Analysis of PRMs for Computational Biology

1
Analysis of PRMs for Computational Biology

• Shawna ThomasRandomized Algorithms12/5/03

2
Motion Planning and PRMs

• The motion planning problem
• Probabilistic Roadmap Methods (PRMs)

Configuration space
goal
C-obst
C-obst
C-obst
C-obst
C-obst
start
3
Application to Computational Biology

• Protein Folding how does a protein folding from
  an unstructured configuration to its final
  native/most stable structure?
• Ligand Binding what are potential binding sites
  on a protein for the ligand/drug molecule?
• RNA Folding what are the folding kinetics of an
  RNA molecule (e.g., population kinetics,
  transition states, folding rates)?

4
Application to Computational Biology

• Protein Folding how does a protein folding from
  an unstructured configuration to its final
  native/most stable structure?
• Ligand Binding what are potential binding sites
  on a protein for the ligand/drug molecule?
• RNA Folding what are the folding kinetics of an
  RNA molecule (e.g., population kinetics,
  transition states, folding rates)?

5
PRMs for Computational Biology

• Use the same PRM technique as for traditional
  robotics problems.
• Replace collision check with energy calculation.
• Nodes are accepted based on the following
  probability

C-space
Potential Energy
6
C-Spaces for Computational Biology
Fuzzy C-space
Traditional C-space
7
Analysis of Failure Probability

• Bound using Min. Path Clearance
• Bound using Varying Path Clearance

8
Analysis of Failure Probability

• Bound using Min. Path Clearance
• Bound using Varying Path Clearance

9
Bound based on Path Clearance

• Idea cover g with a few balls of radius R/2 that
  overlap.
• Put ball centers at x0a, x2, , xnb so that the
  distance between adjacent balls is lt R/2.
• Planner will succeed if it samples at least one
  node in each ball.

• Planner will succeed if it samples at least one
  node in each ball.

10
A Little Geometry

• Let xj and xj1 be two points along g whose
  distance is less than R/2.
• For any two points c Î BR/2(xj) and d Î BR/2, the
  line segment cd is contained inside BR(xj) and
  therefore is valid.

11
A Little Geometry

• Let xj and xj1 be two points along g whose
  distance is less than R/2.
• For any two points c Î BR/2(xj) and d Î BR/2, the
  line segment cd is contained inside BR(xj) and
  therefore is valid.

12
A Little Geometry

• Let xj and xj1 be two points along g whose
  distance is less than R/2.
• For any two points c Î BR/2(xj) and d Î BR/2, the
  line segment cd is contained inside BR(xj) and
  therefore is valid.

13
Failure Probability

• The planner might fail if it doesnt sample at
  least one node in each of the balls at x1 xn-1.
• n 2L/R
• N number of samples
• Prfailure Prdont sample every ball
• Let Ej denote the event that ball j is not
  sampled
• Prfailure PrE1 U E2 U U En-1 S
  PrEj for j1..n-1
• Samples are independent, so this is
• (n-1)(Prdont sample BR/2 )N
• (2L/R)(1 - Prsample BR/2)N

14
Prsample BR/2)

• Three worst-case scenarios

15
Putting it all Together
Using the inequality we can simplify the
expression to
16
Prsample BR/2)

• Three worst-case scenarios

Area of BR-RArea of F
Area of gray regionArea of F
p

17
Putting it all Together
Using the inequality we can simplify the
expression to
18
Bound using Varying Path Clearance

• Idea most of the points along g have clearance
  greater than R. Cover g with as few large balls
  as possible.

19
Bound using Varying Path Clearance

• Idea most of the points along g have clearance
  greater than R. Cover g with as few large balls
  as possible.

20
Conclusion

• This work is a first step in analyzing the
  performance of PRMs on computational biology
  applications.

• Weve presented 2 bounds on the failure
  probability for the probability distribution p(q)
  p in a 2D C-space.

• This analysis extends to higher dimensions and to
  the probability distribution based on clearance.

21
Conclusion

• This work is a first step in analyzing the
  performance of PRMs on computational biology
  applications.

• Weve presented 2 bounds on the failure
  probability for the probability distribution p(q)
  r(q)/R in a 2D C-space.

• This analysis extends to higher dimensions and to
  the probability distribution based on clearance.

22
Future Work

• The dimensionality of the C-space for
  computational biology applications is extremely
  large.
• A protein of length n has C-space dimension 2n
• n can be anywhere from 50 to 50000

• Typically these applications bias their node
  sampling towards more interesting areas of the
  C-space.
• In the future, we want to extend the analysis to
  biased node sampling.

23
References

• J.C. Latombe, Robot Motion Planning. Boston, MA
  Kluwer Academic Publishers, 1991.
• L. Kavraki, P. Svestka, J.C. Latombe, and M.
  Overmars, Probabilistic roadmaps for path
  planning in high-dimensional configuration
  spaces, IEEE Trans. Robot. Automat., vol. 12,
  no. 4, pp. 566-580, August 1996.
• O.B. Bayazit, G. Song, and N.M. Amato, Ligand
  binding with OBPRM and haptic user input
  Enhancing automatic motion planning with virtual
  touch, in Proc. IEEE Int. Conf. Robot. Autom.
  (ICRA), 2001, pp. 954-959.
• N.M. Amato and G. Song, Using motion planning to
  study protein folding pathways, J. Comput.
  Biol., vol 9, no. 2, pp. 149-168, 2002.
• X. Tang, B. Kirkpactrick, S. Thomas, G. Song, and
  N.M. Amato, Using motion planning to study rna
  folding kinetics, PARASOL Lab, Dept. of Computer
  Science, Texas AM University, Tech. Rep. 03-005,
  Oct 2003.
• P. Svestka, On probabilistic completeness and
  expected complexity of probabilistic path
  planning, Dept. of computer Science, Utrecht
  University, Utrecht, the Netherlands, Tech. Rep.
  UU-CS-96-20, May 1996.
• L. Kavraki, M. Kolountzakis, and J.C. Latombe,
  Analysis of probabilistic roadmaps for path
  planning, in IEEE Trans. Robot. Automat., vol.
  14, 1998, pp. 166-171.