The Baryon Field

biochemistries:

“Much of the variation in doubling times seems to reflect differences in the phase of an as-yet-unidentified internal oscillator, rather than stochastic factors, such as ‘noisy’ gene expression”

This line grabbed my attention in a magazine highlight in Nature today - Defiant daughters and coordinated cousins - regarding a letter in the same issue, Sandler et al. ‒ Lineage correlations of single cell division time as a probe of cell-cycle dynamics.

Gene expression noise arises from processes like transcriptional bursting (stochastic initiation of mRNA synthesis), and the degree of noise present can in itself modulate cellular processes. I’ve previously discussed the example of noise “enhancing oscillation and entrainment bandwidth“ in the NF-κB pathway.

Sandler et al. looked closely at normalising conditions across generations of a cell line, and were perplexed at the continued observation that parent cells were less similar to their daughters than cousins were to one another.

_{Intuitively, if one cell in a population has more or less of a particular component than the population average, levels of that component will tend to deviate in the same direction in that cell’s daughters. Owing to subsequent random fluctuations, these deviations will decorrelate over time, such that compositions of genetically identical cells should become less correlated with each generation. Contrary to this expectation, observations indicate that the time it takes one cell to become two — its doubling time — can show a stronger correlation between cousins than between mother–daughter pairs. It has been unclear whether this surprising result reflects the fact that cells born at different times are exposed to different conditions, just as the teenagers of the 1980s behaved differently from those of the 1990s. But the current study demonstrates that, under tightly controlled conditions, the phenomenon persists.}

The study authors propose that a mysterious factor oscillates periodically in cells, such that cousins are born in a similar phase, but mothers and daughters usually are not. It seems tenuous on first reading, but in fact many oscillatory species are actually predicted theoretically in advance of experimental confirmation (there don’t seem to be systematic search efforts for oscillatory species in biochemistry).

As well as using chaos theory, the group reanalysed published data for cyanobacteria, in which growth is coupled to circadian rhythms, meaning they could quantify quite neatly the similarity between family members.

…chaos theory?

For a little context here, the paper has been published from the lab of Nathalie Balaban, who as I saw a researcher on Twitter note just now, has 20 papers and 5,700 citations to her name (via Google Scholar). Housed in the Racah Institute of Physics at The Hebrew University, Jerusalem, her previous work has included antibiotic resistance, genetic toggle switches, and focal adhesion assembly.

Most recently, foreshadowing this paper, she presented “Deterministic vs. Stochastic Variability in the Mammalian Cell Cycle” at a symposium, the proceedings of which were archived in Biophysical Journal. It describes the present work, which “may help predict and, eventually, control cell-to-cell heterogeneity in various systems, such as cancer cells under treatment.”

In this 2011 seminar, she describes sources of variability (heterogeneity), and microfluidics work featured in Science in 2004 that led to systematic observation of antibiotic persistence (PNAS 2008).

Balaban goes on to highlight a couple of thoretical biology papers from 2005, alongside Lachmann and Jablonka’s 1996 “The Inheritance of Phenotypes: an Adaptation to Fluctuating Environments“ in which models are discussed in very general terms, for cyclic biological processes with inheritance…

_{…for the evolution of rates of spontaneous and induced heritable phenotypic variations in a periodically fluctuating environment with a cycle length between two and 100 generations. For the simplest case, the optimal spontaneous transition rate between two states is approximately 1/n (where n is the cycle length). It is also shown that selection for the optimal transition rate under these conditions is surprisingly strong. When n is small, this means that the heritable variations are produced by non-classical inheritance systems, including non-DNA inheritance systems. Thus, it is predicted that in genes controlling adaptation to such environments, non-classical genetic effects are likely to be observed. We argue that the evolution of spontaneous and induced heritable transitions played an important role in the evolution of ontogenies of both unicellular and multicellular organisms. The existence of a machinery for producing induced heritable phenotypic variations introduces a ”Lamarckian” factor into evolution.}

"If you have environments that are bad, and good, bad, and good, with typical time scales,” she told the audience, “the optimised population will be growing a lot more in the favourable times”.

_{"So in principle, just by measuring the time scales of growing and non-growing of your population, you can infer which environment is optimal.”}

The 2005 papers came from Kussel et al. (in Genetics, 2005) and a related piece in Science (shown on the slide above), but are more backstory on the lab’s directions than preamble to this paper in particular. The Science paper considered ‘responsive’ vs. stochastic phenotype switching, “without any direct sensing of the environment”. In the mathematical modelling approach, a growth equation was applied over a population vector, calculating the Lyapunov exponent (Λ) for each, a difficult computation and thus approximated through the eigenvector.

An assumption that the population would settle at equilibrium between changes in environment allowed the authors to take as given that the population vector x(t) in environment j

_{…x(t) will eventually point essentially in the direction of the top eigenvector of the matrix Aj. Upon a change of environment from j to i, there will be a delay time, Tij, during which the population’s composition changes from its old structure (top eigenvector of Aj) to its new one (top eigenvector of Ai).}

_{Thereafter, the population will grow at a rate given by the top eigenvalue of the matrix Ai , λ1(Ai). This simple picture allows for computation of the Lyapunov exponent in the limit of long durations. We find, in this limit, that L depends only on mean environmental durations ti and transition probabilities bij and is independent of other characteristics of environmental fluctuations (for example, the variance of environmental durations). The biological implication is that a stochastic-switching organism is buffered against changes in the distribution of environmental variations, provided its environment does not fluctuate too quickly.}

The outcome of following this model through was that stochastic switching leads to a “diversity cost”, while a “sensing cost“ is seen for responsive switching.

_{In contrast to responsive switching, which senses a new environment, the stochastic-switching organism cannot overcome the entropy of its environment.}

_{The appearance of Ienv [a term for the entropy, or information content, of the fluctuating environment] explicitly in the optimal long-term growth rate of a population points to possibly deeper connections between the fields of population biology and information theory.}

In a footnote they write that: in certain cases, not switching at all, i.e., remaining in a single phenotype at all times, is better than switching phenotype. This can be the case, for example, when environments change very rapidly. When environments last sufficiently long, the solution given in the text is the optimum.

The current article in Nature takes this idea of connections between population biology and information theory, and applies it wholeheartedly to pedigree analysis of cell cycle duration — the sources of variability in which remain unresolved.

_{A central question is whether the variance of the observed distribution originates from stochastic processes, or whether it arises mostly from a deterministic process that only appears to be random. A surprising feature of cell-cycle-duration inheritance is that it seems to be lost within one generation but to be still present in the next generation, generating poor correlation between mother and daughter cells but high correlation between cousin cells. This observation suggests the existence of underlying deterministic factors that determine the main part of cell-to-cell variability. We developed an experimental system that precisely measures the cell-cycle duration of thousands of mammalian cells along several generations and a mathematical framework that allows discrimination between stochastic and deterministic processes in lineages of cells. We show that the inter- and intra-generation correlations reveal complex inheritance of the cell-cycle duration. Finally, we build a deterministic nonlinear toy model for cell-cycle inheritance that reproduces the main features of our data.}

The experimental underpinnings involved Fucci markers (Sakaue-Sawano 2008):

_{[Harnessing] antiphase oscillating proteins that mark cell-cycle transitions [Fucci markers are] genetically encoded fluorescent probes for this purpose. These probes effectively label individual G1 phase nuclei red and those in S/G2/M phases green… Cultured cells and transgenic mice constitutively expressing the cell-cycle probes, in which every cell nucleus exhibits either red or green fluorescence. We performed time-lapse imaging to explore the spatiotemporal patterns of cell-cycle dynamics… [The technique permits] unprecedented spatial and temporal resolution to help us better understand how the cell cycle is coordinated with various biological events.}

_{…The Fucci technology allows dual-color imaging, which can distinguish between live cells in the G1 and the S/G2/M phases. This technology allows for in vivo analysis of spatial and temporal patterns of cell-cycle dynamics, owing to the brightness of the fluorescence and the high contrast between the two colors (red and green).}

Reminiscent of the technique Balaban described in her seminar for automated image analysis for the preceding Science paper on growth variability (ScanLag, described in Levin-Reisman et al., 2010), the Fucci-marked cells were observed under time-lapse microscopy for several days under constant conditions, and Fucci marker-generated fluorescence quantified with a mix of automated MATLAB code and supervised ImageJ/Fiji plugin (neither of which are provided).

Further measurements were taken with microfluidic devices, but the conspicuous correlation in cell-cycle duration between cousins remained (“the cousin–mother inequality”).

To elucidate whether the succession of cell cycle durations along a lineage was stochastic or deterministic, the authors sought to enumerate the factors underlying cell-cycle variability. A few key criteria would indicate determinism, while a great number of contributing factors would point to stochasticity. Fortunately, they were aware of an algorithm used to interrogate dynamical systems in precisely this way.

The Grassberger-Procaccia algorithm extracts a ‘correlation dimension’ from time-series data: able to handle both white noise and coloured noise, with a threshold of ~10 at which its value is interpreted as stochasticity. The result for the lineage pedigree analysis was 3 (the idea sounds vaguely like principal component analysis), meaning around 3 factors should be able to explain the observed patterns.

Their proposal is of a ‘kicked cell cycle’ (described in the supplement):

_{The main inherited factor is phase of the cellular oscillator with period Tosc. This oscillator may be viewed as the circadian clock oscillator, assuming coupling between the phase of the circadian clock (namely if it is “day” or “night”). The circadian phase affects the cell-cycle duration by “kicking” the cell cycle into either a shorter or a longer cell cycle duration, depending on whether the cell is born during the “day” or “night” period. Note that different lineages are not synchronized to the external day-night cycle. The circadian phase is thus transmitted along a lineage but varies between lineages. The phase of this clock is therefore one of the “hidden” deterministic factors that control the cell cycle inheritance.}

‘Sister cells of the same cell cycle inherit the same phase’ of the chemical oscillator, as it’s posed elsewhere in the article.

Figure 1 from the supplementary materials gives a summary plot of the cell cycle durations observed across the experiment. Cell cycle length can vary considerably between species as well as within a population as shown here. These authors observed mouse L1210 lineages (a cell type normally used as a leukaemia model), and the rough range in the data presented here [from standard deviation error bars] is ~11 to 21 hours.

The magazine highlight from Hilfinger and Paulsson (both Dept. Systems Biology at Harvard Medical School) finds the tables to have turned from days of research gone by:

_{A few decades ago, such a deterministic scenario might even have been the first guess. Differences between genetically identical cells were then often explained by nonlinear models, for example oscillations, chaos, or bistable switches. Noise was invoked only to explain infinitesimal perturbations that might eventually cause systems to diverge. Now the pendulum has swung the other way, and physiological heterogeneity is explained by random bursts of gene expression almost by default.}

_{The problem is that both no-noise and all-noise views ignore the interconnectedness of random fluctuations and average dynamical tendencies. In ballistics, the trajectories of projectiles can be considered in the absence of random gusts of wind, but in small chemical systems such as cells, randomness is a consequence of the basic mechanisms by which the system changes. For example, exponential decay at the level of population averages typically reflects exponentially distributed times of individual reaction steps. However, the term ‘random’ in this context does not imply an absence of patterns — the magnitude and time correlations of fluctuating concentrations always depend strongly on the underlying interactions. Processes in single cells are thus always shaped by both chemical noise and average dynamical tendencies, and the question is how to use the information hidden in population averages to understand more about both factors.}

Without quite saying as much, they suggest that in principle the the Balaban lab’s conclusions could be explained by noise.

_{For example, systems in which two random factors affect cell doubling times in opposite ways and decorrelate on different timescales can produce negligible correlations between mother and daughter cells, but substantial correlations between cousins. The tools from chaos theory must then be used very carefully, because they can make random processes seem deterministic depending on the sampling intervals used.}

For further reading on the mathematical groundings, see the supplementary materials of Sander et al.

• Theiler, James. (1990) Estimating fractal dimension. JOSA A 7.6: 1055-1073. [excerpt shown below]

• Grassberger, Peter. (2007) Grassberger-Procaccia algorithm. Scholarpedia 2.5: 3043.

• Kantz, Holger, and Thomas Schreiber. Nonlinear time series analysis. Cambridge University Press, 2004.

(Source: biochemistries)

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