Davidson 2006

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|[[Image:two_pancake_families.jpg|thumb|250px|right|Simulation results for two pancakes, useful for calibrating kinetics of pancake flipping. ]]'''Math''': Our mathematical model for a stack of pancakes is a '''signed permutation''', in which each numerical value represents the pancake size (or desired position in the stack) and the sign represents the orientation. For example, "1, 2, 3, 4" is a stack of four pancakes all in the proper order and orientation. "-2, 4, -1, 3" is a scrambled stack of the same four pancakes, as shown in Figure 1. Here, pancakes 1 and 2 are in the wrong orientation (burnt side up). A population of E. coli cells (10<sup>15</sup> cells, for instance) each carrying ~100 copies of pancake stacks has astounding parallel processing capacity.  
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|[[Image:two_pancake_families.jpg|thumb|250px|right|Simulation results for two pancakes, useful for calibrating kinetics of pancake flipping. ]]'''Mathematical model''': Our mathematical representation of a stack of pancakes is a '''signed permutation''', in which each numerical value represents the pancake size and the sign of the number represents the orientation. For example, (1, 2, 3, 4) is a sorted stack of four pancakes, all in the proper order and orientation, and (-2, 4, -1, 3) is the scrambled stack of four pancakes shown in Figure 1. Note that pancakes 1 and 2 are negative, indicating that these two pancakes are in the wrong orientation (burnt side up).
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We simulated the flipping process by assuming that each segment of DNA flanked by Hix sites is equally likely to be flipped by Hin invertase. In an ''n''-pancake stack, there are <math> \binom{n+1}{2} </math> ways to choose a segment: two spatula locations must be chosen from among the ''n'' + 1 potential locations.  For example, there are <math> \binom{5}{2} = \frac{5!}{3! 2!} = 10 </math> different ways to perform a single flip in a stack of 4 pancakes.
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The random flipping process continues until the stack is sorted in the correct order and orientation, or until we artificially stop the random flipping process.A population of E. coli cells (10<sup>15</sup> cells, for instance) each carrying ~100 copies of pancake stacks has astounding parallel processing capacity.  
<font color='red'>Lance, please work on this section.</font color>
<font color='red'>Lance, please work on this section.</font color>

Revision as of 02:24, 29 October 2006

Logo.gif

Project Overview

[http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Davidson Davidson Parts]

Team Members

Tools and Resources

Check out our [http://www.bio.davidson.edu/courses/synthetic/photos/FlapJack_HotCakes.html Official Team Photo]
Left to Right: Malcolm, Laurie, Sabriya, Erin and Lance
Solving the Pancake Problem with an E. coli Computer


Our goal is to genetically engineer a biological system that can compute solutions to a puzzle called the burnt pancake problem. The EHOP computer is a proof of concept for computing in vivo, with implications for future data storage devices and transgenic systems. Our work was done in collaboration with the [http://2006.igem.org/wiki/index.php/Missouri_Western_State_University_2006 Missouri Western iGEM Team] and an undergraduate research fellow from [http://www.hamptonu.edu/ Hampton University].


The Burnt Pancake Problem
The [http://en.wikipedia.org/wiki/Pancake_sorting pancake problem] is a puzzle in which a scrambled series of units (or stack of pancakes) must be shuffled into the correct order and orientation. You can try solving [http://www.cut-the-knot.org/SimpleGames/Flipper.shtml a simple version of the pancake problem] yourself.

Figure 1 A scrambled stack of four burnt pancakes.
In the burnt pancake problem, each pancake is given an orientation by burning one side. Figure 1 shows a scrambled stack of burnt pancakes. To solve the problem, every unit, or pancake, must be placed in the proper order (largest on bottom, smallest on top) and in the proper orientation (burnt side down, golden side up). The pancakes are flipped with two spatulas: one to lift pancakes off the top of the stack, the other to flip part (or all) of the remaining stack of pancakes. The pancakes lifted by the first spatula are returned to the top of the stack without being flipped. You can watch Media:burnt_pancake.ogg a movie of the stack in Figure 1 being sorted to see how the puzzle is solved.


Approach
Trial and error is one approach to solving the burnt pancake problem, but how could one compute the quickest solution? Our idea is to let E. coli do the work, using each cell as a tiny processor in a massively parallel machine. A mathematical model of the flipping process helps us design the system and interpret the output of our EHOP computer.

Figure 2 3-D structure of a Hin protein complex bound to DNA. View the [http://www.rcsb.org/pdb/explore/explore.do?structureId=1ZR4 interactive 3-D Jmol image].
Biological System: The biological representation of a pancake is a functional unit of DNA such as a promoter or coding region. To flip these units of DNA, we have reconstituted the Hin/ Hix invertase system (Figure 2) from Salmonella typhimurium as a BioBrick compatible system in E. coli. Hin invertase () was cloned from S. typhimurium, Ames strain TA100 and tagged with LVA. We built the recombinational enhancer (RE) and Hin invertase recognition sequence HixC using the publicly available genomic sequence of S. typhimurium and [http://gcat.davidson.edu/IGEM06/oligo.html a dsDNA assembly program we created] for gene synthesis from overlapping oligos.

Every segment of DNA flanked by a pair of HixC sites is a "pancake" capable of being inverted. Hin invertase recognizes pairs of HixC sites and inverts the DNA fragment in between the two HixC sites with the help of the Fis protein bound to the RE. In our system, selectable phenotypes (including antibiotic resistance and RFP expression), depend upon the proper arrangement of a series of HixC-flanked DNA segments in a plasmid. This allows us to select for cells that have successfully solved the puzzle. An example of a sorted stack of two pancakes is shown in Figure 3.

Figure 3 A sorted stack of two biological "pancakes"
Simulation results for two pancakes, useful for calibrating kinetics of pancake flipping.
Mathematical model: Our mathematical representation of a stack of pancakes is a signed permutation, in which each numerical value represents the pancake size and the sign of the number represents the orientation. For example, (1, 2, 3, 4) is a sorted stack of four pancakes, all in the proper order and orientation, and (-2, 4, -1, 3) is the scrambled stack of four pancakes shown in Figure 1. Note that pancakes 1 and 2 are negative, indicating that these two pancakes are in the wrong orientation (burnt side up).

We simulated the flipping process by assuming that each segment of DNA flanked by Hix sites is equally likely to be flipped by Hin invertase. In an n-pancake stack, there are <math> \binom{n+1}{2} </math> ways to choose a segment: two spatula locations must be chosen from among the n + 1 potential locations. For example, there are <math> \binom{5}{2} = \frac{5!}{3! 2!} = 10 </math> different ways to perform a single flip in a stack of 4 pancakes.

The random flipping process continues until the stack is sorted in the correct order and orientation, or until we artificially stop the random flipping process.A population of E. coli cells (1015 cells, for instance) each carrying ~100 copies of pancake stacks has astounding parallel processing capacity.

Lance, please work on this section.


Methods and Results
The following is an outline of what the Davidson team will present at iGEM 2006

Basic parts: Parts used in this project were designed by the [http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Davidson Davidson] and [http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Missouri Missouri Western] iGEM teams
Modeling

  • Modeling the behavior of pancake flipping: deducing kinetics and size biases
  • Using modeling to choose which families of unsolved pancake stacks to start with

Building the Biological System

  • "Single pancake" constructs and initial observations
  • Read-through from the carrier vector backbone leads to uncontrolled Tet expression and uncontrolled flipping
  • New [http://partsregistry.org/Part:BBa_J31009 pSB1A7] vector insulates parts from read-through, but is not compatible with parts carrying double terminators
  • Solution: designing pancake constructs without double terminators
  • "Two-pancake" constructs
  • Dealing with biological equivalence: distinguishing 1,2 from -2,-1 using an RFP reporter
  • Updated design


Conclusions

  • Consequences of devices
    • Practical: proof-of-concept for bacterial computers, data storage
    • Basic research: transgene rearrangement in vivo, insights into evolution that has occurred via DNA rearrangements
  • Next steps: can solve problem but need control over kinetics
  • Lessons learned:
    • Troubleshooting, communication, teamwork, publicity
    • Math and Biology meshed really well and even uncovered a new proof
    • Multiple campuses can increase capacity through communication and cooperation
    • Size of school is not a limiting factor
    • We had a blast and learned heaps
TEAM MEMBERS


Students

  • Sabriya Rosemond is a junior biology major at Hampton University.
  • Erin Zwack is a junior biology major at Davidson College.
  • Lance Harden is a sophomore math major at Davidson College.
  • Samantha Simpson is a sophomore at Davidson College who might design a major in genomics.

Faculty

  • A. Malcolm Campbell [http://www.bio.davidson.edu/campbell Department of Biology]
  • Laurie J. Heyer [http://www.davidson.edu/math/heyer/ Department of Mathematics]
  • Karmella Haynes [http://www.bio.davidson.edu/ Department of Biology - teaching postdoc and visiting assistant professor]
TOOLS AND RESOURCES


iGEM 2006 Jamboree

White Board

Biology Tools (Wet Bench)

Math Tools

  • Pancake Simulators, aka The Lancelator (MATLAB Code)
  • Other Flipping Simulators
    • [http://gcat.davidson.edu/IGEM06/randompancakeflipper.m User-Friendly Version]
    • [http://gcat.davidson.edu/IGEM06/powerflipper.m Power-User Version]
    • [http://gcat.davidson.edu/IGEM06/powerflipperplus.m Power-User Update]
    • [http://gcat.davidson.edu/IGEM06/findmin.m Findmin]
    • [http://gcat.davidson.edu/IGEM06/bigtrial.m Bigtrial]
  • Graphing Tools
    • [http://gcat.davidson.edu/IGEM06/pancakeplotter.m Pancakeplotter]
    • [http://gcat.davidson.edu/IGEM06/permplotter.m Permplotter]
    • [http://gcat.davidson.edu/IGEM06/adjmat.m Adjmat - creates adjacency matrices]
    • [http://gcat.davidson.edu/IGEM06/bioadjmat.m Bioadjmat - Adjmat with biological equivalence]
    • [http://gcat.davidson.edu/IGEM06/make_radial.m Make_radial - creates a radially-embedded graph]
    • [http://gcat.davidson.edu/IGEM06/find_diam.m Find_diam - finds the diameter of a pancake graph]

Bio-Math Tools

  • [http://gcat.davidson.edu/IGEM06/signedperms.m Signedperms - lists all signed permutations for a stack of k pancakes]
  • [http://gcat.davidson.edu/IGEM06/bioperms.m Bioperms - Signedperms with biological equivalence]

Assembly Plans

Progress

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