Modeling Chromatin as a       Computer            Barbara Bryant, ConstellationPharmaceuticals, barbara.bryant@constellati...
Chromatin is a computer•    Protein complexes read and write     chemical symbols on chromatin.•    Chromatin modification...
Biological ChromatinDrew Berry animation
DNA wraps around histone octamers
Each nucleosome has many  possible modifications                from Allis et al, Epigenetics, 2006
Enzymes modify chromatin                  Kosi Gramatikoff                  via Wikimedia Commons
Readers and writers are joinedtogether in protein complexes                 Chromatin-modifying complex             histon...
Examples of chromatin-modifying          complexes from PINDB   Complex     Erasers (HDMs,   Writers (HMTs, DNMTs, HATs)  ...
Chromatin Computational Model                                                                  Chromatin                  ...
Chromatin Computer
Chromatin Computer                                                        XX** BB**  ---- XX--                           ...
Hamiltonian Path ProblemFind a path from vertex 0to vertex 6 visiting eachvertex exactly once.
DNA computer solution   EdgesAdleman 2004, Science
DNA computer solution                        hybridization   Edges                                PathsAdleman 2004, Science
DNA computer solution                        hybridization   Edges                                   Paths                ...
Chromatin Computer Solution000000 BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB                              Initial chromatin...
Building the path000000 BBBBBB   BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB0***** B*****------ 1-----000000 1BBBBB BBBBBB   BBBBBB...
Rules enforcing single visit to        each vertex## Check for one and only one visit to vertex 1*0**** 1B**** --> ------ ...
(Show visualization of running CC)Animated gif of this CC
Hamiltonian Path II: with                backtracking   000000 BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB IIIIII           ...
Turing Machine           B   y   x   B                   ^                   1
Turing Machine           B   y   x   B               ^               2
Turing Machine           B   z   x   B                   ^                   3
Turing Machine           B   z   z   B                       ^                       4
Chromatin computers are           computationally universalProof: simulation of a Turing Machine using a Chromatin Compute...
CC programs can handle input of any size              Rules for destructive-copy-and-add-one          BB*   H00   BB*    -...
1   1   1 H00          |H00|          |   |H111   1   1    0 H11
1   1   1   0 H11             |H11|          H1 |    |1   1   1 H10   1
1   1   1 H10    1         |H10|      H2 |   |1   1 H21   0    1
1   1 H21    0   1     |H21|     |H30|1   1 H30    0   1
1   1 H30    0   1      |H30|      |   |H31   1    0 H30   1
...
Almost done  0   0 H20     0   1   1   1   1       |H20|    H2 |    |  0 H20   0     0   1   1   1   1
Almost done  0 H20     0   0   1   1   1   1   |H20|H2 |    |H20   0     0   0   1   1   1   1
Done     H20     0   0   0   1   1   1   1   |H20|H2 |     |H2     0     0   0   0   1   1   1   1
Biological chromatin is a massively     parallel, random access, self-modifying           stored procedure computer•   Non...
Enhancerenhancer                            TSS           Inspirations:           •Okazaki fragments           •Active chr...
Inspirations: ACH, Okazaki• Show replication video from Drew Berry
Enhancerenhancer          TSS
Enhancerenhancer          TSS
Enhancer     TSS
Enhancer  TSS
EnhancerTSS
EnhancerTSS
Copying1111
Copying1111
Copying1111
Multiplying1111
Hamiltonian Path III: arbitrary         graph as input on chromatin tape                             Input: Initial state,...
Sketch of HamPath III• Set up the vertex-visited section of tape.• Add the next valid edge; check the vertex has not yet  ...
Program design: state transition graph
Hamiltonian Path III: graph as input     S         L                  G             V                             P       ...
What does biological chromatin        actually compute?• State, subroutines, variables, recursion?• Self-modifying program...
Silencing in S. cerevisiae• Silencing in Saccharomyces cerevisiae  heterochromatin domain has been well  studied, and a mo...
Proposed Nucleosome• H4K16(A/B) Core(I/B) Protein(s/C/B)    State(H/E)•   A: acetylation, B: no modification•   I: end of ...
Silencing steps• (Nucleation) Signal for start, series of protein  interactions and finally SIR complex binds to  silencer...
Initial chromatin & rules        ... - NNNN -ABBE -ABBE -ABBE -... - AIBE -IIII -... NNNN   ABBE   -->   NNNN   BBBE    # ...
Sample run with simulator
Can we build a computer from         molecular parts?• To build a biological chromatin  computer, we need to create the wr...
Making the tape+             Ligation of             ssDNA ends
Proof of concept for CC rule               engineeringFrom Fig. 1 of Haynes and Silver 2011, “Synthetic reversal of gene s...
Engineering of read/write rules•   Start with list of existing parts: existing reader, writer    and scaffolding/adaptor p...
CC software engineering• State, subroutines, variables, recursion• Self-modifying program (through changed  expression of ...
What we’ve done• Defined chromatin computing closely  modeled on biology• Built a simulator• Wrote CC programs• Proved CC ...
Related WorkProhaska, Stadler and Krakauer, “Innovation in gene regulation: The case ofchromatin computation” Journal of T...
Why it’s important to model   chromatin as a computer• Understand chromatin biology• Fix chromatin when it’s broken• Engin...
Thanks• David Yee           • Bob Tepper• Rich Ferrante       • Yang Shi• David Allis         • Adam Rudner• Keith Robison...
Extra slides
• Different modifications have different time scales.  Acetylation marks have a lifetime of  minutes, phosphorylation hour...
How much writable chromatin memory       is there in each human cell?        Size Units          Item3,000,000,00 base pai...
Size of program compared to     biological chromatin• Hamiltonian Path program allowing any  sized input graph  • Number o...
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
Chromatin Computing talk at ISMB 2012
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Chromatin Computing talk at ISMB 2012

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ISMB 2012 talk by Barbara Bryant, Li Chenhao and Greg Tucker-Kellogg, "Modeling chromatin as a computer"

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  • Edges in the graph were encoded by ssDNA in which the the two halves of DNA represented the two vertices of the edge with different sequences.When a vertex was a “TO” vertex in an edge, its sequence was the reverse complement of the sequence used in an edge where it was the “FROM” vertex.
  • By mixing these ssDNA edges, the vertex sequences found their reverse complement to string together longer dsDNA pieces representing paths through the graph.
  • In order to find the correct path, the DNA was run out on gels to find pieces that were the correct size (7 vertices), and that piece of the gel was cut out and further assayed to make sure that each vertex appeared.
  • How would we solve this Hamiltonian Path problem with a chromatin computer?We’ll do it by having rules encode edges.Each nucleosome represents one vertex in the path. The first position at each nucleosome indicates which vertex is at that point in the path; it is either blank or a digit from 0 to 6. The remaining 5 positions will be used to check whether vertices 1 through 5 are visited exactly once, and take one of the values {B, 0, 1, F}. A “0” indicates that the corresponding vertex has not been seen in the path so far; a “1” indicates that it has been seen once; and an “F” indicates that it has been seen more than once. The 7thnucleosome of a valid path should have the configuration 611111, because the path should end at vertex 6 and each of the vertices 1 through 5 should have be seen exactly once.
  • Application of several edge traversal rules is shown here, and the path is being built up.Of course, the wrong path might be followed; how do you make sure you visit every vertex once?
  • Adleman solved the problem of checking that every node had been visited by doing affinity purification to check for the presence of each vertex. We can perform this check directly in the CC program. These rules check that we have one and only one instance of vertex 1in the path, and propagate the necessary information from left to right along the nucleosomes representing the path. The “F” (for “Fail”) in the one rule below indicates that too many ones have been visited. The second position in each nucleosome holds the information of whether we have previously visited vertex number 1.Note: real readers are “degenerate” in that they can often recognize more than one mark.If we get to the F symbol, then this is not a good path. By starting with a chromatin tape having many copies of the graph, or many little pieces of chromatin, we can use parallelism to solve the problem.
  • Tape before application of the ruleTape after application of the rule.Read and write portions of the rule.
  • So, real chromatin programs can theoretically operate on inputs of any size. Now, Turing Machines in their original definition are awkward to program and take a long time to run programs. Real silicon computers that you and I program have a much richer instruction set. People have developed more efficient TMs with characteristics like multiple tapes, multiple r/w heads, stored programs, etc. What efficiencies does biological chromatin provide beyond what we’ve modeled so far?
  • Karmella Haynes and Pamela Silver at HarvardAs evidence that engineering of these complexes can be done successfully, researchers have reported using human polycomb chromatin protein and homologs from other species to construct modular synthetic transcription factors that recognize H3K27me3 and switch silenced genes on [47]. This designed complex re-expressed tumor suppressor p16 (CDKN2A) and other loci in U2OS osteosarcoma cells.Fuse CBX8 chromodomain, which recognizes H3K27me3 (normally a repressive mark) to a potent transcriptional activation domain, resulting in activation of genes normally silenced.Emphasizes that the mark is just a mark, not an effector!
  • Prohaska, Stadler and Krakauer, “Innovation in gene regulation: The case of chromatin computation” Journal of TheorBiol 2010 265(1):27-44. Talks about chromatin computation; similar formalism; doesn’t show TM completeness (computational universality) though does compare to more limited cis-regulatory modules (Boolean functions based on pattern of presence/absence of transcription factors). Authors are from Univ of Leipzig, Max Planck Institute, Fraunhofer Institute, University of Vienna, and Santa Fe InstituteRohlf: http://www.izbi.uni-leipzig.de/izbi/mitarbeiter/Binder/Rohlf.pdf also mentions chromatin computation.
  • Chromatin Computing talk at ISMB 2012

    1. 1. Modeling Chromatin as a Computer Barbara Bryant, ConstellationPharmaceuticals, barbara.bryant@constellationpharma .com; barb.bryant@gmail.com Li Chenhao, National University of Singapore Greg Tucker-Kellogg, National University of Singapore, dbsgtk@nus.edu.sg ISMB 2012
    2. 2. Chromatin is a computer• Protein complexes read and write chemical symbols on chromatin.• Chromatin modification is computationally universal.• Hard problems can be solved on biologically-sized chromatin computers.• It is useful to model chromatin as a computer. Reference: Barbara Bryant, "Chromatin Computation", PLoS ONE 2012 PMID 22567109
    3. 3. Biological ChromatinDrew Berry animation
    4. 4. DNA wraps around histone octamers
    5. 5. Each nucleosome has many possible modifications from Allis et al, Epigenetics, 2006
    6. 6. Enzymes modify chromatin Kosi Gramatikoff via Wikimedia Commons
    7. 7. Readers and writers are joinedtogether in protein complexes Chromatin-modifying complex histone tail scaffolding histone tail chromatinDNA reader reader protein writer remodeler
    8. 8. Examples of chromatin-modifying complexes from PINDB Complex Erasers (HDMs, Writers (HMTs, DNMTs, HATs) Readers (bromodomains, PHD, HDACs) PWWP, chromodomains, MBD) NUMAC CARM1 SMARCA4 GST-Smad2 CREBBP, NCOA3 CREBBP, SMARCA4, TRIM33 DNMT3B HDAC1 DNMT3B DNMT3B AF9.com HDAC1, HDAC2 DOT1L, TAF1, TAF5 CBX8, MLLT10, TAF1, TAF3 CtBP HDAC1, HDAC2, EHMT1, EHMT2 CBX4, CDYL KDM1A DMAP1 EPC1, ING3, SRCAP BRD8, ING3 PCAF KAT2B, SUPT3H, TADA2A, KAT2B TADA3, TAF10, TAF12, TAF5L, TAF6L, TAF9 ING5-TAP KAT6A, KAT6B, KAT7 BRD1, BRPF1, BRPF3, ING5, KAT6A, KAT6B, PHF15, PHF16, PHF17 MLL1-WDR5 KAT8, MLL, TAF1, TAF9 CHD8, MLL, PHF20, TAF1 HDAC2 HDAC1, HDAC2, CHD3, CHD4, PHF21A KDM1A, MTA2http://pin.mskcc.org/ Luc & Tempst, Bioinformatics 2004 20(9):1413 Next: the CC model
    9. 9. Chromatin Computational Model Chromatin modification sites Chromatin tape BBXX DNA histone tail scaffolding histone tail chromatin Chromatin-modifying reader reader protein writer remodeler complexes Read/write rules BB**  XX--
    10. 10. Chromatin Computer
    11. 11. Chromatin Computer XX** BB**  ---- XX-- BBXX XXXX BBXX BBBB XX** BB** ---- XX— BBXX XXXX XXXX BBBBSee Bryant 2012, PLoS ONE Next: an example CC program
    12. 12. Hamiltonian Path ProblemFind a path from vertex 0to vertex 6 visiting eachvertex exactly once.
    13. 13. DNA computer solution EdgesAdleman 2004, Science
    14. 14. DNA computer solution hybridization Edges PathsAdleman 2004, Science
    15. 15. DNA computer solution hybridization Edges Paths Sizing, probe hybridizationAdleman 1994, Science Correct path
    16. 16. Chromatin Computer Solution000000 BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB Initial chromatin state 2***** B*****  ------ 3----- Rule encoding traversal of edge from vertex #2 to vertex #3 (10,6,2)-CC 10 marks: {B,0,1,2,3,4,5,6,F,S} 6 modification sites in each nucleosome 2-nucleosome rules
    17. 17. Building the path000000 BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB0***** B*****------ 1-----000000 1BBBBB BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB 1***** B***** ------ 2-----000000 1BBBBB 2BBBBB BBBBBB BBBBBB BBBBBB BBBBBB 2***** B***** ------ 3-----000000 1BBBBB 2BBBBB 3BBBBB BBBBBB BBBBBB BBBBBB But, how do you make sure you visit every vertex once?
    18. 18. Rules enforcing single visit to each vertex## Check for one and only one visit to vertex 1*0**** 1B**** --> ------ -1----*0**** 2B**** --> ------ -0----*0**** 3B**** --> ------ -0----*0**** 4B**** --> ------ -0----*0**** 5B**** --> ------ -0----*0**** 6B**** --> ------ -0----*1**** 1B**** --> ------ -F----*1**** 2B**** --> ------ -1----*1**** 3B**** --> ------ -1----*1**** 4B**** --> ------ -1----*1**** 5B**** --> ------ -1----*1**** 6B**** --> ------ -1---- But, what if it chooses the wrong path and fails?
    19. 19. (Show visualization of running CC)Animated gif of this CC
    20. 20. Hamiltonian Path II: with backtracking 000000 BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB BBBBBB IIIIII Insulator ## If stuck at 6, erase 6***** BBBBBB --> X----- ------ ## Bounce back if hit right end X***** IIIIII --> H----- ------ ## If hit wall, turn around X***** BBBBBB --> H----- ------ 1***** IIIIII --> H----- ------ 2***** IIIIII --> H----- ------ ## Walk left to 0, erasing as you go 3***** IIIIII --> H----- ------ G***** H***** --> H----- BBBBBB 4***** IIIIII --> H----- ------ X***** H***** --> H----- BBBBBB 5***** IIIIII --> H----- ------ 1***** H***** --> H----- BBBBBB 2***** H***** --> H----- BBBBBB ## Walk failure right to the I wall. 3***** H***** --> H----- BBBBBB X***** 1***** --> G----- X----- 4***** H***** --> H----- BBBBBB X***** 2***** --> G----- X----- 5***** H***** --> H----- BBBBBB X***** 3***** --> G----- X----- 6***** H***** --> H----- BBBBBB X***** 4***** --> G----- X----- 0***** H***** --> 000000 BBBBBB X***** 5***** --> G----- X----- X***** 6***** --> G----- X-----• Click to run Ham Path backtracking animation
    21. 21. Turing Machine B y x B ^ 1
    22. 22. Turing Machine B y x B ^ 2
    23. 23. Turing Machine B z x B ^ 3
    24. 24. Turing Machine B z z B ^ 4
    25. 25. Chromatin computers are computationally universalProof: simulation of a Turing Machine using a Chromatin Computer See Bryant 2012, PLoS ONE
    26. 26. CC programs can handle input of any size Rules for destructive-copy-and-add-one BB* H00 BB* --> --- BB- H11 BB* H11 BB* --> H1- BB- --- BB* H10 BB* --> H2- BB- --- BB* H20 BB* --> H2- BB- --- BB* H21 BB* --> --- H30 --- BB* H30 BB* --> --- BB- H3- BB* H31 BB* --> --- H4- --- BB* H41 BB* --> --- BB- H4- BB* H4B BB* --> --- H11 --- Start: B 1 1 1 0 B B B B B End: B 0 0 0 0 1 1 1 1 B
    27. 27. 1 1 1 H00 |H00| | |H111 1 1 0 H11
    28. 28. 1 1 1 0 H11 |H11| H1 | |1 1 1 H10 1
    29. 29. 1 1 1 H10 1 |H10| H2 | |1 1 H21 0 1
    30. 30. 1 1 H21 0 1 |H21| |H30|1 1 H30 0 1
    31. 31. 1 1 H30 0 1 |H30| | |H31 1 0 H30 1
    32. 32. ...
    33. 33. Almost done 0 0 H20 0 1 1 1 1 |H20| H2 | | 0 H20 0 0 1 1 1 1
    34. 34. Almost done 0 H20 0 0 1 1 1 1 |H20|H2 | |H20 0 0 0 1 1 1 1
    35. 35. Done H20 0 0 0 1 1 1 1 |H20|H2 | |H2 0 0 0 0 1 1 1 1
    36. 36. Biological chromatin is a massively parallel, random access, self-modifying stored procedure computer• Non-deterministic / parallel computation• DNA methylation• DNA sequence readers (transcription factors)• Gene expression: program output, and self-modifying programs• Recognition of multiple marks by one reader• Different rate constants or affinities• Move left/right along the chromatin (remodeler)• Remove/replace histone (remodeler)• RNA processing systems operating on nascent RNA transcripts
    37. 37. Enhancerenhancer TSS Inspirations: •Okazaki fragments •Active chromatin hubs •Organizer factory
    38. 38. Inspirations: ACH, Okazaki• Show replication video from Drew Berry
    39. 39. Enhancerenhancer TSS
    40. 40. Enhancerenhancer TSS
    41. 41. Enhancer TSS
    42. 42. Enhancer TSS
    43. 43. EnhancerTSS
    44. 44. EnhancerTSS
    45. 45. Copying1111
    46. 46. Copying1111
    47. 47. Copying1111
    48. 48. Multiplying1111
    49. 49. Hamiltonian Path III: arbitrary graph as input on chromatin tape Input: Initial state, Graph Working memory: State, Vertices, Path Output: Path L G V P State Blank PathLastvertex Vertices Graph
    50. 50. Sketch of HamPath III• Set up the vertex-visited section of tape.• Add the next valid edge; check the vertex has not yet been used.• Check whether we found a Hamiltonian path (visits all vertices and ends at the last vertex)• If there are no next valid edges, backtrack by removing the last edge.• Repeat 1 S L G V P Encoding: 2^12 combinations 5 30 2 2 2 2 Unary representation for vertex number.
    51. 51. Program design: state transition graph
    52. 52. Hamiltonian Path III: graph as input S L G V P Click on the complex for the animation… L G V P State Blank PathLastvertex Vertices Graph
    53. 53. What does biological chromatin actually compute?• State, subroutines, variables, recursion?• Self-modifying program (through changed expression of chromatin modifying complex components)• How do CC programs evolve?
    54. 54. Silencing in S. cerevisiae• Silencing in Saccharomyces cerevisiae heterochromatin domain has been well studied, and a model with sequential histone modification involved is well defined in literature.• Here we map this model with a piece of chromatin and rules to implement the propagation of heterochromatin in our Chromatin Computer.
    55. 55. Proposed Nucleosome• H4K16(A/B) Core(I/B) Protein(s/C/B) State(H/E)• A: acetylation, B: no modification• I: end of heterochromatin (H3K56ac or H3K79me), B: no modification• s: Sir3p, C: SIR complex (2-3-4), B: none• H: heterochromatin, E: euchromatin• N:nucleation center
    56. 56. Silencing steps• (Nucleation) Signal for start, series of protein interactions and finally SIR complex binds to silencer.• (Propagation) 1. Sir2 deacetylate H4K16ac on adjacent nucleosome. 2. Sir3 binds to the deacetylated nucleosome. 3. Sir3 recruits Sir4-Sir2 complex. 4. Goto step 2.• (Termination) Boundary forms when H3K56ac or H3K79me is encountered
    57. 57. Initial chromatin & rules ... - NNNN -ABBE -ABBE -ABBE -... - AIBE -IIII -... NNNN ABBE --> NNNN BBBE # Nucleation with the first SIR BBBE **** --> BBsE ---- # Recruit Sir3p **sE **** --> --CH ---- # Recruit Sir2 -4 complex & deactivation **C* A*** --> ---- B--- # Deacetylation(BIBE **** --| BIsE ----) # Inhibition not required in simulation
    58. 58. Sample run with simulator
    59. 59. Can we build a computer from molecular parts?• To build a biological chromatin computer, we need to create the writable tape and the read/write rules.
    60. 60. Making the tape+ Ligation of ssDNA ends
    61. 61. Proof of concept for CC rule engineeringFrom Fig. 1 of Haynes and Silver 2011, “Synthetic reversal of gene silencing” JBC
    62. 62. Engineering of read/write rules• Start with list of existing parts: existing reader, writer and scaffolding/adaptor protein domains across organisms• Protein engineering for additional functionality.• Apply combinatorial remixing to engineer chromatin- modifying complexes.• Generate a physical library of complexes, then test on nucleosome arrays with known modifications to determine read/write functionality.• Write CC programs using these components; test with simulator.
    63. 63. CC software engineering• State, subroutines, variables, recursion• Self-modifying program (through changed expression of chromatin modifying complex components)• Compiler.• Correctness proofs for CC programs.• What kinds of CC programs are easy to evolve?
    64. 64. What we’ve done• Defined chromatin computing closely modeled on biology• Built a simulator• Wrote CC programs• Proved CC is computationally universal• Expanded the CC instruction set for more efficient computing
    65. 65. Related WorkProhaska, Stadler and Krakauer, “Innovation in gene regulation: The case ofchromatin computation” Journal of Theor Biol 2010 265(1):27-44.Benecke, "Chromatin code, local non-equilibrium dynamics, and theemergence of transcription regulatory programs," Eur Phys J E Soft Matter.2006 Mar;19(3):353-66.Rohlf et al, "Modeling the dynamic epigenome: from histone modificationstowards self-organizing chromatin," Epigenomics. 2012 Apr;4(2):205-19.
    66. 66. Why it’s important to model chromatin as a computer• Understand chromatin biology• Fix chromatin when it’s broken• Engineer chromatin-based computers
    67. 67. Thanks• David Yee • Bob Tepper• Rich Ferrante • Yang Shi• David Allis • Adam Rudner• Keith Robison • Lisa Tucker-Kellogg• Sebastian Hoersch • Larry Hunter• Jim Audia • Phil Bourne• Mark Goldsmith • Constellation
    68. 68. Extra slides
    69. 69. • Different modifications have different time scales. Acetylation marks have a lifetime of minutes, phosphorylation hours, whereas methylation can last for days and even through generations of cell cycle. (Pubmed ID 21818411) The time scale is quite different from that of DNA sequence changes: chromatin modifications are written and erased many times during the life of a cell, whereas DNA sequence is mostly static, varying just slightly over generations.
    70. 70. How much writable chromatin memory is there in each human cell? Size Units Item3,000,000,00 base pairs Size of genome 0 300 base pairs length of region covered by a nucleosome, allowing for some nucleosome-free regions 10,000,000 nucleosomes Therefore, number of nucleosomes in genome 64 locations Number of modifiable locations on each nucleosome 2 marks Number of possible marks at each position (marked or not marked) 264 mark Number of possible different values (mark combinations) combinations taken by one nucleosome. 64 bits Number of bits per nucleosome 8 bytes Number of bytes per nucleosome 80,000,000 bytes Number of bytes per human cell 152,000 bytes Amount of onboard memory in the Apollo mission that got astronauts to the moon (www.doneyles.com/LM/Tales.html)
    71. 71. Size of program compared to biological chromatin• Hamiltonian Path program allowing any sized input graph • Number of rules: 150 • Upper bound on required number of nucleosome states: 2^12 (12 modification sites, 2 states each)• Biological chromatin • 100s-1000s of rules (or more) • > 64 nucleosome states

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