-relevant mean parameters, PR and the stoichiometry, SPR, as well as their transition rate constants, demonstrating FRET-relevant conformational transitions, as well as FRET-irrelevant photo-physical transitions.
conformational changes, along with mpH2MM derived PR, and SPR values and the accompanying transition rates.
2 Results
2.1 DNA hairpin exhibiting millisecond dynamics
Tsukanov et. al. introduced a DNA hairpin with a loop containing 31 adenines and a six base-pair stem[38]. The opening and closing rate constants of the hairpin varied as a function of the GC content of the stem as well as the sodium chloride concentration. When appropriately labeled with a FRET donor and acceptor pair (ATTO 550 and ATTO 647N, respectively), the open and closed hairpin sub-populations resulted in low and high mean PR values, respectively. The hairpin containing two GCs out of the six stem bases, exhibited opening and closing rates of a few milliseconds. Taken together with the knowledge of the opening and closing transition rates at each sodium chloride concentration, this DNA construct serves as an ideal model system to test the performance of H2MM.
We perform two-color nsALEX (2c-nsALEX) experiments[29] with this construct at a concentration of 300 mM sodium chloride, in which the open and closed hairpin sub-populations have been shown as equally populated[38]. The burst variance analysis (BVA)[39] plot shows clear evidence of within-burst dynamics occurring while the molecule crosses the confocal volume (figure 2a). However, it is not necessarily clear how many distinct sub-populations are actually involved in the within-burst dynamics leading to the outcome shown in the BVA plot. In visual examination of the 2D E-S plot, three sub-populations are apparent: 1) an open hairpin sub-population with mean PR ~0.2 and 2) a closed hairpin sub-population, with a PR ~0.65, both of which have mean SPR values ~0.5, and 3) a third sub-population with a mean PR ~0, and mean SPR ~1, where the acceptor is either in a dark state, possibly a triplet state (i.e., dark acceptor sub-population), or missing altogether (Figure 2b). The 2D E-S plot also exhibits bursts with intermediate PR values bridging between the open and closed hairpin sub-populations, and corresponding to bursts where the hairpin is undergoing opening and closing transitions while crossing the confocal volume.
a) Burst variance analysis (BVA), the standard deviation of PR values of bursts is displayed versus their PR values. Bursts with standard deviations higher than expected solely from shot noise (semicircle), are ones that include dynamic heterogeneity, such as within-burst FRET dynamics. Triangles indicate the average of standard deviation values per PR bin. b) 2D histogram of PR and SPR (E-S plots, colloquially) of bursts. The PR and SPR values of sub-populations derived from mpH2MM are marked by red circles, and the standard deviation of these values, derived from the Viterbi dwell time analysis, are marked by black crosses. c) Comparison of values of the integrated complete likelihood (ICL) of spH2MM (top panel) and mpH2MM (bottom panel) of optimized models with different state-models. The ideal state-model is marked in red. d) A sample burst trajectory, with photons represented as colored vertical bars, with donor excitation photons colored green or red for donor and acceptor, respectively, and acceptor excitation photons colored purple. PR (top panel) and SPR (bottom panel) of sub-populations determined from dwells using the Viterbi algorithm, are overlayed on the photon bars. e,f) E-S scatter plots of data processed by the Viterbi algorithm. mpH2MM sub-populations and viterbi-derived standard deviations are overlayed as red circles and black crosses, respectively. e,f) E-S scatter plot of bursts (e) or dwells within bursts (f), color coded by which states are present in the bursts (e) or according to the state of the dwell (f), according to Viterbi algorithm.
Analysis with spH2MM and mpH2MM can be performed using any given state model. However, we first must select the ideal state model from several ones, where the models differ in the number of states present, and hence also the number of mean PR values and number of transition rate constants. Discriminating over- and under-fit state models from the ideal model has been difficult using H2MM[7, 40]. Lerner et al. have previously introduced the modified Bayes information criterion (BIC’), that does not provide an extremum-based decision on the most likely state-model[7]. We now implement the integrated complete likelihood (ICL)[41, 42], which gets a minimum value for the most-likely state-model. While it should not be viewed as an absolute discriminator, using smFRET simulated data, where the ground truth is known, we find that the ICL consistently chooses the correct number of states (see jupyter notebooks in supplementary dataset[43]). Based on the ICL, we find that for spH2MM analysis of the DNA hairpin data, the two-state model is the state-model that best explains the donor-excitation data. However, after using mpH2MM, it turns out that the four-state model is the state-model that best explains all of the data, including the acceptor-excitation data.
Visual inspection of the one-dimensional projection of burst data onto the PR parameter immediately suggests an explanation for this discrepancy, as it appears as only two sub-populations. The dark acceptor sub-population and the open hairpin FRET sub-population exhibit similar low PR values, and are hard to distinguish. This projection reflects the data accessible to spH2MM, the donor excitation streams, and thus the open hairpin and dark acceptor subpopulations are expected to have nearly identical signatures with regard to the streams accessible to spH2MM, thus leading to the false inference of only two states. The open hairpin FRET sub-population and the dark-acceptor subpopulation are, however, very much distinct with regard to the acceptor excitation stream, which is accessible to mpH2MM. In the ICL-selected four-state model retrieved by mpH2MM, two out of the four sub-populations match nicely with the sub-populations in the ICL-selected spH2MM model, having similar PR values. The SPR values are ~0.5 (figure 2b red circles), as expected for molecules undergoing FRET. The third and fourth sub-populations in the model can be matched to dark acceptor and dark donor sub-populaions, respectively. The third state has a PR value ~0 and a SPR value ~1 (figure 2b, top left red circle). This state has a clear sub-population of bursts associated with it in the 2D E-S plot. The fourth sub-population has an intermediate PR value, and a very low SPR value of 0.17 (figure 2b, bottom red circle). There is no obvious subpopulation visually observed in the E-S plots to which this would correspond, but the PR and SPR values are consistent with this dark donor sub-population. More importantly, comparing the parameters of the state models retrieved by spH2MM and mpH2MM, we find that the transition rate constants derived using mpH2MM are closer to those found by Tsukanov et. al.[38] than those extracted using spH2MM (table S2, and supplementary csv files of all state-models found by H2MM analysis[43]). The transition rate constants provide a clue as to why the fourth sub-population does not show up in the E-S plots, as they predict rare transitions to it, and fast transitions away. Thus, populating the fourth sub-population occurs only briefly and rarely in bursts undergoing rapid dynamics, such that it does not appear as a clear sub-population in the E-S plots (table S2, and supplementary csv file[43]).
The Viterbi algorithm finds the most-likely state path through each burst, given a state-model and its parameter values (figure 2d). We use this to classify bursts by which sub-populations are present within each burst (figure 2e), and separate photons into dwells, for which PR and SPR can be defined (figure 2f). Visual examination of this plot shows that the Viterbi algorithm reasonably classifies most bursts that have PR and SPR values close to the predicted value of a given sub-population as only having that sub-population present, as well as bursts with intermediate PR and SPR that are predicted to include dwells of multiple sub-populations. Notably, there are only a few bursts classified as having dwells solely in the dark donor sub-population (figure 2e), keeping with what is predicted by the transition rates, and indeed, few dwells are even found in this sub-population (figure 2f). In his study of photo-bleaching pathways in smFRET experiments, Kong et. al.[44] found that transitions of the donor to dark states are rare, while such transitions of the acceptor are frequent. The scarcity of the donor dark state in the Viterbi analysis serves to both confirm this observation, and prove the sensitivity of mpH2MM at the same time.
2.2 RNAP-promoter open complex
Using two-color microsecond ALEX experiments, Robb et. al. has shown that the transcription bubble in the RNA polymerase (RNAP)-promoter open complex (RPo) undergoes sub-millisecond FRET dynamics[6]. Later, using several double labeled RPo constructs, Lerner et. al. used spH2MM to show that the transcription bubble exhibits two FRET sub-populations associated with two transcription initiation complex forms, exhibiting sub-millisecond transitions, as well as free DNA and dark acceptor sub-populations[7]. We analyze 2c-nsALEX data using mpH2MM on one of the constructs previously used by Lerner et. al., where an ATTO 647N acceptor dye labels the base at register (−8) in the template strand, (−8)TA, and a Cy3B donor dye labels the base at register (−6) in the non-template strand, (−6)NTD, in a lacCONS promoter[45, 46]. Accordingly, we term this construct (−8)TA-(−6)NTD. The results of the analysis show that the four-state model minimizes the ICL (figure 3c). Based on the mean PR and SPR values, the sub-populations resolved by mpH2MM are similar to the ones previously observed by Lerner et al.[7]: 1) a dark acceptor sub-population, 2) a high mean PR free DNA sub-population, 3) an RNAP-bound open-bubble sub-population with intermediate mean PR (RPo), and 4) the previously-observed RNAP-bound scrunched-bubble sub-population with a lower mean PR(scrunched RPo) (figure 3b). These results compare quite well with the ones previously observed by Lerner et al., but with the added benefit that the SPR values are also recovered. The SPR values are particularly interesting, as the RNAP-bound open-bubble sub-population exhibits an anomalously low SPR, relative to that of the scrunched-bubble sub-population (figure 3e,f orange points). The half-life of the RNAP-bound open-bubble sub-population is shorter than the burst duration, and thus only a few transcription initiation complexes are expected to traverse the confocal volume while being solely in this sub-population. This is born out by the Viterbi analysis (figure 3e), and thus the anomalous SPR would not be expected to be obvious from the E-S plot (figure 3b). Because the SPR values are sensitive to the ratio between the donor and acceptor fluorescence quantum yields, this result might show that not only does the distance between the dyes change between open- and scrunched-bubble conformations, but also does the dyes’ immediate environment, influencing the fluorescence quantum yields of one of the dyes. In that context, the substantial decrease in the SPR value of the open-bubble sub-population relative to the value of the scrunched-bubble sub-population points to a decrease in the donor fluorescence quantum yield or an increase in the acceptor fluorescence quantum yield, or both. Whether it is the donor or the acceptor that change their fluorescence quantum yield can be assessed by inspecting whether the acceptor fluorescence decay after acceptor direct excitation has changed or not when comparing between the sub-populations. Visual examination of the fluorescence decays of the open bubble RPo and scrunched bubble RPo sub-populations, indicate that the anomalous SPR is certainly not due to an increase in the acceptor fluorescence quantum yield, as the normalized acceptor fluorescence decays after acceptor excitation of the two sub-populations are nearly identical, while there is a small difference in the donor decays (figure S13). Therefore, the anomalously lower SPR value of the open bubble RPo sub-population is due to a decreased fluorescence quantum yield of the donor labeling register (−6) of the nontemplate strand at the lacCONS promoter. This additional piece of information can be used when modeling the structure of RPo, since the most common reasons for the change in fluorescence quantum yield are the rate of dynamic quenching of fluorescence by the solvent, and hence solvent exposure, and fluorescence quenching or enhancement by specific groups within RPo.
a) Burst variance analysis (BVA), the standard deviation of PR values of bursts is displayed versus their PR values. Bursts with standard deviations higher than expected solely from shot noise (semicircle), are ones that include dynamic heterogeneity, such as within-burst FRET dynamics. Triangles indicate the average of standard deviation values per PR bin. b) 2D histogram of PR and SPR (E-S plots, colloquially) of bursts. The PR and SPR values of sub-populations derived from mpH2MM are marked by red circles, and the standard deviation of these values, derived from the Viterbi dwell time analysis, are marked by black crosses. c) Comparison of values of the integrated complete likelihood (ICL) of spH2MM (top panel) and mpH2MM (bottom panel) of optimized models with different state-models. The ideal state-model is marked in red. d) A sample burst trajectory, with photons represented as colored vertical bars, with donor excitation photons colored green or red for donor and acceptor, respectively, and acceptor excitation photons colored purple. PR (top panel) and SPR (bottom panel) of sub-populations determined from dwells using the Viterbi algorithm, are over-layed on the photon bars. e,f) E-S scatter plots of data processed by the Viterbi algorithm. mpH2MM sub-populations and viterbi-derived standard deviations are overlayed as red circles and black crosses, respectively. e) E-S scatter plot of bursts (e) or dwells within bursts (f), color coded by which states are present in the bursts (e) or according to the state of the dwell (f), according to Viterbi algorithm.
3 Conclusions
Multi-parameter H2MM increases both the information content of the results, and the sensitivity of the H2MM algorithm to differences in the photon steams that are too subtle when examining only as single parameter. mpH2MM is able to disentangle dark acceptor molecules from the very low FRET open hairpin conformation, and detect the low in SPR of the RPo open bubble conformation. With the added analysis of the fluorescent lifetimes, we determine that this is due to a decrease in donor fluorescence quantum yield, showing that the donor has a unique interaction with the protein in this conformation. This finding itself serves as additional information on the immediate microenvironment of the donor dye within the RPo open bubble conformation, which can then be used in modelling the underlying structure.
As ALEX or PIE setups are now ubiquitous, the acceptor excitation stream is almost always available, and ignoring it is essentially throwing away useful data. mpH2MM therefore maximizes the use of available data for characterizing rapidly interconverting sub-populations.
We have demonstrated mpH2MM with 2c-nsALEX measurements, but it is by no means restricted to these setups. The most obvious application of mpH2MM beyond 2c-nsALEX, is with the multiple photon streams in multi-color smFRET-based measurements[30–37]. Here, three or even four spectrally-distinct dyes are attached to the biomolecule of interest, and each produces a distinct photon stream. This enables the simultaneous observation of multiple inter-dye proximity ratios at once. If qualitative tests indicate that such a system is undergoing within-burst dynamics, mpH2MM is well-suited to extract the transfer efficiencies relevant to the underlying dynamically-interconverting sub-populations. Applying these methods is as simple as assigning an index to each photon stream. The transition probability matrix will be unaffected, while the emission probability matrix will expand with more photon streams. Conversion of the emission probability matrix into useful PRi,j values can be achieved by taking the values of the corresponding streams, and treating them as intensities. The supplied Jupyter notebooks provide examples for how to execute mpH2MM using FRETbursts. Experimenters using other platforms must utilize their knowledge of the peculiarities of their data to properly filter and cast their data into the simple and general format that the H2MM_C package[47] accepts.
4 Methods
4.1 Experimental setup
We performed the 2c-nsALEX smFRET measurements of the doubly-labeled DNA hairpin construct[38] in the presence of 300 mM sodium chloride, using a confocal-based setup (ISS™, USA) assembled on top of an Olympus IX73 inverted microscope stand. We used a pulsed picosecond fiber laser (λ=532 nm, pulse width of 100 ps FWHM, operating at 20 MHz repetition rate and 100 μW measured at the back aperture of the objective lens) for exciting the Cy3B donor dye (FL-532-PICO, CNI, China), and a pulsed picosecond diode laser (λ=642 nm, pulse width of 100 ps FWHM, operating at 20 MHz repetition rate and 60 μW measured at the back aperture of the objective lens) for exciting the ATTO 647N acceptor dye (QuixX® 642-140 PS, Omicron, GmbH), delayed by 25 ns. The laser beams pass through a polarization-maintaining optical fiber and then further shaped by a linear polarizer and a halfwave plate. A dichroic beam splitter with high reflectivity at 532 and 640 nm (ZT532/640rpc, Chroma, USA) reflect the light through the optical path to a high numerical aperture (NA) super apochromatic objective (60X, NA=1.2, water immersion, Olympus, Japan), which focuses the light onto a small confocal volume. The microscope collects the fluorescence from the excited molecules through the same objective, and focuses it with an achromatic lens (f = 100 mm) onto a 100 μm diameter pinhole (variable pinhole, motorized, tunable from 20 μm to 1 mm), and then re-collimates it with an achromatic lens (f = 100 mm). Then, donor and acceptor fluorescence are split between two detection channels using a dichroic mirror with a cutoff wavelength at λ=652 nm (FF652-Di01-25×36, Semrock Rochester NY, USA). We further filter the donor and acceptor fluorescence from other light sources 585/40 nm (FF01-585/40-25, Semrock Rochester NY, USA) and 698/70 nm (FF01-698/70-25, Semrock Rochester NY, USA) band-pass filters, respectively, and detect the donor and acceptor fluorescence signals using two hybrid photomultipliers (Model R10467U-40, Hamamatsu, Japan), routed through a 4-to-1 router to a time-correlated single photon counting (TCSPC) module (SPC-150, Becker & Hickl, GmbH) as its START signal (the STOP signal is routed from the laser controller). We perform data acquisition using the VistaVision software (version 4.2.095, 64-bit, ISS™, USA) in the time-tagged time-resolved (TTTR) file format. After acquiring the data, we transform it into the photon HDF5 file format[48] for easy dissemination of raw data to the public, and easy input in the FRETbursts analysis software. The 2c-nsALEX measurements of the (−8)TA-(−6)NTD lacCONS promoter construct, in the absence and presence of the E. coli RNAP holoenzyme, was performed using the setup in Ingargiola et al.[49].
4.2 Data Processing
All data processing and analysis is performed using Jupyter Notebooks available in supplementary dataset, along with the accompanying photon-HDF5 files containing the raw data[43]. We perform burst search and selection using the FRETbursts analysis software[50]. Background is assessed per each 30 seconds of acquisition, and bursts are identified as time periods were the instantaneous photon count rate of a sliding window of m = 10 consecutive photons is at least F = 6 times higher than the background rate. Bursts in the normal selection are selected if they include at least 30 photons in total between all streams. Visualizations are performed using FRETbursts’ dplot function. Bursts identified by FRETbursts are then converted into a format readable by the H2MM_C software[47], by a simple function supplied in the Jupyter notebooks available in supplementary dataset[43]. In spH2MM, only photons arising from donor excitation are considered, assigned to either donor or acceptor streams, identified by index 0 or 1 respectively, depending on at which detector they arrived. mpH2MM also considers photons arriving during acceptor excitation, assigning these photons an index of 2. All H2MM calculations were performed within the Jupyter notebooks, available in supplmentary dataset[43], using the Python package by Paul David Harris[47]. We use the H2MM algorithm (both single- and multi-parameter) to test how well 2, 3 or 4 state-models describe the photon data. To choose the best model, we use the ICL[41, 42], where the state-model reaching a minimal ICL is generally considered the one that describes the data best, with minimal free parameters. If the ICL is minimized for the 4 state model, a 5 state model is also tested to confirm minimization of the ICL. The ICL parameter is defined in Eq. 1:
where
is the posterior probability of the most likely state path, as determined by the Viterbi algorithm, K is the number of free parameters in the model, and n is the number of photons in all bursts in the data set. K is calculated as in Eq. 2:
where q is the number of states the state-model represents, and r is the number of photon streams used for the calculation of all of the parameters that are assessed. For spH2MM, r = 2, while for 2c-nsALEX mpH2MM, r = 3. The ICL is preferable as an extremum-based criterion over the previously proposed threshold based on the modified Bayes Information Criterion (BIC’)[7]. See supplementary dataset[43] for Jupyter notebooks testing the reliability of ICL with simulated data sets generated using PyBroMo[51] (https://github.com/OpenSMFS/PyBroMo/releases/tag/0.8.1; was utilized in previous works[7, 48, 52]). We optimize the H2MM model with state-models having successively greater numbers of states, until the ICL ceases to improve, and always optimizing two-, three-, and four-state models. We use the Viterbi algorithm to find the most likely state path based on the posterior probability. From the state path, photons are separated into dwells, each of which can be assigned a duration, a mean PR, and for mpH2MM, a mean SPR. Dwells with fewer than 5 photons are discarded. This also allows bursts to by classified by which and how many states are present. We use the weighted standard error of the PR and SPR as a proxy for the standard error of the H2MM model (see SI for full derivation). The fluorescent decays of sub-populations are also analyzed based on sorting photons by the Viterbi algorithm, see SI for further details, and a proposed method for mapping the non-binomially distributed lifetime to a binomially-distributed parameter ammenable to mpH2MM.
5 Author Contributions
E.L. performed 2c-ALEX measurements. The measurements of the RNA polymerase-promoter open complex were performed by E.L. in the laboratory of S.W. P.D.H. analyzed data and contributed analytical tools. P.D.H. & E.L. designed and performed the research and wrote the manuscript.
6 Acknowledgements
This paper was supported by the National Institutes of Health (NIH, grant R01 GM130942 to S.W. and E.L. as a subaward), the National Science Foundation (NSF, grants 1818147 and 1842951 to S.W.), the Human Frontiers Science Program (HFSP, grant RGP0061/2019 to S.W.), the Israel Science Foundation (ISF, grant 3565/20 to E.L., within the KillCorona – Curbing Coronavirus Research Program), the Milner Fund (to E.L.), and the Hebrew University of Jerusalem (start-up funds to E.L.). We would also like to acknowledge the great help of Bill Harris in making the H2MM_C package cross platform.