ABSTRACT. With the advancement in nucleic-acid-based technology in general, and strand-displacement DNA computing in particular,
a large class of abstract biochemical networks may be physically realized using nucleic acids.
Mathematical and experimental methods for designing abstract biochemical circuits, and then physically realizing them, respectively,
have been predominantly developed at the (less-detailed) deterministic level, when the circuits involve molecules in high-abundance and
operate in well-mixed environments. A proof-of-concept is a recently in-vitro man-made
chemical oscillator, called the displacillator. However, molecular circuits involving species in low-abundance,
and operating in fluctuating environments, are increasingly found to play an important role in applications,
such as when molecular computers are interfaced with cell-like vesicles, and when they are utilized in
nanotechnology. In such circumstances,
methods for controlling the intrinsic noise in the system are necessary for a successful network design
at the (more-detailed) stochastic level.

To bridge the gap, the noise-control algorithm for designing biochemical networks will be presented in
this talk. The algorithm structurally modifies any given reaction network under mass-action kinetics,
in such a way that (i) controllable state-dependent noise is introduced into the stochastic dynamics
(the chemical master equation), while (ii) the deterministic dynamics (reaction-rate equations) are preserved.
The structural modification involves appropriately enlarging the input network, by adding suitable
auxiliary species and chemical reactions. This allows for a hybrid approach when constructing reaction networks:
the deterministic model may be used to guide the design, while the noise-control algorithm may be applied to
favorably reprogram the intrinsic noise in the stochastic model. The deterministic-stochastic hybrid approach
allows one to reshape the probability distributions of target chemical species, and gain control over their sample-paths,
while inheriting the fixed mean-field behaviors. The capabilities of the algorithm are demonstrated by redesigning
test reaction systems, enriching them with stochastic phenomena, such as noise-induced multimodality/multistability
(coexistenceof multiple maxima in the probability distributions) and oscillations.

ABSTRACT. We propose a systematic approach to approximate the behaviour of models of polymers synthesis/degradation. Our technique consists in discovering time-dependent lower and upper bounds for the concentration of some patterns of interest. These bounds are obtained by approximating the state of the system by a hyper-box, with differential equations defining the evolution of the coordinates of each hyper-face. The equation of each hyper-face is obtained by pessimistically bounding the derivative with respect to the corresponding coordinate when the system state ranges over this hyper-face.

In order to synthesise these bounds, we use Kappa to describe our models of polymers. This provides symbolic equalities and inequalities which intentionally may be understood as algebraic constructions over patterns, and extensionally as sound properties about the concentration of the bio-molecular species that contain these patterns.