Supplementary Information

CBP] Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287, USA  MGH] Massachusetts General Hospital, Boston, MA 02114 (Current Address)     [UCSF] Department of Cellular and Molecular Pharmacology, University of California, San Francisco, CA 94158  UTK] Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA  CBP] Center for Biological Physics, Department of Physics, Arizona State University, Tempe, AZ 85287, USA     [SMS]School of Molecular Sciences, Arizona State University, Tempe, AZ 85287, USA

S1. Verification of Sampler Using Synthetic Data

The figure below shows a bivariate density plot for rates and excitation probabilities learned using the sampler described in the main manuscript for synthetically generated data. The six rates that were used are 0.01, 0.1, 0.5, 2.0, 4.0, and 10.0 ns−1, and the corresponding excitation probabilities are 0.2, 0.3, 0.1, 0.2, 0.1, and 0.1. These ground truth values are shown as orange dots in the plot. Our Markov-Chain Monte-Carlo sampler accurately predicts most of these rates and probabilities, however, rates that are not well separated in magnitude have much larger spread in their distributions. For instance, the learned distribution around the rates corresponding to 2.0 and 4.0 ns−1 is much wider in the plot below.

Robustness Test. Bivariate density plot for a system with six photophysical states with escape rates 0.01, 0.1, 0.5, 2.0, 4.0, and 10.0 ns−1, and the corresponding excitation probabilities are 0.2, 0.3, 0.1, 0.2, 0.1, and 0.1. The ground truth is shown as dots. 


S2. Results Using Metropolis-Hastings Algorithm

Experimental Data Analysis Using Metropolis-Hastings. In row (a), we have three 256×256 pixels raster-scanned images of samples containing three cubes labeled with ATTO 647N dyes, each. The images on the left and the right are for single-dye cubes and fluorocubes illuminated with 80% maximum laser power, respectively. However, the center image for large-cubes was produced using only 30% laser power. The laser power is varied in order to obtain approximately equal photon numbers per probe per experimental run. Histograms for photon arrival times (microtimes) recorded from these images are shown in row (b). In row (c), each panel shows a bivariate posterior for escape rates λi (log scale) and corresponding probabilities πi as in Eq. 5 of the main manuscript. We smooth out the bivariate plots using the kernel density estimation (KDE) tool available in Julia. The left panel shows the distribution for single-dye cubes. The distribution is concentrated about λi≈0.25 ns−1 or a lifetime of around 1/λi=4.0 ns. πb≈0.10 and π0≈0.49 in this case. The plot for large cubes in the center panel shows no change in lifetime, suggesting insignificant interactions. πb≈0.08 and π0≈0.37 for these cubes. In the rightmost panel for fluorocubes, the peak moves towards a significantly shorter lifetime of around 1.54 ns. The background probability πb here is 0.12, slightly higher than that of non-interacting cubes, and π0≈0.59. 

]


S3. Results For Cy3 Fluorocubes

Experimental Data Analysis. Here, we show raster scanned images (a), microtime histograms (b), and learned posterior distributions (c) for probes labeled with Cy3 dyes. Each panel in row (c) shows a bivariate posterior for escape rates λi (log scale) and corresponding probabilities, learned from a single experimental run with samples containing multiple fluorescent probes of the same type (fluorocubes or large cubes) in the field of view. In the left panel, we show the distribution for large cubes and illuminated with 85% of the maximum laser power. The distribution is concentrated about λi≈0.465 ns−1 or a lifetime of around 1/λi=2.15 ns. πb≈0.03 and π0≈0.09 in this case. In the second panel on the right, for fluorocubes labeled with six dyes illuminated with 85% laser power, the peak moves towards a significantly shorter lifetime of around 1.31 ns. The background probability is about 0.15 here and 27 % of the pulses do not lead to any photon detections. 


S4. Experimental Details


Table: Sequences for Assembly of DNA FluoroCubes and Large Cubes.

  Name	    Sequence and modification	  	Vendor
	FC_SC_01_Cy3	/5Cy3/ATGAGGTGTATGTGTAGAGTGATGGATGTAGT/3Cy3/	IDT
	FC_SC_02_Cy3	/5Cy3/AGGATGAGTGAGAGTGAGATGAGAGTAGATGT/3Cy3/	IDT
	FC_St_02_Cy3	/5Cy3/CACTCTCACACCTCATACATCTACCATCACTC/3Cy3/	IDT
	FC_SC_01_ATTO647N	/5ATTO647NN/ATGAGGTGTATGTGTAGAGTGATGGATGTAGT/3ATTO647NN/	IDT
	FC_SC_02_ATTO647N	/5ATTO647NN/AGGATGAGTGAGAGTGAGATGAGAGTAGATGT/3ATTO647NN/	IDT
	FC_St_02_ATTO647N	/5ATTO647NN/CACTCTCACACCTCATACATCTACCATCACTC/3ATTO647NN/	IDT
	SD_SC_01_UN	TTATGAGGTGTATGTGTAGAGTGATGGATGTAGTTT	IDT
	SD_SC_02_UN	TTAGGATGAGTGAGAGTGAGATGAGAGTAGATGTTT	IDT
	SD_St_02_ATT0647N	/5ATTO647NN/CACTCTCACACCTCATACATCTACCATCACTCTT	IDT
	SD_St_02_Cy3	/5Cy3/CACTCTCACACCTCATACATCTACCATCACTCTT	IDT
	FC_St_01_Bio	Biotin TACACATACTCATCCTACTACATCTCTCATCTTT	IDT
	LC_SC_01	TTGAAAATTATCTCGATAAGCAGAAGGACCTGTATAACTGGCAAGAGACAAGGCCGCTTCAGAA	IDT
	LC_SC_02	AGGATAGCCGGACCGTATTAATGCCGCGCCAACGGTTTCCCGGACCTAGTGTCTATCAAGTCTA	IDT
	LC_SC_03	TTCTATGAAACCATTCTCGGGTCGAGCGGGTCACTGTTGTGACCTACGAGAAGCGTATAGATGT	IDT
	LC_SC_04	TCCGCGCGAATAGCTCACAGGCGAACTACGTATGAATTGGTTTAAACGCTCCTCGGGAATTAAT	IDT
	LC_SC_05	ACGACAGGTGGCAAACCACCTCCGATGTCAGCGCCGCATACCCATTCACTGTGAATTTCCACAC	IDT
	LC_SC_06	CGAGGATTCGCAGGTCCATGGGATTCACCAAGCTCGTATACACCCTGATTCTCCATGGCAGCGC	IDT
	LC_SC_07	GTAAGTTGAAGTAGGAAGCTTTTTCTAGCCATAGCATCGACACTACGACCTGCTTTTCGACACA	IDT
	LC_SC_08	GGACTGCATTCTGGACAGTAACTGCATTAACTACGTGCTCCCAACATAAGTGACGTCCTCAGCA	IDT
	LC_St_01	TTTGCTGAGGTGGAAATTTT	IDT
	LC_St_02	TTCCGGGAAACCGTTGGCCCTTCTGCTCGCCTGTCGTAGGTCGGGT	IDT
	LC_St_03_ATTO647N	GGTTCTGCGAATCCTCGGTGACGTCACTTCCTACTTCAACTTACTT/ATTO647N	IDT
	LC_St_03_Cy3	GGTTCTGCGAATCCTCGGTGACGTCACTTCCTACTTCAACTTACTT/Cy3	IDT
	LC_St_04	GTATGCGGCGCTCAGTTACTTCGTAGTGTCGATGCTTT	IDT
	LC_St_05_Bio	/5Biosg/TTCCATGGACTCATAGAATT	IDT
	LC_St_06	TTGAGAATCAACAACAGTTT	IDT
	LC_St_07_ATTO647N	TTACATCTATCACTAGGTTT/ATTO647N	IDT
	LC_St_07_Cy3	TTACATCTATCACTAGGTTT/Cy3	IDT
	LC_St_08	TTTACGTAGTTTATCGAGTT	IDT
	LC_St_09_ATTO647N	ATTO647N/TTCCGGCTATCCTTTCTGAAGCGGCCCCGAGGAGGAAT	IDT
	LC_St_09_Cy3	Cy3/TTCCGGCTATCCTTTCTGAAGCGGCCCCGAGGAGGAAT	IDT
	LC_St_10	GTATACGAGCTTAAAAAGCTTATGTTGGGAGCACGTTT	IDT
	LC_St_11	TTTGCCAGTTATACAGGTGCGGCATTCGACCCGACGTTTAAAATGG	IDT
	LC_St_12	TTAGGTGGTTCGCGCGGATT	IDT
	LC_St_13	TTAGTTAATGGACATCGGTT	IDT
	LC_St_14_ATTO647N	ATTO647N/TTATAATTTTCAATAGACTTGATAGAACGCTTCTGAGCTATTTGCC	IDT
	LC_St_14_Cy3	Cy3/TTATAATTTTCAATAGACTTGATAGAACGCTTCTGAGCTATTTGCC	IDT
	LC_St_15	TTGACCCGCTAATACGGTTT	IDT
	LC_St_16_ATTO647N	ACCTGTCGTGCGAAAGCAGGGTCCAGAATGCAGTCCTT/ATTO647N	IDT
	LC_St_16_Cy3	ACCTGTCGTGCGAAAGCAGGGTCCAGAATGCAGTCCTT/Cy3	IDT
	LC_St_17	TTATGGCTAGGGTGAATCTT	IDT
	LC_St_18	TTCACAGTGACCAATTCATT	IDT
	LC_St_19	TTTGTGTCGACTGCCATGTT	IDT
	LC_St_20_ATTO647N	TTATTAATTCTTGTCTCTTT/ATTO647N	IDT
	LC_St_20_Cy3	TTATTAATTCTTGTCTCTTT/Cy3	IDT


Supplementary Information
Saurabh Ayush, Niekamp Stefan, Sgouralis Ioannis, Pressé Steve
Supplementary Information