Structural and Functional Analysis of Proteins Using Rigidity Theory

Over the past two decades, we have witnessed an unprecedented explosion in available biological data. In the age of big data, large biological datasets have created an urgent need for the development of bioinformatics methods and innovative fast algorithms. Bioinformatics tools can enable data-driven hypothesis and interpretation of complex biological data that can advance biological and medicinal knowledge discovery. Advances in structural biology and computational modelling have led to the characterization of atomistic structures of many biomolecular components of cells. Proteins in particular are the most fundamental biomolecules and the key constituent elements of all living organisms, as they are necessary for cellular functions. Proteins play crucial roles in immunity, catalysis, metabolism and the majority of biological processes, and hence there is significant interest to understand how these macromolecules carry out their complex functions. The mechanical heterogeneity of protein structures and a delicate mix of rigidity and flexibility, which dictates their dynamic nature, is linked to their highly diverse biological functions. Mathematical rigidity theory and related algorithms have opened up many exciting opportunities to accurately analyse protein dynamics and probe various biological enigmas at a molecular level. Importantly, rigidity theoretical algorithms and methods run in almost linear time complexity, which makes it suitable for high-throughput and big-data style analysis. In this chapter, we discuss the importance of protein flexibility and dynamics and review concepts in mathematical rigidity theory for analysing stability and the dynamics of protein structures. We then review some recent breakthrough studies, where we designed rigidity theory methods to understand complex biological events, such as allosteric communication, large-scale analysis of immune system antibody proteins, the highly complex dynamics of intrinsically disordered proteins and the validation of Nuclear Magnetic Resonance (NMR) solved protein structures.

Sparse graphs and an augmentation problem

For two integers $$k>0$$ k > 0 and $$\ell $$ ℓ , a graph $$G=(V,E)$$ G = ( V , E ) is called $$(k,\ell )$$ ( k , ℓ ) -tight if $$|E|=k|V|-\ell $$ | E | = k | V | - ℓ and $$i_G(X)\le k|X|-\ell $$ i G ( X ) ≤ k | X | - ℓ for each $$X\subseteq V$$ X ⊆ V for which $$i_G(X)\ge 1$$ i G ( X ) ≥ 1 , where $$i_G(X)$$ i G ( X ) denotes the number of induced edges by X. G is called $$(k,\ell )$$ ( k , ℓ ) -redundant if $$G-e$$ G - e has a spanning $$(k,\ell )$$ ( k , ℓ ) -tight subgraph for all $$e\in E$$ e ∈ E . We consider the following augmentation problem. Given a graph $$G=(V,E)$$ G = ( V , E ) that has a $$(k,\ell )$$ ( k , ℓ ) -tight spanning subgraph, find a graph $$H=(V,F)$$ H = ( V , F ) with the minimum number of edges, such that $$G\cup H$$ G ∪ H is $$(k,\ell )$$ ( k , ℓ ) -redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is $$(k,\ell )$$ ( k , ℓ ) -tight. For general inputs, we give a polynomial algorithm when $$k\ge \ell $$ k ≥ ℓ and show the NP-hardness of the problem when $$k<\ell $$ k < ℓ . Since $$(k,\ell )$$ ( k , ℓ ) -tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

Risk framing and business model adaptation: A conceptualization based on threat-rigidity theory

Firm leaders’ inclination to adapt their business model is sensitive to how risk is framed (as an external threat or an opportunity) in the macro-economic environment. We apply threat-rigidity theory to examine the relationship between risk framing and business model adaptation. We also investigate if emotionality has explanatory value for how managers adapt to business models. We test our hypotheses in a field experiment involving 134 Scandinavian managers. Here, we relate managers’ inclinations to adapt to different business models to different risk scenarios. The results reveal that, in general, managers are more risk seeking in gain scenarios than in loss scenarios. This finding is in line with the threat-rigidity theory. Emotionality was found to relate more to risk aversion than to risk seeking in the domain of potential gain. We argue that emotionality has explanatory value for how managers adapt to business models, because emotions are key influences on risk perception.