Biomod/2011/UTAustin/Hook'em Hybridizers:Simulation

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Simulation

A signal pulse counter can be constructed using delays and amplifying AND gates, as shown below. Hence, a counting device can be implemented using only delay units and amplifying AND gates made from DNA strands. We have designed a delay unit with two intermediate signals in addition to its final output using 8 molecules: 3 transducers, two thresholds, and three fuels. The structures of these molecules are given in Rationale. A good design for an amplifying and gate can be found in (1); we use this design (see also Rationale).

It is reasonable to assume that this system may function only when the delay units and and gates are present in certain concentration ranges. Hence, we applied simulation techniques to determine a safe range of system concentrations where the system will behave as designed. Unfortunately, no existant simulation software was applicable to our situation. Hence, we developed TripleSim, a general purpose nucleic acid system simulation application. We applied TripleSim to this DNA system to answer the question: What should the concentration of the threshold components be to give a certain signal delay at each delay unit?

This article proceeds first by showing the results of this investigation via TripleSim. Technical implementation details involving TripleSim are in section 3.

Section 2: TripleSim was used to simulate the behavior of the system containing a single delay unit. In the DNA System specification format accepted by CircDesigNA (http://cssb.utexas.edu/circdesigna),

---------
DOMAINS
---------
X	20
x	1
T	7
A	20
a	1
A1	20
a1	1
A2	20
a2	1
---------
MOLECULES
---------
INPUT [T X x}
G_X [X(x(T(A a}[T*)x*)X*)T*}
G_A [A(a(T(A1 a1}[T*)a*)A*)T*}
G_A1 [A1(a1(T(A2 a2}[T*)a1*)A1*)T*}
X- [X x T}
A- [A a T}
A1- [A1 a1 T}
KILLER [X(x(}[x*)X*)T*}
T_A1 [A1(a1(}[a1*)A1*)T* a*}
T_A2 [A2(a2(}[a2*)A2*)T* a1*}

The 10 molecules in this system (including input) are rendered in line notation below:

We can investigate this system by varying the concentration of any of these 10 molecules, keeping the rest of the system constant. However, we assume that gates should be present at excess concentrations. Furthermore, the fuel strands X-, A-, and A1- should be present at concentrations even higher than the gate strands - hence, at excess. The Killer should also be present at high concentrations, to ensure that it can "kill" even large pulses of the input strand, INPUT. Using TripleSim, we investigate the kinetics of this system based on the concentration of Input strand and based on the concentration of the thresholds.

In the following investigations, the kinetics of the production of two intermediate signals (XTA and ATA1) and the delayed output signal (A1TA2) are recorded.

Section 2.1: Increasing the concentration of the input strand has the effect of reducing the delay of time before the signal strand is produced at noticable concentration. However, for low concentrations of input, the system reliably delays the production of the signal strand A1TA2 for 30 minutes.

Section 2.2 Only when the thresholds are at high concentration does the system behave as a delay unit.

File:DelayEffects.png

Section 2.3: Investigation of AND gate behavior. Source (1) has studied the AND gate design we use (see [Rationale]). However, TripleSim can be used to verify that this AND gate does amplify the signal.

(Figure pending).

Section 3: Technical details of TripleSim: TripleSim attempts to give as complete a representation of the possible reactions the components of a DNA system can undergo, given initial concentrations. This is complicated by the fact that a system can produce an exponential number of possible products, sometimes even an infinite number. However, we can assume safely that in a useful DNA system, only a small number of these products are produced in noticeable concentrations. Hence, TripleSim is based on the concept of Dynamic Compilation: Only include a possible system reaction if its reactants are sufficiently important. TripleSim can enumerate all possible association, disassociation, and branch migration a DNA molecule can undergo. TripleSim can enumerate the reaction graph of a DNA system by enumerating these reactions involving each DNA molecule in the system, starting with the initial reactants, and then enumerating the reactions involving all the products of those initial reactions. TripleSim only enumerates the reactions involving species which occur above some specified threshold concentration over a specified machine running time in simulations of the incomplete system. This information comes at the cost of efficiency; simulating a large system of chemical reactions can be slow.

TripleSim converts an incomplete model of the interactions in a system of DNA molecules to a system of ordinary differential equations. These ODEs are integrated using the backwards Euler method. While this system is usually stiff, the backwards Euler method appears to lead to handle this system efficiently even at a strict error tolerance.

Bibliography: 1: DY Zhang, AJ Turberfield, B Yurke, et.al. "Engineering entropy-driven reactions and networks catalyzed by DNA." Science, 2007.