# Research

## Areas of Interest:

### 1. Physics of Living Systems

• Effects of variability at the single-cell level on population growth
• Coupling of gene expression and cell cycle
• Evolutionary origin of the asymmetric segregation of protein during cell division

### 2. Physics of disordered materials

• Plasticity of disordered solids
• Rheology of yield-stress materials
• Mechanical behaviors of jammed packing

### Effects of cell-to-cell variability on population fitness

Establishing a quantitative connection between the population growth rate and stochastic growth of single cells is a prerequisite for understanding evolutionary dynamics of microbes. However, existing theories fail to account for the experimentally observed correlations between mother-daughter generation times that are unavoidable when cell size is controlled for, which is essentially always the case. We study population-level growth in the presence of cell size control and corroborate our theory using experimental measurements of single-cell growth rates. We derive a closed formula for the population growth rate and demonstrate that it only depends on the single-cell growth rate variability, not other sources of stochasticity. Our work provides an evolutionary rationale for the narrow growth rate distributions often observed in nature: when single-cell growth rates are less variable but have a fixed mean, the population will exhibit an enhanced population growth rate as long as the correlations between the mother and daughter cells’ growth rates are not too strong.

Relevant publications:

1. Lin J, Amir A. Population growth with correlated generation times at the single-cell level. (2018).

2. Ho P-Y, Lin J, Amir A. Modeling cell size regulation: From single-cell level statistics to molecular mechanisms and population level effects. Annual Review of Biophysics (2018).

3. Lin J, Amir A. The Effects of Stochasticity at the Single-Cell Level and Cell Size Control on the Population GrowthCell Systems (2017)

### Coupling gene expression to cell volume growth: understanding the universal proportionality between gene expression and cell volume

Many experiments show that the numbers of mRNA and protein are proportional to the cell volume in growing cells. However, models of stochastic gene expression often assume constant transcription rate per gene and constant translation rate per mRNA, which are incompatible with these experiments. Here, we construct a minimal gene expression model to fill this gap. Assuming ribosomes and RNA polymerases are limiting in gene expression, we show that the numbers of proteins and mRNAs both grow exponentially during the cell cycle and that the concentrations of all mRNAs and proteins achieve cellular homeostasis; the competition between genes for the RNA polymerases makes the transcription rate inde- pendent of the genome number. Furthermore, by extending the model to situations in which DNA (mRNA) can be saturated by RNA polymerases (ribosomes) and becomes limiting, we predict a transition from exponential to linear growth of cell volume as the protein-to-DNA ratio increases.

Relevant publications:

1. Lin J, Amir A. Homeostasis of protein and mRNA concentrations in growing cellsNature Communications (2018)

### Evolutionary origin of the asymmetric segregation of protein during cell division

Asymmetric segregation of key proteins at cell division—be it a beneficial or deleterious protein—is ubiquitous in unicellular organisms and often considered as an evolved trait to increase fitness in a stressed environment. Here, we provide a general framework to describe the evolutionary origin of this asymmetric segregation. We compute the population fitness as a function of the protein segregation asymmetry a, and show that the value of a which optimizes the population growth manifests a phase transition between symmetric and asymmetric partitioning phases. Surprisingly, the nature of phase transition is different for the case of beneficial proteins as opposed to deleterious proteins: a smooth (second order) transition from purely symmetric to asymmetric segregation is found in the former, while a sharp transition occurs in the latter. Our study elucidates the optimization problem faced by evolution in the context of protein segregation, and motivates further investigation of asymmetric protein segregation in biological systems.

Relevant publications:

1. Lin J, Min J, Amir A. Optimal segregation of proteins: phase transitions and symmetry breaking. Physics Review Letters (2019)

### Plasticity of disordered solids

Failure and flow of amorphous materials are central to various phenomena including earthquakes and landslides. There are accumulating evidences that the yielding transition between the flowing and solid phase is a critical phenomenon, but the associated exponents and their scaling relations are not well understood. In contrast to crystals, the elementary rearrangements in amorphous solids are those local irre- versible rearrangements of a few particles, called shear transformations. A central feature that distinguishes the yielding transition from other non-equilibrium phase transitions is the long range and anisotropic interaction between shear transformations. I have shown that this peculiar interaction leads to a singular density of shear transformations, $$P(x)\sim x^{\theta}$$ at small $$x$$, where $$x$$ is a local measure of stability, namely, the extra stress one needs to add locally to reach the elastic instabilities. We denote such a singular distribution as a pseudo gap. It turns out that the plastic avalanche rates, i.e., number of avalanche per unit strain, during quasi-static shear is not proportional to system size implies the existence of a finite pseudo gap exponent. In the flowing phase above and at the yield stress, I have constructed a scaling description of the yielding transition of soft amorphous solids at zero temperature.

On the other hand, in the solid phase, the pseudo gap also plays a significant role as one increases the shear stress slowly after a quench. I have shown that the entire solid phase below the yield stress is critical, and plasticity always involves system-spanning events because of the finite pseudo gap exponent.  Eventually, a mean field description of plastic flow in amorphous solids are proposed and solved analytically that captures the broad distribution of mechanical noise generated by plasticity, whose behavior is related to biased Levy flights near an absorbing boundary.