Lectures will be held in presence only. The location is
Aula: Careri
Schedule: Tuesday 10-12 Thursday 12-14
Ricevimento: Lun 15-16 - Stanza 504 Ed. Fermi (scrivere per appuntamento / write an email for appointment)
PLEASE NOTE: all students are kindly requested to send an email to irene.giardina@uniroma1.it to subscribe to the course mailing list.
Detailed Program and Lectures
in black: what has been already done up to the present day
in grey: what will be done in the next lectures
in yellow: last year program that will not be done this year
in pink: new subjects
in green: reference bibliography
video-recordings can be found in the lectures of past academic years, see lectures 21-22 or 20-21 (link above)
PART I
THEORETICAL BACKGROUND: Stochastic equations and diffusion processes
Lecture 1 26/2/2026
Introduction
Diffusion processes I:
Random walks: diffusion coefficient, drift velocity, distribution p(X,t), standard diffusion vs anomalous diffusion; Langevin equation
Lecture 2 3/3/26
Langevin Equation, FDT theorem, Stores-Einstein relation
Overdamped limit, Wiener process
Diffusion Equation, Fick's law
Diffusion equation, derivation.
Lecture 3 5/3/26
Metabolism in bacteria.
Langevin equation in presence of external forces. Fokker Planck equation, Nerst-Planck formula and membrane potential.
Equilibrium limit
Ornstein Uhlenbeck model - effect of confinement
Lecture 4 - 10/3/26
Ornstein Uhlenbeck model (stochastic hamronic oscillator)- mean-square displacement - different dynamical regimes
Arrhenius law, quick derivation
For a more rigorous approach: introduction to the problem as a first passage problem (Gardiner, chap. 5 & 3.6)
Summary and final comments on noise vs inertia vs force
Biblio:
Zwanzig, Non-equilibrium statistical mechanics (Oxford University Press, 2001), chap.3
Gardiner, Handbook of stochastic methods (Springer, 1997) (Fokker-Planck eq. & Arrhenius (chap. 5, cap 3.6)
Nelson, chap. 4 (biological examples)
H. Berg, Random walks in biology (Princeton UP, 1993)
Bouchaud JP & Georges A. "Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications." Physics reports 195.4-5 (1990): 127-293. (introduction)
Lecture 5 - 12/3/26
BACTERIAL CHEMOTAXIS
http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/home.htm
Chemotaxis - Introduction to the problem
Experiments of Adler - phenomenology
Estimate of thermal effects
Experiments of Berg and Brown and characterization of run & tumble
Phenomenology of chemotaxis
papers Adler 1973; Berg (general) 2005; Berg&Brown 71 (see Dropbox link below); Bialek chap. 4
A few words on bacterial movement on mesoscale: Navier-Stokes and Reynolds number (scallop theorem - flagellar motors NOT DONE - see Di Leonardo in Biophysics)
Berg & Purcell: chemoreception and signal vs noise discrimination in bacteria: What does `counting' mean
Computation of T: signal integration time necessary to discriminate the gradient of the nutrient (using diffusion arguments and assuming chemoreceptor=perfect counter)
Computation of T using correlation functions of the arrival process of molecules on the receptor
Lecture 6 - 17/3/26
Connection between the probability of occupation of the receptor and arrival rate of molecules
Bialek chap 4 (simplified computation)
Berg & Purcell, articolo 1977 (physics of chemoreception)
Signal-transduction pathway: general description and role of the enzimes
Lecture 7 - 19/3/26
Biochemical signal amplification - MWC two model state to explain regulation of flagellar motors (equilibrium computation). Bialek cap 4 - Berg 2005
Dynamical equations for chemotaxis: model in d=1
Bialek chap 4, problems 49-50 (model 1d)
Lecture 8 24/3/26
Coarse grained approach - Keller-Segel model (pdf file on keller-segel)
Tumbling rate as a Response function (linear approximation, comparison to dynamical response in physics)
Review M. Cates (non-linear response kernel)
Relevant papers quoted above can be found in the directory
https://www.dropbox.com/sh/domuijulp605en9/AAA5-M9aPJMlvSygdIweGg4Xa?dl=0
Interesting websites
http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/home.htm
http://www.rowland.harvard.edu/labs/bacteria/movies/index.ph
PHOTORECEPTION
Lecture 9 26/3/26
Photoreception. First experiments on photoreception at low light intensity
(Hechte-Pirenne). Experiments on single photoreceptor. Intracellular current and response to single photons.
Counting and signal discrimination - optimal threshold values.
From individual photoreceptors back to Hechte-Pirenne. Dark noise and SNR.
Response at high intensity: superposition of independent events and linear response.
Input-output relations: membrane potential in response to a modulation of light intensity.
Lecture 10 28/3/2026
Response at high intensity:
Input-output relations: Computation of the average potential, of fluctuations, transfer function and SNR.
Comparison with experiments.
Lecture 11 30/3/26
Biochemical amplification I - general scheme
Rhodopsin isomerization. BO approximation and retinol absorption spectrum. Simple model with quadratic potential. Fluorescence red-shift. Coupling retinol-opsin---> beyond BO.
Lecture 12 31/3/26
Enzymatic reactions. Michaelis-Menten formula.
Complete enzymatic reactions for cGMP with production and degradation enzymes. Comments on the efficient regime (return time, gain, amplification factor).
Lecture 13 9/4/26
Electric current through an ionic channel.
Statistics of open/closed channels (dynamical model). Connection with the concentration of cGMP (MWC model for the opening statistics of ionic channels).
Lecture 14 23/4/26
Photoreception: final considerations. The problem of filtering: transmission of the signal from photoreceptors to bipolar cells. Optimal non-linear static filter - experiments on bipolar cells.
linear dynamic filters (not done)
All photoreception lectures/material can be found in Bialek chap. 2
PART II
COLLECTIVE PHENOMENA AND INTERACTING SYSTEMS
THEORETICAL BACKGROUND IN STATISTICAL PHYSICS
Brief reminder of Statistical Mechanics of equilibrium processes: canonical ensemble - average quantities -
The Ising model as an archetype of cooperative behavior: order parameter, correlation functions (simple and connected)
Mean-field solution, qualitative description of the ordering transition.
Mean-field free energy; free energy landscape vs transition vs ergodicity breaking.
Lecture 15 21/4/26
Ising model:
Connection with the Gibbs potential. Phase transitions as points of maximal response (divergence of the susceptibility)
Response functions - fluctuation dissipation theorems.
Critical exponents and universality classes.
Computing correlations via response functions.
Lecture 16 23/4/26
Correlation length and correlated regions
Scale free behavior and long range collective behavior.
Models with continuous symmetry. Goldstone modes. Power-laws vs criticality vs soft modes.
Texts: any text on critical phenomena
eg: Binney et al, Critical Phenomena, Oxford Science Publications - chap 1-2-3-6 (only the sections on the Ising model)
Huang, Statistical Mechanics, Wiley, Chap. 14-16-17
Toda-Kubo-Saita Statistical Physics chap.4
NEURAL NETWORKS
Lezione 17 28/4/26
Neurons: ionic channels and single neuron dynamics with activation channels
Neuroni II: il ruolo dei canali calcio voltage-gated - problema della regolazione dei parametri e ruolo del calcio
Example with single leak channel
Role of deactivation and Hodgkin-Huxley equations
Lezione 18 30/4/25
Neural networks: memory retrieval and Hopfield model
Maximum Entropy Models: general approach and derivation of the ME distribution
Lezione 19 5/5/26
Application to neural networks: single-site MEM
MEM: role of correlations - comparison with experimental data.
Learning dynamics and update of synapses - plasticity - comparison with experimental data.
References: Bialek chap. 5.2 (canali ionici e dinamica di singolo neurone - NO), chap. 5.4 (Hopfield and maximum entropy models)
PROTEINS
Proteins: simple modeling of a heteropolymer - link with spin glasses: frustration/complex landscape vs funnel landscape.
Problem of protein folding - Go-models and folding mechanism. NO
Protein design - minimal model with 2 aminoacids: exhaustive computations of compact folded structures - connection between sequences and structures.
Lezione 20 7/5/26
Data analysis of natural protein families: relevance of one point and two points statistics.
Proteins: statistical inference - Maximum Entropy models.
References: Bialek chap. 5.1
Lecture by Prof. Zamponi su Generative Models for protein evolution and design.
Slides at the link below:
https://drive.google.com/drive/folders/1E_MkmnI8WykjT7QpArLSXUcZk6GLpR3K?usp=sharing
ACTIVE MATTER
Lezione 21 12/5/26
Active matter: definition - role of motility - connection with the dynamics of passive particles.
Simple models of active particles: Active Brownian Particles;
Active Ornstein Uhlenbeck.
Lezione 22 14/5/26
Vicsek model - general equations.
Vicsek model - d=2.
Kinetic transition to the polar state - comparison with equilibrium orientational models (XY) - predictions.
Lezione 23 19/5/26
Phase diagram, role of the control parameters (noise and density).
Comparison with experimental results on various kinds of active matter: giant density fluctuations in granular active matter.
Lezione 24 21/5/26
Flocks: topological interactions - long-range correlations. Maximum Entropy approach to flocks.
Propagation of information and dispersion relation.
Lezione 25 26/5/26
Swarms of midges: critical behavior and dynamic scaling
What does it mean to be critical at finite size
Dynamic scaling hypothesis
Scaling in the correlation functions and new dynamic critical exponent
Texts:
Vicsek et al, Phys Rev Lett 75 (1995): 1226.
http://rocs.hu-berlin.de/complex_sys_2015/resources/Seminarpapers/Vizcek_Model_2006.pdf
Ginelli F., The Physics of the Vicsek model:
http://link.springer.com/article/10.1140/epjst/e2016-60066-8
nota per gli studenti: articolo di rassegna sul Vicsek model, molto leggibile. c'e' un errore pero' nel calcolo della linea critica rumore/densita' (vedere i miei appunti per il calcolo corretto)
Narayan et al. Science 2007, https://www.researchgate.net/profile/Sriram_Ramaswamy/publication/6223264_Long-lived_giant_number_fluctuations_in_a_swarming_granular_nematic/links/00b7d529d79e592a62000000.pdf
Deseigne et al, PRL 2010
http://iramis.cea.fr/spec/Docspec/articles/s10/035/public/publi.pdf
sugli uccelli,
www.cobbs.it
PNAS 2012 http://www.pnas.org/content/109/13/4786.full
Lezioni - NON FATTE
Flusso di informazione e rappresentazioni efficienti, trasmissione di informazione e risposta collettiva - ruolo dell'entropia come informazione - concetto di efficienza in biologia
esempi biologici
articoli su argomenti interessanti (Infotaxis - neural encoding etc)
problema generico dell'inferenza statistica