Dynamics of infectious diseases

In population biology, at the microscopic level, stochasticity comes from the discrete nature of individuals and the disordered interaction networks. Fluctuations and finite size effects in general are much more important than in typical physical systems. Disease transmission in particular follows a network of contacts related to the geographical distribution of the population and to its mobility patterns, involving a combination of local transmission and long-distance infection. However, traditional analytic models in population dynamics assume homogeneously mixed infinite populations, and are unable to model fluctuations and correlation effects. In this framework, disease incidence patterns should be explained as steady states of deterministic models, and the role of stochasticity is just that of generating the observed superimposed noise.

Recently, a stochastic theory based on a mechanism of resonance with internal noise has shifted the role of stochasticity closer to the center stage, by showing that dominant dynamic patterns found in the incidence data can be explained as resonant fluctuations, whose behaviour is largely independent of the amplitude of seasonal forcing, and by contrast very sensitive to the basic epidemiological parameters [1, 2]. Apart from stochasticity, another missing ingredient of the standard epidemic models that has been attracting increasing attention is the host population contact structure, or ’mixing network’. Most of the research connecting networks and epidemiology dates from the last ten years, when many fundamental results of network theory became widely known while new ones have been derived. These ideas originated in the mathematics and physics communities were applied in an epidemiological setting from the beginning prompting the interest of epidemiologists and several theoretical and field epidemiology results.

It turns out that these two missing ingredients 'interfere constructively' in the sense that both spatial correlations, arising as a consequence of interactions between individuals, and time correlations, due to well defined recovery periods, have a major effect in the enhancement of the amplitude and the coherence of the resonant stochastic fluctuations, providing ordered patterns of recurrent epidemics, whose period may differ significantly from that of the small oscillations around the deterministic equilibrium [3].

We are studying the combined effect of demographic stochasticity and correlations in the context of the simplest epidemiologic models, the Susceptible-Infected-Recovered models where renewal of susceptibles takes place either through births or through immunity waning. Our approach is based on the use of stochastic simulations together with analytical methods.

References

1. Predator-prey cycles from resonant amplification of demographic stochasticity
   A. J. McKane and T. J. Newman, Phys. Rev. Lett. 94, 218102 (2005).


2. Stochastic amplification in epidemics
   D. Alonso, A. J. McKane and M. Pascual, J. R. Soc. Interface 4, 575-582 (2007).


3. Stochastic fluctuations in epidemics on networks
   M. Simões, M.M. Telo da Gama and A. Nunes, J. R. Soc. Interface 5, 555-566 (2008).

Dynamics of competition and evolution

Another challenge of complex population dynamics is the understanding of pathogen variation in endemic infections, modulated by host immunity. This requires changing from traditional models to broader approaches that allow for pathogen evolution and include the coupling between the host and the pathogen population [1, 2]. Within this programme, the role of the host population contact network structure is just starting to be considered [3, 4].

Recent results show that complex dynamics and unexpected effects may arise in an 'adaptive' network, in which links between nodes may be created or removed according to rules that depend on the state of the nodes [5, 6]. Infection dynamics is a natural setting to investigate the coupled evolution (co-evolution) of the interacting elements and the network of interactions, and to test the results against real data.

In different contexts, cultural evolution and biological evolution share the same fundamental mechanims of inheritance, variation and selection, and a number of quantitative models of opinion dynamics and of language change inspired by evolutionary biology have been proposed in the recent literature [7, 8]. We shall also consider how variation and competition on an iteraction network shape the long term evolution of language and culture.

References

1. Unifying the epidemiological and evolutionary dynamics of pathogens
   B.T. Grenfell, O.G. Pybus, J.R. Gog, J.L.N. Wood, J.M. Daly, J.A. Mumford and E.C. Holmes, Science 303, 327-332 (2004).


2. A simple model of epidemics with pathogen mutation
   Michelle Girvan, Duncan S. Callaway, M. E. J. Newman, and Steven H. Strogatz, Phys. Rev. E 65, 031915 (2002).


3. The effects of host contact structure on pathogen diversity and strain structure
   C.O'F. Buckee, K. Koelle, M.J. Mustard and S. Gupta, Proc. Natl. Acad. Sci. USA 101, 10839-10844 (2004).


4. Localized contacts between hosts reduce pathogen diversity
   A. Nunes, M.M. Telo da Gama and M.G.M. Gomes, J. Theor. Biol. 241, 477-487 (2006).


5. Epidemic dynamics on an adaptive network
   T. Gross, C.J. Dommar D'Lima and B. Blasius, Phys. Rev. Lett. 96, 208701 (2006).


6. Adaptive coevolutionary networks: a review
   T. Gross and B. Blasius, J. R. Soc. Interface 5, 259-271 (2008).


7. Stochastic models of evolution in genetics, ecology and linguistics
   R. A. Blythe and A. J. McKane, J. Stat. Mech.: Theor. Exp, P07018 (2007).


8. Coherence thresholds in models of language change and evolution: The effects of noise, dynamics, and network of interactions
   J. M. Tavares, M. M. Telo da Gama and A. Nunes, Phys. Rev. E 77, 046108 (2008).

Protein folding, misfolding and aggregation

I. Kinetics & mechanisms of protein folding

The folding of small proteins is two-state, topology-dependent & highly cooperative

For the vast majority of small (i.e., proteins with less than 100 aa), single domain proteins, folding kinetics is typically well-described by a two-state model, suggesting that the only relevant milestons along the folding reaction are the unfolded state, the transition state, and the native fold. Furthermore, the folding rates of these small proteins span a remarkable million-fold range (see top panel in the figure) and are mostly determined by the topology of the native fold, as measured, e.g., by the contact order [1, 2]. A major goal of our simulations-based studies is that of understanding how native topology drives and controls the kinetics and mechanisms of protein folding. For example, we have investigated what is the relative role of local and long-range (LR) interactions in the folding kinetics of different native topologies. In doing so we have found that while LR interactions play a dominant role in determining the folding rates, irrespective of native topology, the dispersion of folding times, observed upon unbalancing the relative contribution of local and LR interactions to the protein's native energy, is critically dependent on the fold's topology [3].

Another characteristic trait of the folding of small proteins is its extraordinary thermodynamic cooperativity, which gives rise to the S-shaped form of protein experimental denaturation curves, and microscopically leads to a bi-modal distribution of molecules over any observable parameter at the so-called transition temperature (see middle panel in the figure). While it is well accepted that such behaviour must rely on highly unusual energetics (involving non-additive multi-body effects, where the formation of one bond favours the formation of additional bonds - in fact this is the reason why the folding transition is termed cooperative - the exact nature of the interactions underlying protein folding cooperativity and the relation of the latter with topology-dependent kinetics remain to be elucidated. In one of our initial efforts to understand protein folding cooperativity we have found that structures with predominantly local contacts (i.e., low CO and alpha-helix rich) are generally associated with a lack of cooperativity in the formation of tertiary bonds during folding [4]. More recently, we have shown that the folding kinetics of lattice polymers modified to render their folding more cooperative are, like those of small proteins, rapid and single-exponential, which underlies the existence a smooth energy landscape (see bottom pannel in the figure) [5].

In a related effort, we have investigated the role of protein sequence as a modulator of the nucleation mechanism. In order to do so we have compared a nucleation scenario that is exclusively driven by native topology with one that is driven by both primary structure (i.e. sequence) and topology. We found that the sequence's finer details tune the formation of the nucleus but its overall position along the protein chain is mostly driven by the fold's topology [6].

References

1. Topology, Stability, Sequence, and Length: Defining the Determinants of Two-State Protein Folding Kinetics
   Kevin W. Plaxco, Kim T. Simons, Ingo Ruczinski, and David Baker, Biochemistry 39, 11177-11183 (2000).


2. Topological complexity, contact order and protein folding rates
   P.F.N. Faísca & R.C. Ball, Journal of Chemical Physics117, 8587-8592 (2002) & Virtual Journal of Biological Physics Research 4 (9), November-1 (2002).


3. The Go model revisited: Native structure and the geometric coupling between local and long-range contacts
   P.F.N. Faísca, M.M. Telo da Gama & A. Nunes, Proteins: Structure, Function and Bioinformatics 60, 712-722 (2005).


4. Folding and form: Insights from lattice simulations
   P.F.N. Faísca, M.M. Telo da Gama & R.C. Ball, Physical Review E 69, 051917, 8 pp. (2004) & and Virtual Journal of Biological Physics Research 7 (11), June-1 (2004).


5. Cooperativity and the origins of rapid, single-exponential kinetics in protein folding
   P.F.N. Faísca & K.W. Plaxco, Protein Sci. 15, 1608-1618 (2006).


6. Nucleation phenomena in protein folding: The modulating role of protein sequence
   Rui D.M. Travasso, Patrícia F.N. Faísca & Margarida M. Telo da Gama, J. Phys.: Cond. Matt. 19, 15 pp. (2007) 285212.

II. The role of intermediate states in folding, misfolding & disease

There is plenty of evidence that intermediates are involved in the onset of the so-called amyloidoses

Folding intermediates are metastable conformations that are transiently populated during the folding reaction. Since the early 1990s, when is was firstly shown that the folding of small proteins follows a two-state kinetic paradigm, intermediates have been typically seen as relatively dull species mainly associated with the folding of large (>120 amino acids) proteins. Their exact role as folding helpers or folding breakers is, however, a matter of substantial debate as there is plentiful evidence for both scenarios. More recently, the recognition that intermediate species play a pivotal role in the onset of amyloidoses (e.g., Alzheimer's and Parkinson's diseases) by participating in the early stage of the aggregation process that leads to amyloid fibrils, has contributed to a renewed interest in these conformers. We are currently using Monte Carlo lattice simulations to explore the role of intermediate species in the folding of large proteins, and we are also using off-lattice Langevin Molecular Dynamics simulations to investigate the genesis and universal character of the early amyloid precursor.

III. The physico-chemical triggers of amyloid formation

While in many cases there is an underlying genetic defect that facilitates, or triggers, the formation of amyloid, the existence of sporadic forms, which are indeed the most common forms of the amyloidogenic diseases, poses the challenge of understanding their molecular basis. In the last few years research in the field of protein misfolding established some sequence properties (e.g. high hydrophobicity, low net charge, ability to form beta strands) as important determinants of protein aggregation. We will use full atomistic Molecular Dynamics to explore the impact of amyloidogenic sequence properties in the unfolding dynamics of several target proteins with medical relevance.