3.7 Appendix
3.7.18 Overview of all fluorescence data sets collected at Caltech
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T21 (nM) Open tweezers (nM) T21 (nM) Open tweezers (nM)
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T21 (nM) Open tweezers (nM) T21 (nM) Open tweezers (nM)
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T21 (nM) Open tweezers (nM) T21 (nM) Open tweezers (nM)
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Figure 3.41: Mode I. Left: Oscillator traces. Center: Load traces. Right: Oscillator and load traces.
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Figure 3.42: MODE II. Left: Oscillator traces. Center: Load traces. Right: Oscillator and load traces.
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Figure 3.43: Mode II∗. Left: Oscillator traces. Center: Load traces. Right: Oscillator and load traces.
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Figure 3.44: Mode III. Left: Oscillator traces. Center: Load traces. Right: Oscillator and load traces.
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Figure 3.45: Mode IV. Left: Oscillator traces. Center: Load traces. Right: Oscillator and corresponding load traces.
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Figure 3.46: Mode V. Left: Oscillator traces. Center: Load traces. Right: Oscillator and corresponding load traces. The TwCls DNA strand was always added in 50 nM excess of TWV load.
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Figure 3.47: Mode V∗. Left: Oscillator traces. Center: Load traces. Right: Oscillator and corresponding load traces. The DNA strand TwCls (closing the mode V tweezers) was always added in 50 nM excess of TWV load.
Chapter 4
Structural robustness in molecular networks
4.1 Introduction
In this chapter, a simple question will be asked: are there dynamical models for biological systems that have structurally stable equilibria and preserve this property robustly with respect to their specific parameters? This question has been considered before in the literature. For instance, in [94], through numerical exploration of the Jacobian eigenvalues for two-, three- and four-node networks, the authors isolated a series of interconnections which are stable, robustly with respect to the specific parameters. Such structures also turned out to be the most frequent topologies in existing biological networks databases. In another recent work [75], through extensive numerical analysis on three-node networks, the authors have shown that adaptability (defined as a significant step response followed by relaxation to the pre-stimulus equilibrium) of these systems can be investigated solely based on their structure, regardless of the chosen reaction parameters. Numerical simulation has arguably been the most popular tool to investigate robustness of biological networks [68, 45, 46, 58, 5, 127].
Analytical approaches to the study of robustness have been used in specific contexts. A series of recent papers [116, 115] focused on input/output robustness of ODE models for phosphorylation cascades. In particular, the theory of chemical reaction networks is used in [115] as a powerful tool to demonstrate the property of absolute concentration robustness. Indeed, the so-called deficiency the- orems [34] are to date some of the most general results to establish robust stability of a (bio)chemical reaction network. Monotonicity is also a structural property that is useful in demonstrating robust dynamic behaviors of a class of biological models [121, 10]. Robustness has also been investigated in the context of compartmental models, which are often encountered in biology and chemistry [54].
Here, I will follow the paper I co-authored with Prof. Franco Blanchini [19] and present a simple and general theoretical tool kit for the analysis of bio-molecular systems. Such tool kit is constituted by Lyapunov and set-invariance methods. Provided that certain standard properties are verified, a
number of well-known biological networks are demonstrated to be asymptotically stable, robustly with respect to the model parameters. In some cases, robust bounds on the system performance are found. This approach does not require numerical simulation efforts.
The framework suggested here aims at formulating qualitative models without the need of exact mathematical expressions and parameters. The utilized analytical methods rely only on qualitative interactions between network components. The properties that can be derived from the models we formulate are, consequently, structurally robust because they are not inferred from specific math- ematical formulas chosen to fit data. The techniques suggested are based on set-invariance and Lyapunov theory, in particular piecewise-linear functions, and show that such models are amenable for robust investigation byengineers and mathematicians. These techniques are effective and promis- ing in dealing with biological robustness [3, 32]. Several models from the literature are considered, reporting the original equations, and rephrasing them in our setup as case studies. Robust certifica- tions can be given to important properties (some of which have been established based on specific models).