Many complex systems display a surprising degree of tolerance against errors. For example, relatively simple organisms grow, persist and reproduce despite drastic pharmaceutical or environmental interventions, an error tolerance attributed to the robustness of the underlying metabolic network1. Complex communication networks2 display a surprising degree of robustness: although key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these and other complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. Here we demonstrate that error tolerance is not shared by all redundant systems: it is displayed only by a class of inhomogeneously wired networks, called scale-free networks, which include the World-Wide Web3,4,5, the Internet6, social networks7 and cells8. We find that such networks display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected even by unrealistically high failure rates. However, error tolerance comes at a high price in that these networks are extremely vulnerable to attacks (that is, to the selection and removal of a few nodes that play a vital role in maintaining the network's connectivity). Such error tolerance and attack vulnerability are generic properties of communication networks.

Error and attack tolerance of complex networks

The most highly connected proteins in the cell are the most important for its survival. Proteins are traditionally identified on the basis of their individual actions as catalysts, signalling molecules, or building blocks in cells and microorganisms. But our post-genomic view is expanding the protein's role into an element in a network of protein–protein interactions as well, in which it has a contextual or cellular function within functional modules1,2. Here we provide quantitative support for this idea by demonstrating that the phenotypic consequence of a single gene deletion in the yeast Saccharomyces cerevisiae is affected to a large extent by the topological position of its protein product in the complex hierarchical web of molecular interactions.

Lethality and centrality in protein networks

1. Light and Energy. 2. Organization and Structure of Photosynthetic Systems. 3. History and Development of Photosynthesis. 4. Photosynthetic Pigments-Structure and Spectroscopy. 5. Antenna Complexes and Energy Transfer Processes. 6. Reaction Center Complexes. 7. Electron Transfer Pathways and Components. 8. Chemiosmotic Coupling and ATP Synthesis. 9. Carbon Metabolism. 10. Genetics, Assembly and Regulation of Photosynthetic. Systems. 11. Origin and Evolution of Photosynthesis. Appendix 1. Light, Energy and Kinetics

Molecular mechanisms of photosynthesis

Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher-order interactions among large groups of elements have an important role. Here we show, in the vertebrate retina, that weak correlations between pairs of neurons coexist with strongly collective behaviour in the responses of ten or more neurons. We find that this collective behaviour is described quantitatively by models that capture the observed pairwise correlations but assume no higher-order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behaviour. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons.

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Weak pairwise correlations imply strongly correlated network states in a neural population.

Engineered systems are often built of recurring circuit modules that carry out key functions. Transcription networks that regulate the responses of living cells were recently found to obey similar principles: they contain several biochemical wiring patterns, termed network motifs, which recur throughout the network. One of these motifs is the feed-forward loop (FFL). The FFL, a three-gene pattern, is composed of two input transcription factors, one of which regulates the other, both jointly regulating a target gene. The FFL has eight possible structural types, because each of the three interactions in the FFL can be activating or repressing. Here, we theoretically analyze the functions of these eight structural types. We find that four of the FFL types, termed incoherent FFLs, act as sign-sensitive accelerators: they speed up the response time of the target gene expression following stimulus steps in one direction (e.g., off to on) but not in the other direction (on to off). The other four types, coherent FFLs, act as sign-sensitive delays. We find that some FFL types appear in transcription network databases much more frequently than others. In some cases, the rare FFL types have reduced functionality (responding to only one of their two input stimuli), which may partially explain why they are selected against. Additional features, such as pulse generation and cooperativity, are discussed. This study defines the function of one of the most significant recurring circuit elements in transcription networks.

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Structure and function of the feed-forward loop network motif

We have surveyed proteins with known atomic structure whose function involves electron transfer; in these, electrons can travel up to 14 A between redox centres through the protein medium. Transfer over longer distances always involves a chain of cofactors. This redox centre proximity alone is sufficient to allow tunnelling of electrons at rates far faster than the substrate redox reactions it supports. Consequently, there has been no necessity for proteins to evolve optimized routes between redox centres. Instead, simple geometry enables rapid tunnelling to high-energy intermediate states. This greatly simplifies any analysis of redox protein mechanisms and challenges the need to postulate mechanisms of superexchange through redox centres or the maintenance of charge neutrality when investigating electron-transfer reactions. Such tunnelling also allows sequential electron transfer in catalytic sites to surmount radical transition states without involving the movement of hydride ions, as is generally assumed. The 14 A or less spacing of redox centres provides highly robust engineering for electron transfer, and may reflect selection against designs that have proved more vulnerable to mutations during the course of evolution.

Natural engineering principles of electron tunnelling in biological oxidation-reduction.

Iron(III) protoporphyrin IX (Fe(III)PP) and iron(III) hematoporphyrin (Fe(III)HP) were selectively and covalently attached to dimercaptoalkane-modified gold electrodes. Reaction of their vinyl or hydroxyethyl groups with the surface-immobilized thiols produced thioether linkages, reminiscent of the heme macrocycle attachment in c-type cytochromes. Cyclic voltammetry revealed reversible electrochemistry of self-assembled monolayers (SAMs) of FePPs and FeHPs on the thiol-modified gold substrates. The surface coverage estimated from the charges transferred corresponds to 30% of a monolayer. The heterogeneous rate constant of electron transfer between the Fe(III)PPs and the gold substrate decreases exponentially with the length of the spacer, demonstrating a value of 1.0 A-1 for the tunneling length coefficient, β. At pH 8, a linear increase of the formal redox potential (E°‘) with the length of the linker was also observed. This suggests that in the film, there is a close contact between the porphyrins and t...

Electrochemistry of Self-Assembled Monolayers of Iron Protoporphyrin IX Attached to Modified Gold Electrodes through Thioether Linkage

Nature has engineered a universe of redox proteins to efficiently control the oxidation and reduction of substrates and to convert redox energy into a delocalized transmembrane proton gradient power source. Some rapid physiologically relevant electron transfers are rate limited by electron tunneling. Distance appears to be the principle means naturally selected to control the speed of electron tunneling; free energy and reorganization energy can play important auxiliary roles. Thus, an electron from a biological redox center can tunnel in any direction and is likely to reduce the closest redox center with a favorable free energy. Although it is clearly possible to facilitate electron tunneling by designing covalent bridges in the regions between donors and acceptors, this does not seem to be a strategy that evolution has used. Evolutionary mutagenic adjustment of a bridge-like quality of the amino acid medium may be difficult in the face of heavy selection on the folding, stability and other properties of the protein medium. Repositioning cofactors by even a few angstroms has more profound effects on promoting and retarding rates, independent of the structure of the amino acid medium.

Biological electron tunneling through native protein media

Experimental explorations of functional mechanisms in natural electron-transfer proteins are often frustrated by their fragility and extreme complexity. We have designed and synthesized four-α-helix-bundle redox proteins, maquettes, that are much simplified and more robust than natural redox proteins and can be designed to bind onto electrode surfaces to facilitate systematic investigations. The points of interest that can be now assessed are not only the processes that govern biological assembly of equilibrium structures, electrochemistry, and electron tunneling rates but also how these factors are coupled together to effect redox driven catalysis. Here we describe maquettes that bis-histidine ligate protoporphyrin IX (heme), much like native b cytochromes, as well as contain charged surface patches, much like native cytochrome c. The positively charged residues aid adsorption to negatively charged surfaces, such as gold electrodes modified by 11-mercaptoundecanoic acid, and facilitate cyclic voltammetry...

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De Novo Design of a Cytochrome b Maquette for Electron Transfer and Coupled Reactions on Electrodes

A new technique for the determination of the orientation distribution of a porphyrin within a Langmuir monolayer, in terms of the mean tilt angle relative to the monolayer normal and the width, is described. The technique utilizes the measurement of polarized fluorescence, excited with the electric field both parallel and perpendicular to the monolayer plane. The main difference between this technique and the existing ones [Fraaije et al., Biophys. J. 1990, 57, 965; Edmiston et al., J. Am. Chem. Soc. 1997, 119, 560; Bos et al., Biophys. J. 1995, 68, 2573.] is that the fluorophores are excited directly by the incident beam rather than an evanescent field, which greatly facilitates its application to Langmuir monolayers at the air−water interface. The technique was applied to the study of two systems with different porphyrin orientations:  a Langmuir monolayer of a dihelical synthetic peptide BBC16 containing Zn(II)protoporphyrinIX and a monolayer of tetrakis (N-methyl-4-pyridil) porphyne electrostatically ...

Xiaoxi Chen

Papers

Natural engineering principles of electron tunnelling in biological oxidation-reduction.

Electrochemistry of Self-Assembled Monolayers of Iron Protoporphyrin IX Attached to Modified Gold Electrodes through Thioether Linkage

Biological electron tunneling through native protein media

De Novo Design of a Cytochrome b Maquette for Electron Transfer and Coupled Reactions on Electrodes

Determination of the Porphyrin Orientation Distribution in Langmuir Monolayers by Polarized Epifluorescence