We work on problems at the interface of biology and physics, with emphasis on populations, evolution, and stochastic dynamics.

Stochastic gene expression in the fly retina

Measuring cell identity in noisy biological systems

Stochastic switches and selection in bacterial populations

Population genetics of simple sequence repeats

Lineage dynamics in aging and branching processes

Current Projects

We combine theoretical biophysical approaches with experiments and bioinformatics to explore systems that exhibit complex, population-level phenomena. Current experiments in the lab focus on population dynamics of microorganisms in fluctuating conditions.  

Theoretical work is focused on statistical mechanical descriptions of population- and evolutionary-dynamics.  Several projects address the evolution of stochastic switches in bacteria, as well as bacterial genome evolution.  We also collaborate with other labs to understand stochastic gene expression in higher organisms, including plants and flies.

Stochastic switches – What they are, and where to find them.

Horizontal gene transfer in bacteria

Stochastic switches are genetic circuits that allow cells to maintain or switch their phenotypic state spontaneously, without directly sensing their environment. Surprisingly, this behavior can be advantageous over sensor-mediated responses in certain environments.  For example, if environmental changes are rare, or if certain changes are particularly catastrophic, stochastic switching can be selected over sensing.  The cost of sensing and response (or responsive switching) is an important parameter that determines which adaptive mode is preferred.

Pathogenic bacteria and fungi exhibit a multitude of stochastic switching mechanisms – often known as phase variation mechanisms – many of which are critical for pathogenicity and evasion of the host’s immune response.  Stochastic switches are also intimately involved in antibiotic persistence. 

Signature of stochastic switching – How to tell when bacteria are doing it.

Cartoon: Stochastic vs. Responsive Switching

The cartoon shows a growing population composed of cells in different phenotypic states (colors) in a fluctuating environment. The adapted phenotype in each environment is shown with matching color.  The population vector, p(t), specifying the number of cells in each phenotype,  is governed by differential equations that can be solved analytically in different regimes (fast or slow fluctuations).  These solutions allow the optimal phenotypic switching rates in fluctuating environments to be determined. Under slow fluctuations, we have shown that the amount of information (i.e. Shannon information) present in the fluctuating environment directly determines the long-term growth rate of stochastic switching organisms.  In this regime, the more accurately an organism’s own switching rates mimic the statistics of its environment, the better adapted it is over long time-scales.  This behavior undergoes a sharp transition the fluctuation frequency increases,

Is there a general approach to detect stochastic switching?  Alternatively, is it possible to distinguish stochastic from responsive switching?  Because problems concerning stochastic gene expression are fundamentally quantitative, classical genetics approaches without a major input from theory are likely to mislead.  We have developed a theoretical framework, based on a statistical mechanical analogy, which provides a rigorous basis for quantifying switches using population-level measurements of individual histories (or cell lineages).  Our experiments apply this approach to bacterial and fungal populations in the lab.

As explained on the right, stochastic switches lead to a large variance in reproductive rates (fitness) between individual histories.  By following dividing cells under the microscope in fluctuating conditions, one can track individual lineages, and use cell divisions as a reporter of fitness. This allows the historical fitness variance to be measured directly from lineage trees. Using simulations, we showed such measurements reliably recapitulate the true fitness variance, and thus allow stochastic switches to be distinguished from responsive ones, by simply “watching” cell divisions.

Our framework maps phenotypic states onto single-cell lineages. To each lineage, or individual history, is associated a historical fitness (see Figure). This mapping gives meaning in populations to thermodynamic quantities, like free energy (long-term growth rate), enthalpy (average historical fitness), and heat capacity (variance of historical fitness). Population dynamics of stochastic switching is dominated by selective sweeps, which are the main determinant of survival.  Sweeps lead to a large variance of historical fitness between individual histories. Responsive switches do not rely on sweeps – instead, each individual maintains an adapted state, and the historical fitness variance is therefore small. The historical fitness variance detects stochastic switching:

stochastic (theory)

responsive (theory)


The figure presents theoretical predictions and comparison with simulations of stochastic population dynamics. A pronounced peak is predicted for stochastic switches, at an environmental fluctuation

historical fitness variance


τ  (environmental duration, hr)

period that is significantly longer than the generation time.  Measurements using simulated lineage trees recapitulate the theoretical predictions.

History Formulation of Population Dynamics

Stochastic gene expression in the fly retina – Quantifying complex phenotypes.

Color vision depends on specialized light-sensing rhodopsin (rh) proteins. Six different rhodopsin proteins are used in the compound eye of the fruit fly, each tuned to measure light of different wavelengths. The R7 photoreceptor cells in the center of each of the ~800 ommatidial bundles present in a fly retina, express one and only one type of rhodopsin protein. The type of rhodopsin expressed depends on a stochastic process that leads to a random choice: approximately 30% of R7 cells express rh3, while the other 70% express rh4. A complex genetic network underlies this bistable stochastic behavior, and the behavior of its components are the subject of intensive study.

We have developed image analysis algorithms that allow every aspect of the system to be quantified in molecular detail. In a collaboration with Robert Johnston and Claude Desplan at NYU, we analyze whole-retina confocal microscopy images, using our algorithms to automatically recognize each ommatidial bundle and its constituent photoreceptors across the retina. This allows rhodopsin levels to be automatically determined within specific cell types, allowing complex stochastic phenotypes to be studied in quantitative detail.

Single ommatidia are computationally extracted from confocal image stacks.

Whole-retina confocal imaging (3D view)

Computational recognition of ommatidia

Each ommatidium recognized by the algorithm is shown in a different color.

Photoreceptor Recognition

Within each ommatidium, photoreceptors are computationally identified and labelled.


Leibler S and Kussell E.  Individual histories and selection in heterogeneous populations. PNAS 107:13183–13188 (2010).

Natalie de Souza.  Research Highlight: Learning from history. Nature Methods 7: 672–673 (2010).


Kussell E and Leibler S.  Phenotypic diversity, population growth, and information in fluctuating environments.

Science 309: 2075-2078 (2005).


Johnston R, Otake Y, Sood P, Vogt N, Behnia R, Vasiliauskas D, McDonald E, Xie B, Koenig S, Wolf R, Cook T, Gebelstein B, Kussell E, Nagakoshi H, Desplan C.  Interlocked feedforward control cell-type specific expression in the Drosophila eye.

Cell, 145:956-968 (2011).

Sood P, Johnston R and Kussell E. Stochastic de-repression of rhodopsins in single photoreceptors of the fly retina.       PLoS Computational Biology, 8(2):e1002357 (2012).