Lectures will be held in presence only. The location is
Aula: Careri
Schedule: Tuesday 8:30-10 Friday 14-16
Ricevimento: Lun 14:30-16 - Stanza 504 Ed. Fermi (scrivere per appuntamento / write 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 (link above)
PART I
THEORETICAL BACKGROUND: Stochastic equations and diffusion processes
Lecture 1 28/2/2025
Introduction
Diffusion processes I:
Random walks: diffusion coefficient, drift velocity, distribution p(X,t), standard diffusion vs anomalous diffusion; Langevin equation
Lecture 2 4/3/25
Langevin Equation, FDT theorem, Stores-Einstein relation
Overdamped limit, Wiener process
Lecture 3 7/3/25
Diffusion Equation, Fick's law
Diffusion equation, derivation.
Metabolism in bacteria.
Langevin equation in presence of external forces. Fokker Planck equation, Nerst-Planck formula and membrane potential.
Lecture 4 - 11/3/25
Equilibrium limit
Ornstein Uhlenbeck model - mean-square displacement - different dynamical regimes
Lecture 5 - 14/3/25
End of discussion of OU model
Arrhenius law, quick derivation - introduction to the problem as a first passage problem
Summary and final comments on noise vs inertia vs force
Notes on OU model
Notes on Arrhenius law
Biblio:
Zwanzig, Non-equilibrium statistical mechanics (Oxford University Press, 2001), chap.3
Gardiner, Handbook of stochastic methods (Springer, 1997) (Fokker-Planck eq. e Arrhenius (cap. 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)
BACTERIAL CHEMOTAXIS
http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/home.htm
Lecture 6 - 18/3/25
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)
Lecture 7 - 21/3/25
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
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 8 - 25/3/25
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
Lecture 9 28/3/25
Dynamical equations for chemotaxis: model in d=1 (continue from 25/3)
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)
Bialek chap 4, problems 49-50 (model 1d)
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 10 1/4/25
Photoreception. First experiments on photoreception at low light intensity
(Hechte-Pirenne). Experiments on single photoreceptor. Intracellular current and response to single photons.
Lecture 11 4/4/25
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 12 8/4/2025
Response at high intensity:
Input-output relations: Computation of the average potential, of fluctuations, transfer function and SNR.
Comparison with experiments.
Lecture 13 11/4/25
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.
Enzymatic reactions. Michaelis-Menten formula.
Lecture 14 15/4/25
Complete enzymatic reactions for cGMP with production and degradation enzymes. Comments on the efficient regime (return time, gain, amplification factor).
Electric current through an ionic channel.
******EASTER VACATIONS ********
Lecture 15 23/4/25
Statistics of open/closed channels (dynamical model). Connection with the concentration of cGMP (MWC model for the opening statistics of ionic channels).
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
Lecture 16 24/4/2024
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)
Lecture 17 29/4/204
Ising model:
Mean-field solution, qualitative description of the ordering transition.
Mean-field free energy; free energy landscape vs transition vs ergodicity breaking.
Connection with the Gibbs potential. Phase transitions as points of maximal response (divergence of the susceptibility)
Critical exponents and universality classes.
Lecture 18 8/5/2024
Response functions - fluctuation dissipation theorems.
Computing correlations via response functions:
Lecture 19 9/5/2025
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 20 13/5/25
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
Lezione 21 16/5/25
Example with single leak channel
Role of deactivation and Hodgkin-Huxley equations
Neural networks: memory retrieval and Hopfield model
Lezione 22 20/5/25
Learning dynamics and update of synapses - plasticity - comparison with experimental data.
Maximum Entropy Models: general approach and derivation of the ME distribution
Application to neural networks: single-site MEM
Lezione 23 23/5/25
MEM: role of correlations - 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 24 27/5/24
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
ACTIVE MATTER
Lezione 25 30/5/24
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 26 3/6/25
Vicsek model - general equations.
Vicsek model - d=2.
Kinetic transition to the polar state - comparison with equilibrium orientational models (XY) - predictions.
Lezione 27 10/6/25
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. Flocks: topological interactions - long-range correlations. Maximum Entropy approach to flocks.
Propagation of information and disperision relation.
Lezione 28 11/6/24
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