This document presents a mathematical model for competitive DNA hybridization reactions on microarrays. It introduces competing interactions that can occur, such as specific and non-specific hybridization and intramolecular folding. The model is described through a system of ordinary differential equations considering multiple probe-target interactions and unimolecular folding. Future work to improve the model by including diffusion effects and a parallel solver for large systems is discussed.

1. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Modeling of Competitive Kinetics of DNA
Hybridization Reactions
Vladimir Gantovnik and Cynthia Gibas
Department of Computer Science
Bioinformatics Research Center
University of North Carolina at Charlotte
March 8, 2008

2. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Outline
1 Introduction
2 Competing Interactions on Microarrays
3 Mathematical Model
4 Example
5 Conclusion

3. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Microarray

4. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Microarray Hybridization
The ultimate goal:
Accurate modeling of hybridization reactions on microarray taking
into account competitive probe-target interactions and
unimolecular folding.

5. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Competing Interactions on Microarrays
+ Specific
hybridization

6. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Competing Interactions on Microarrays
Bulk Non-specific
dimerization hybridization
+ Specific
hybridization

7. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Competing Interactions on Microarrays
Bulk Non-specific
dimerization hybridization
+ Specific
hybridization
Intramolecular
folding

8. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Hybridization Models
1 Two-state models: One probe - one target duplex formation.

9. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Hybridization Models
1 Two-state models: One probe - one target duplex formation.
2 Multi-state models: Single duplex formation competing with
unimolecular folding. Mathews,Zuker,Turner. 1999. (Unifold,
VisualOMP).

10. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Hybridization Models
1 Two-state models: One probe - one target duplex formation.
2 Multi-state models: Single duplex formation competing with
unimolecular folding. Mathews,Zuker,Turner. 1999. (Unifold,
VisualOMP).
3 Multiplex competitive models: Competitive reactions of
duplex formations
(University of Pennsylvania) Zhang, Hammer, Graves. 2005.
(University of Utah) Bishop, Blair, Chagovetz. 2006.
(Portland State University) Horne, Fish, Benight. 2006.
(Russian Academy of Sciences) Chechetkin. 2007.

11. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Hybridization Models
1 Two-state models: One probe - one target duplex formation.
2 Multi-state models: Single duplex formation competing with
unimolecular folding. Mathews,Zuker,Turner. 1999. (Unifold,
VisualOMP).
3 Multiplex competitive models: Competitive reactions of
duplex formations
(University of Pennsylvania) Zhang, Hammer, Graves. 2005.
(University of Utah) Bishop, Blair, Chagovetz. 2006.
(Portland State University) Horne, Fish, Benight. 2006.
(Russian Academy of Sciences) Chechetkin. 2007.
4 Multiplex-multi-state models: Competitive reactions of duplex
formations and unimolecular folding of probes and targets.

12. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Chemical System
Table: Chemical reactions of microarray hybridization
Reaction Equilibrium Constant Rate Constants
pi + tj pi tj Kpi tj a d
kpi tj , kpi tj
pi + pj pi pj Kpi pj a d
kpi pj , kpi pj
ti + tj ti tj Kti tj a , kd
kti tj ti tj
pi f
pi Kp f a d
kpf , kpf
i i i
ti tif Kt f ktaf , ktdf
i i i

13. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Kinetic Model
Assumptions
System consists of two species types: probes pi and targets tj .
Each pi can react with every tj forming pi tj duplex.
Each pi and tj can react with each of its counterparts of the
same species, forming pi pj and ti tj complexes.
The forward rate constants are assumed to be same for all
species: kpi tj = kpi pj = ktai tj = k a .
a a
There are no diffusion barriers to the reaction process.
Initial conditions:
At τ = 0
(Cpi , Ctj , Cpf , Ct f , Cpi pj , Cpi tj , Cti tj ) = (Cpi , Ct0 , 0, 0, 0, 0, 0)
0
j
i j

14. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Kinetic Model
Nt Np
dCpi d a
= kpi tj Cpi tj − k Cpi Ctj + kpi pj Cpi pj − k a Cpi Cpj +
d
dτ
j=1 j=1
d a
+ kpf Cpf − kpf Cpi , i = 1, . . . , Np
i i i
Np Nt
dCtj
= kpi tj Cpi tj − k a Cpi Ctj +
d
ktd tj Cti tj − k a Cti Ctj +
i
dτ
i=1 i=1
+ ktdf Ct f − ktaf Ctj , j = 1, . . . , Nt
j j j
dCpi pj dCpi tj
= kpi pj Cpi pj − k a Cpi Cpj
d
= kpi tj Cpi tj − k a Cpi Ctj
d
dτ dτ
dCti tj
= ktd tj Cti tj − k a Cti Ctj
i
dτ
dCpf dCt f
d a
i
= kpf Cpf − kpf Cpi j
= ktdf Ct f − ktaf Ctj
dτ i i i dτ j j j

15. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Problem Complexity
Total number of equations
1
(5 + Np + Nt ) (Np + Nt )
2
Np Nt Number of equations
1 1 7
5 5 75
10 10 250
100 100 20500
1000 1000 2005000
For larger systems the simulation is computationally expensive.
(Need for parallel ODE solver)

16. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Problem-Size Reduction
1 Equation describing the potential interactions between distinct
species of probes could be eliminated from the system
dCpi pj
= kpi pj Cpi pj − k a Cpi Cpj
d
dτ

17. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Problem-Size Reduction
1 Equation describing the potential interactions between distinct
species of probes could be eliminated from the system
dCpi pj
= kpi pj Cpi pj − k a Cpi Cpj
d
dτ
2 Not all probes and targets have equal potential to react and
form stable duplexes.
Sequence filtering and screening based on different criteria.
(>15 consecutive base-pairs, >75% similarity, etc., Kane,
2000).
Subsystem of reactions which can occur in a given mixture.

18. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Kinetic Model
Inputs
Number of probes and targets: Np and Nt .
Temperature: T .
Initial concentrations of probes and targets: Cpi and Ct0 .
0
i
Forward rate constants: k a , kpf , ktaf .
a
i j
Thermodynamic parameters for all DNA complexes.
Outputs
Concentrations of probes, targets, and duplexes at any given
time τ including equilibrium concentrations.

19. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Duplex Thermodynamics
Thermodynamic transition parameters ∆H, ∆S, and ∆G are
determined from published sequence-dependent
thermodynamic parameters (SantaLucia’s group).
Total free energy
∆G = ∆H − T ∆S
Tm = ∆H/∆S
∆G = ∆H (1 − T /Tm )
Equilibrium constant and reverse rate constant
∆G ka
K eq = exp − kd =
RT K eq

20. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Example: Probe + Specific Target + Nonspecific Target

21. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Conclusion
1 A general analytical description of multiplex-multi-state model
is developed.
2 This model describes the kinetic behavior of system with
competitive reactions between target mixture, probe set, and
corresponding conformations of probes and targets.

22. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Future Work
1 Introduce diffusion of target strands across the probe surface
as governed by Second Ficks law:
∂Ct 2 ∂ 2 Ct ∂ 2 Ct ∂ 2 Ct
=D Ct = D + +
∂τ ∂x 2 ∂y 2 ∂z 2
2 Implementation of parallel solver for the large ODE systems.

23. Introduction Competing Interactions on Microarrays Mathematical Model Example Conclusion
Acknowledgements
Bioinformatics Research Center at UNCC:
Prof. Cynthia Gibas
Prof. Jennifer Weller
Dr. Raji Balasubramaniyan
GS. Vladyslava Ratushna
GS. Raad Gharaibeh