Pair approximation for a model of vertical and horizontal transmission of parasites.

We apply Stochastic Dynamics method for a differential equations model, proposed by Marc Lipsitch and collaborators (Proc. R. Soc. Lond. B 260, 321, 1995), for which the transmission dynamics of parasites occurs from a parent to its offspring (vertical transmission), and by contact with infected host (horizontal transmission). Herpes, Hepatitis and AIDS are examples of diseases for which both horizontal and vertical transmission occur simultaneously during the virus spreading. Understanding the role of each type of transmission in the infection prevalence on a susceptible host population may provide some information about the factors that contribute for the eradication and/or control of those diseases. We present a pair mean-field approximation obtained from the master equation of the model. The pair approximation is formed by the differential equations of the susceptible and infected population densities and the differential equations of pairs that contribute to the former ones. In terms of the model parameters, we obtain the conditions that lead to the disease eradication, and set up the phase diagram based on the local stability analysis of fixed points. We also perform Monte Carlo simulations of the model on complete graphs and Erdös-Rényi graphs in order to investigate the influence of population size and neighborhood on the previous mean-field results; by this way, we also expect to evaluate the contribution of vertical and horizontal transmission on the elimination of parasite. Pair Approximation for a Model of Vertical and Horizontal Transmission of Parasites.

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Stochastic models and dynamic measures for the characterization of bistable circuits

During the last few years, a great deal of interest has risen concerning the applications of stochastic methods to several biochemical and biological phenomena. Phenomena like gene expression, cellular memory, bet-hedging strategy in bacterial growth and many others, cannot be described by continuous stochastic models due to their intrinsic discreteness and randomness. In this thesis I have used the Chemical Master Equation (CME) technique to modelize some feedback cycles and analyzing their properties, including experimental data. In the first part of this work, the effect of stochastic stability is discussed on a toy model of the genetic switch that triggers the cellular division, which malfunctioning is known to be one of the hallmarks of cancer. The second system I have worked on is the so-called futile cycle, a closed cycle of two enzymatic reactions that adds and removes a chemical compound, called phosphate group, to a specific substrate. I have thus investigated how adding noise to the enzyme (that is usually in the order of few hundred molecules) modifies the probability of observing a specific number of phosphorylated substrate molecules, and confirmed theoretical predictions with numerical simulations. In the third part the results of the study of a chain of multiple phosphorylation-dephosphorylation cycles will be presented. We will discuss an approximation method for the exact solution in the bidimensional case and the relationship that this method has with the thermodynamic properties of the system, which is an open system far from equilibrium.In the last section the agreement between the theoretical prediction of the total protein quantity in a mouse cells population and the observed quantity will be shown, measured via fluorescence microscopy.

Ecological modelling for next generation sequencing data

Le tecniche di next generation sequencing costituiscono un potente strumento per diverse applicazioni, soprattutto da quando i loro costi sono iniziati a calare e la qualità dei loro dati a migliorare. Una delle applicazioni del sequencing è certamente la metagenomica, ovvero l'analisi di microorganismi entro un dato ambiente, come per esempio quello dell'intestino. In quest'ambito il sequencing ha permesso di campionare specie batteriche a cui non si riusciva ad accedere con le tradizionali tecniche di coltura. Lo studio delle popolazioni batteriche intestinali è molto importante in quanto queste risultano alterate come effetto ma anche causa di numerose malattie, come quelle metaboliche (obesità, diabete di tipo 2, etc.). In questo lavoro siamo partiti da dati di next generation sequencing del microbiota intestinale di 5 animali (16S rRNA sequencing) [Jeraldo et al.]. Abbiamo applicato algoritmi ottimizzati (UCLUST) per clusterizzare le sequenze generate in OTU (Operational Taxonomic Units), che corrispondono a cluster di specie batteriche ad un determinato livello tassonomico. Abbiamo poi applicato la teoria ecologica a master equation sviluppata da [Volkov et al.] per descrivere la distribuzione dell'abbondanza relativa delle specie (RSA) per i nostri campioni. La RSA è uno strumento ormai validato per lo studio della biodiversità dei sistemi ecologici e mostra una transizione da un andamento a logserie ad uno a lognormale passando da piccole comunità locali isolate a più grandi metacomunità costituite da più comunità locali che possono in qualche modo interagire. Abbiamo mostrato come le OTU di popolazioni batteriche intestinali costituiscono un sistema ecologico che segue queste stesse regole se ottenuto usando diverse soglie di similarità nella procedura di clustering. Ci aspettiamo quindi che questo risultato possa essere sfruttato per la comprensione della dinamica delle popolazioni batteriche e quindi di come queste variano in presenza di particolari malattie.

Sistemi dinamici stocastici su network

L'obiettivo della tesi è studiare la dinamica di un random walk su network. Essa è inoltre suddivisa in due parti: la prima è prettamente teorica, mentre la seconda analizza i risultati ottenuti mediante simulazioni. La parte teorica è caratterizzata dall'introduzione di concetti chiave per comprendere i random walk, come i processi di Markov e la Master Equation. Dopo aver fornito un esempio intuitivo di random walk nel caso unidimensionale, tale concetto viene generalizzato. Così può essere introdotta la Master Equation che determina l'evoluzione del sistema. Successivamente si illustrano i concetti di linearità e non linearità, fondamentali per la parte di simulazione. Nella seconda parte si studia il comportamento di un random walk su network nel caso lineare e non lineare, studiando le caratteristiche della soluzione stazionaria. La non linearità introdotta simula un comportamento egoista da parte di popolazioni in interazioni. In particolare si dimostra l'esistenza di una Biforcazione di Hopf.

Proceedings of the Seventeenth Annual Biochemical Engineering Symposium

This is the seventeenth of a series of symposia devoted to talks by students about their biochemical engineering research. The first, third, fifth, ninth, twelfth, and sixteenth were at Kansas State University, the second and fourth were at the University of Nebraska-Lincoln, the sixth was in Kansas City and was hosted by Iowa State University, the seventh, tenth, thirteenth, and seventeenth were at Iowa State University, the eighth and fourteenth were at the University of Missouri–Columbia, and the eleventh and fifteenth were at Colorado State University. Next year's symposium will be at the University of Colorado. Symposium proceedings are edited by faculty of the host institution. Because final publication usually takes place elsewhere, papers here are brief, and often cover work in progress. ContentsThe Effect of Polymer Dosage Conditions on the Properties of ProteinPolyelectrolyte Precipitates, K. H. Clark and C. E. Glatz, Iowa State University An Immobilized Enzyme Reactor/Separator for the Hydrolysis of Casein by Subtilisin Carlsberg, A. J. Bream, R. A. Yoshisato, and G. R. Carmichael, University of Iowa Cell Density Measurements in Hollow Fiber Bioreactors, Thomas Blute, Colorado State University The Hydrodynamics in an Air-Lift Reactor, Peter Sohn, George Y. Preckshot, and Rakesh K. Bajpai, University of Missouri–Columbia Local Liquid Velocity Measurements in a Split Cylinder Airlift Column, G. Travis Jones, Kansas State University Fluidized Bed Solid Substrate Trichoderma reesei Fermentation, S. Adisasmito, H. N. Karim, and R. P. Tengerdy, Colorado State University The Effect of 2,4-D Concentration on the Growth of Streptanthus tortuosis Cells in Shake Flask and Air-Lift Permenter Culture, I. C. Kong, R. D. Sjolund, and R. A. Yoshisato, University of Iowa Protein Engineering of Aspergillus niger Glucoamylase, Michael R. Sierks, Iowa State University Structured Kinetic Modeling of Hybidoma Growth and Monoclonal Antibody Production in Suspension Cultures, Brian C. Batt and Dhinakar S. Kampala, University of Colorado Modelling and Control of a Zymomonas mobilis Fermentation, John F. Kramer, M. N. Karim, and J. Linden, Colorado State University Modeling of Brettanomyces clausenii Fermentation on Mixtures of Glucose and Cellobiose, Max T. Bynum and Dhinakar S. Kampala, University of Colorado, Karel Grohmann and Charles E. Yyman, Solar Energy Research Institute Master Equation Modeling and Monte Carlo Simulation of Predator-Prey Interactions, R. 0. Fox, Y. Y. Huang, and L. T. Fan, Kansas State University Kinetics and Equilibria of Condensation Reactions Between Two Different Monosaccharides Catalyzed by Aspergillus niger Glucoamylase, Sabine Pestlin, Iowa State University Biodegradation of Metalworking Fluids, S. M. Lee, Ayush Gupta, L. E. Erickson, and L. T. Fan, Kansas State University Redox Potential, Toxicity and Oscillations in Solvent Fermentations, Kim Joong, Rakesh Bajpai, and Eugene L. Iannotti, University of Missouri–Columbia Using Structured Kinetic Models for Analyzing Instability in Recombinant Bacterial Cultures, William E. Bentley and Dhinakar S. Kompala, University of Colorado

Simple model of protein folding kinetics.

A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.

Effects of the environment on the electric conductivity of double-stranded DNA molecules

We present a theoretical analysis of the effects of the environment on charge transport in double-stranded synthetic poly(G)-poly(C) DNA molecules attached to two ideal leads. Coupling of the DNA to the environment results in two effects: (i) localization of carrier functions due to static disorder and (ii) phonon-induced scattering of the carriers between the localized states, resulting in hopping conductivity. A nonlinear Pauli master equation for populations of localized states is used to describe the hopping transport and calculate the electric current as a function of the applied bias. We demonstrate that, although the electronic gap in the density of states shrinks as the disorder increases, the voltage gap in the I-V characteristics becomes wider. A simple physical explanation of this effect is provided.

Gaussian quantum operator representation for bosons

We introduce a Gaussian quantum operator representation, using the most general possible multimode Gaussian operator basis. The representation unifies and substantially extends existing phase-space representations of density matrices for Bose systems and also includes generalized squeezed-state and thermal bases. It enables first-principles dynamical or equilibrium calculations in quantum many-body systems, with quantum uncertainties appearing as dynamical objects. Any quadratic Liouville equation for the density operator results in a purely deterministic time evolution. Any cubic or quartic master equation can be treated using stochastic methods.

Quantum dynamics with stochastic gauge simulations

The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite stochastic time-evolution equations, equivalent to master equations, for many systems including quantum time evolution. The method is illustrated with a variety of simple examples ranging from astrophysical molecular hydrogen production, through to the topical problem of Bose-Einstein condensation in an optical trap and the resulting quantum dynamics.

Charge transport in a quantum electromechanical system

We describe a quantum electromechanical system comprising a single quantum dot harmonically bound between two electrodes and facilitating a tunneling current between them. An example of such a system is a fullerene molecule between two metal electrodes [Park et al., Nature 407, 57 (2000)]. The description is based on a quantum master equation for the density operator of the electronic and vibrational degrees of freedom and thus incorporates the dynamics of both diagonal (population) and off diagonal (coherence) terms. We derive coupled equations of motion for the electron occupation number of the dot and the vibrational degrees of freedom, including damping of the vibration and thermo-mechanical noise. This dynamical description is related to observable features of the system including the stationary current as a function of bias voltage

Population inversion of a driven two-level system in a structureless bath

We derive a master equation for a driven double quantum dot damped by an unstructured phonon bath, and calculate the spectral density. We find that bath-mediated photon absorption is important at relatively strong driving, and may even dominate the dynamics, inducing population inversion of the double-dot system. This phenomenon is consistent with recent experimental observations.

Relational time for systems of oscillators

Using an elementary example based on two simple harmonic oscillators, we show how a relational time may be defined that leads to an approximate Schrodinger dynamics for subsystems, with corrections leading to an intrinsic decoherence in the energy eigenstates of the subsystem.

Spin-detection in a quantum electromechanical shuttle system

We study the electrical transport of a harmonically bound, single-molecule C-60 shuttle operating in the Coulomb blockade regime, i.e. single electron shuttling. In particular, we examine the dependance of the tunnel current on an ultra-small stationary force exerted on the shuttle. As an example, we consider the force exerted on an endohedral N@C-60 by the magnetic field gradient generated by a nearby nanomagnet. We derive a Hamiltonian for the full shuttle system which includes the metallic contacts, the spatially dependent tunnel couplings to the shuttle, the electronic and motional degrees of freedom of the shuttle itself and a coupling of the shuttle's motion to a phonon bath. We analyse the resulting quantum master equation and find that, due to the exponential dependence of the tunnel probability on the shuttle-contact separation, the current is highly sensitive to very small forces. In particular, we predict that the spin state of the endohedral electrons of N@C-60 in a large magnetic gradient field can be distinguished from the resulting current signals within a few tens of nanoseconds. This effect could prove useful for the detection of the endohedral spin-state of individual paramagnetic molecules such as N@C-60 and P@C-60, or the detection of very small static forces acting on a C-60 shuttle.

Quantum Information Processing with spin systems: from modeling to possible implementations

The physical implementation of quantum information processing is one of the major challenges of current research. In the last few years, several theoretical proposals and experimental demonstrations on a small number of qubits have been carried out, but a quantum computing architecture that is straightforwardly scalable, universal, and realizable with state-of-the-art technology is still lacking. In particular, a major ultimate objective is the construction of quantum simulators, yielding massively increased computational power in simulating quantum systems. Here we investigate promising routes towards the actual realization of a quantum computer, based on spin systems. The first one employs molecular nanomagnets with a doublet ground state to encode each qubit and exploits the wide chemical tunability of these systems to obtain the proper topology of inter-qubit interactions. Indeed, recent advances in coordination chemistry allow us to arrange these qubits in chains, with tailored interactions mediated by magnetic linkers. These act as switches of the effective qubit-qubit coupling, thus enabling the implementation of one- and two-qubit gates. Molecular qubits can be controlled either by uniform magnetic pulses, either by local electric fields. We introduce here two different schemes for quantum information processing with either global or local control of the inter-qubit interaction and demonstrate the high performance of these platforms by simulating the system time evolution with state-of-the-art parameters. The second architecture we propose is based on a hybrid spin-photon qubit encoding, which exploits the best characteristic of photons, whose mobility is exploited to efficiently establish long-range entanglement, and spin systems, which ensure long coherence times. The setup consists of spin ensembles coherently coupled to single photons within superconducting coplanar waveguide resonators. The tunability of the resonators frequency is exploited as the only manipulation tool to implement a universal set of quantum gates, by bringing the photons into/out of resonance with the spin transition. The time evolution of the system subject to the pulse sequence used to implement complex quantum algorithms has been simulated by numerically integrating the master equation for the system density matrix, thus including the harmful effects of decoherence. Finally a scheme to overcome the leakage of information due to inhomogeneous broadening of the spin ensemble is pointed out. Both the proposed setups are based on state-of-the-art technological achievements. By extensive numerical experiments we show that their performance is remarkably good, even for the implementation of long sequences of gates used to simulate interesting physical models. Therefore, the here examined systems are really promising buildingblocks of future scalable architectures and can be used for proof-of-principle experiments of quantum information processing and quantum simulation.