Fourier laplace difference between mitosis

We also show that the variability in added size and the strength of size control in fission yeast depend weakly on the temperature but strongly on the culture medium. More importantly, we find that stronger size homeostasis and larger added size variability are required for fission yeast to adapt to unfavorable environmental conditions. Go to: Author summary Advances in microscopy enable us to follow single cells over long timescales from which we can understand how their size varies with time and the nature of innate strategies developed to control cell size.

These data show that in many cell types, growth is exponential and the distribution of cell size has one peak, namely there is a single characteristic cell size. However data for fission yeast show remarkable differences: growth is non-exponential and the distribution of cell sizes has two peaks, corresponding to different growth phases.

Here we construct a detailed stochastic mathematical model of this organism; by solving the model analytically, we show that it is able to predict the two peaked distributions of cell size seen in data and provide an explanation for each peak in terms of various growth phases of the single-celled organism.

Furthermore, by fitting the model to the data, we infer values for the rates of all microscopic processes in our model. This method is shown to provide a much more reliable inference than current methods and shed light on how the strategy used by fission yeast cells to control their size varies with external conditions. Go to: Introduction The fission yeast Schizosaccharomyces pombe is a single-cell eukaryote whose shape is well approximated by a cylinder with hemispherical ends [ 1 — 3 ].

The length of the rod-shaped cell increases during the G2 phase of the cell cycle, while its width diameter remains almost constant. In experiments, length, area, and volume have all been used to characterize cell size. The recent advent of microfluidic techniques allows the tracking of thousands of individual cells over hundreds of cell cycles which potentially enables a detailed investigation of cell growth and size control strategies [ 4 ].

It has been reported that cell size grows exponentially in many cell types such as bacteria, cyanobacteria, archaea, budding yeast, and mammalian cells [ 5 — 17 ]. However, fission yeast undergoes a complex non-exponential growth pattern in each cell cycle, as illustrated by the time-course data of cell size along a typical cell lineage Fig 1A.

At the beginning of the cell cycle, the rod-shaped cell starts to grow by extension at its old cell end the end that existed before the last division. Later in mid G2 phase, the cell exhibits a transition in cell polarization, and growth is also initiated at the new cell end the end created during the last division , in a process called new end take-off NETO [ 1 , 3 ]. Fig 1 Cell size dynamics in fission yeast. Here the size of a cell is characterized by its length.

The data shown are published in [ 4 ]. The green dots show cell sizes at birth. B: Histogram of cell sizes along all cell lineages. The cell size distribution of lineage measurements has a bimodal shape. C: Scatter plot of the birth size versus the division size and the associated regression line.

There has been a long-standing controversy about the growth pattern of fission yeast before mitosis [ 18 — 20 ]. In earlier studies, exponential [ 19 , 21 — 25 ], linear [ 26 , 27 ], and bilinear [ 1 , 3 , 28 — 30 ] growth models have been proposed. Since we have very small number of samples, we estimated the p-values of the scores at the normalized cell-division-cycle frequency by a bootstrap method similar to [ 36 ], that is, for each gene, we fix the sampling times and permute the expression values.

A total of permutations are generated, and for each permuted time series the score is computed. The p-value is simply the ratio of the permuted sequences that produced scores higher than from the original time-course expression.

By permuting the expression values with respect to their sampling times, we aimed to destroy the periodicity that may exist within the time series. If periodicity exists within the time series, it is unlikely that a randomly permuted sequence will recover that periodicity, thus the magnitude of the spectrum at the tested period will be reduced and unlikely to achieve greater magnitude than the magnitude of the original time series.

If on the other hand periodicity does not exist in the time series, then neither the original nor the permuted sequences will likely to have large magnitude at the test period, therefore, the possibility of the permuted sequence having higher magnitude than the original sequence is quite high. Thus the lower the p-value, meaning that only a small ratio of the permuted sequences resulted in higher magnitude than by the original sequence, the more likely that the gene is periodically expressed.

Since the three benchmark sets discussed above include genes that are known to be or potentially periodically expressed, we will evaluate their performance by searching for the genes in these benchmark sets from amongst the highly ranked genes. We search within the top K-ranked genes for those genes that are present in the benchmark sets B1, B2, and B3.

Thus we have a total of 9 "dataset — benchmark" combinations. The plots for the 9 "dataset — benchmark" combinations are given in Figures 1 , 2 , 3. Figure 1 Detection rate in the top scoring genes by the Fourier-score-based algorithm [ 31 ], M-estimator [ 29 ], and the Laplace periodogram for the Alpha dataset with no random impulse added for a B1, b B2, and c B3 benchmark sets. Full size image Figure 2 Detection rate in the top scoring genes by the Fourier-score-based algorithm [ 31 ], M-estimator [ 29 ], and the Laplace periodogram for the CDC15 dataset with no random impulse added for a B1, b B2, and c B3 benchmark sets.

Full size image Figure 3 Detection rate in the top scoring genes by the Fourier-score-based algorithm [ 31 ], M-estimator [ 29 ], and the Laplace periodogram for the CDC28 dataset with no random impulse added for a B1, b B2, and c B3 benchmark sets. Full size image The figures plot the ratio of periodic genes as indicated by the B1, B2, and B3 benchmark sets discovered in a subset of the top scoring genes scored by the three algorithms.

As the subset of top scoring genes number of genes in the subset increases, the ratio of benchmark periodic genes contained in these subsets also increases. From Figures 1 , 2 , 3 , we can see that the Fourier score method consistently produces the best performance out of the three methods.

The Laplace periodogram is able to detect periodically expressed genes at detection ratio that is comparable to the Fourier score for some experiment-benchmark combinations, and less for others. For the other combinations, the Laplace periodogram either has comparable detection performance for all values of K, or is able to bridge the performance gap as K approaches For each of the combinations, the M-estimator-based method achieves the worst performance, with large drop-offs in performance from both the Fourier score and the Laplace periodogram.

Also note that the results in Figures 2 and 3 meet with our expectation that most of the genes in B2 and B3 are not periodic, leading to a very low ratio of genes from those two sets being detected as periodic. Arabidopsis dataset For this comparison, we use the experimental data provided by [ 33 ], which includes the expression of genes in Arabidopsis, of which are determined to follow a circadian fluctuation in mRNA abundance. Total of 12 samples of the time-series gene expression is taken for each gene, which covers approximately two cycles of the circadian oscillation.

Comparison similar to the Saccharomyces cerevisiae datasets is also performed for the Arabidopsis dataset. The prediction results for the Fourier-score-based algorithm and the Laplace periodogram are shown in Figure 4. As we can see from the figure, the performance of the two algorithms are virtually the same up to the top scoring genes.

It would seem at first easier to rank the gene expression correctly for Arabidopsis due to the higher ratio of genes in the dataset following circadian oscillation, at 7. However, the Arabidopsis dataset also contains more genes than the yeast Alpha dataset, which increases the ranking difficulty. Figure 4 Detection rate in the top scoring genes by the Fourier-score-based algorithm [ 31 ] and the Laplace periodogram for the Arabdopsis dataset.

Full size image Periodic gene detection in the presence of outliers We now compare the detection performances of the Fourier score, Laplace periodogram, and M-estimator on the same Saccharomyces cerevisiae Alpha, CDC15, and CDC28 datasets from [ 2 , 3 ], but with added impulsive noise to the dataset. For each gene, there is a 0. Note that the range of the peaks of the magnitude in these experiments varies between 3 and 5. We generate for each of the three experiments 50 such generated datasets with randomly placed impulse noises.

We then perform periodicity detection with the three methods on these randomly generated datasets and averaged the results for each experiment. However, in our simulations, we observed that the performance for each of the algorithms when ranked using the estimated p-values is no better than if we had randomly ranked the genes.

Therefore, instead of ranking the genes by their p-value, here we rank them by using their magnitude instead of using the estimated p-values. This observation can be explained by the presence of the outlier impulsive noise, whose influence on the spectrum is not removed by the random permutations, thus resulting magnitude for a periodically expressed time series and its permutations do not differ by a significant amount.

The results of these plots are then given in Figures 5 , 6 , 7. Figure 5 Detection rate in the top scoring genes by the Fourier-score-based algorithm [ 31 ], M-estimator [ 29 ], and the Laplace periodogram for the Alpha dataset with random impulse added for a B1, b B2, and c B3 benchmark sets. Full size image Figure 6 Detection rate in the top scoring genes by the Fourier-score-based algorithm [ 31 ], M-estimator [ 29 ], and the Laplace periodogram for the CDC15 dataset with random impulse added for a B1, b B2, and c B3 benchmark sets.

Mitosis between laplace fourier difference crypto sha1 library android

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First of all, you can't really excite your system with a pure sinusoidal. It's too late, you should have started at the big bang. The best you can do is use a causal sinusoidal, which has extra frequency components. But let's say that what you want to know is the response of the system to an arbitrary input in the time domain. You don't really need Fourier or Laplace to know this. A convolution will do. What do you have in hand, really?

You measured the impulse response. Somehow you plotted it out, let's say continuously, as opposed to an ADC that sampled the signal - which is usually what happens, and you'd be asking about the Z-transform vs FFT instead. Let's also assume that the bang you gave it was a good delta: strong but short. Since your system is RLC, it is linear, so superposition principles work we wouldn't be talking about this otherwise anyway.

Any input can be constructed by adding attenuated impulses offset in time sort of - it's a limit thing. So the total response is just adding all these individual responses together. Once the simpler model is solved, the inverse integral transform is applied, which would provide the solution to the original model. For example, since most of the physical systems result in differential equations, they can be converted into algebraic equations or to lower degree easily solvable differential equations using an integral transform.

Then solving the problem will become easier. What is the Laplace transform? The inverse transform can be made unique if null functions are not allowed. The following table lists the Laplace transforms of some of most common functions. What is the Fourier transform? Fourier transform is also linear, and can be thought of as an operator defined in the function space.

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