Results| Exploratory data analysis – branching angles 57

Branching angle is one of the key growth indicators for Neurospora crassa. Exploratory data analysis for the recorded branching angle values was performed separately within parent, daughter, and further generation hyphae sub-populations. The illustration of branching angle definition for various generations of hypha is given in Figure 29 (C).

The analysis revealed that the range of recorded values slightly differs between parents and other generation hyphae. The maximum values are significantly higher (almost doubled) for the daughter and further generation hyphae compared to the parent hypha.

Figure 29 Exploratory data analysis of branching angles displayed by Neurospora crassa growing on the agar substrate (part I). Time of the observations varies from 32 minutes 46seconds (movies I-IV) up to 1h 05minutes 31 seconds (movies V-VI). (A) Relative frequency counts of branching angle values, where: green- indicates parents, blue – daughters, red – further generation hyphae. (B) Scatter plot of branching angle values, where: green indicates parents, blue-daughters, red-further generation hyphae. The plot shows that although various hyphae sub-populations have similar range of branching angle values, the maximum branching angle values are higher for daughters and further generations compared to the parent hyphae. Also, it is clear that the data is clustered. This is because various hyphae sub-populations dominate at different times: parents (first 10 minutes), daughters (from t=10 min to t=30 min), further generations for t > 30minutes. (C) Illustration of the branching angle definitions for parent, daughter, and further generation hyphae (D) Final frames from the movies I-VI with branching angles indicated by different colours for various hyphae

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This feature of Neurospora crassa is well reflected in the scatter plot in Figure 29(B). Also, in Figure 29 (A) the relative frequency counts plot for parents (green) is shifted to the left, towards low branching angle values. Additionally, various sub-populations of hyphae dominate at various time intervals (Figure 29 (B)). The relative number of parent branches is the highest at the time interval from t=0 to t~10 minutes. The daughters are a dominant sub-population at a time interval from t ~15 min to t~ 30 min while the further generation hyphae are observed at the highest number at the time interval between t~40 min to t~65 minutes. The tables below show the measurement outcomes based on two different approaches. The first of the two consecutive measurement series is based on capturing the absolute values of branching angles while the second one takes into account the information on whether a branch was sent on the right (positive values) or the left side (negative values) of the initial hypha.

The number of measured branching angles in the first series of measurements is as follows: 70 for the parents, 52 for the daughters, and 33 for further generation hyphae. It gives in total 161 branching angle values. Similar mean and median values for all cases suggest normally distributed data collected in the first series. The maximum recorded values are significantly higher for daughter and further generation hyphae (153° and 113° respectively) compared to the parent subpopulation (88°). Interestingly, the minimum recorded value, 33°, is higher than the values recorded for daughters (26°) and further generation hyphae (24°). There is no mode value for the parents. The mode value for daughters is 83°, for further generations 79°, and for all generations of hyphae it is 94°. These results are consistent with the measurement results reported by Moore et al. (Moore, 2011). Interestingly, the value 94° gives a characteristic geometrical pattern of branching in silico fungus (please, see fungi animations examples).

Branching angle Ntotal Mean [deg] SD Min 1stQ Median [deg] 3rdQ Max Mode [deg] Parents 70 59 13 33 51 58 68 88 - Daughters 52 83 18 26 74 82 94 153 83 Further 33 78 18 24 70 78 91 113 79 All 161 71 19 24 58 72 84 153 94

Table 4Branching angles – a breakdown of total numbers of the measurements taken, means with standard

deviations, and a 5-number data summary that consists of: the minimum value, 1st _{quartile (value at the end}

of the first 25% of the data of the whole data distribution), median (value in the middle of the distribution),

3rd _{quartile (value at the end of the 75% of the data of the whole data distribution), and the maximum value.}

The means, and standard deviations are part of the parametric statistical analysis while the 5-number summary is a non-parametric method and gives correct information even if the data is not distributed normally or when the data distribution is not known

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For the second series of measurements, the total number of the values collected is as follows: 71 for parents, 52 for daughters, and 40 for further generation hyphae. It gives 163 branching angle values altogether.

The mean and median values are significantly different within each sub-population. For the parent hyphae it is -1° (mean) versus -34°(median), for the daughters -18° (mean) versus - 73° (median), and for the further generation hyphae 11.5° versus 51°. For all sub- populations collectively the mean is -3° and the median -34°. In this case, the distribution is bimodal, and this explain why medians give adequate values regarding the description of the real-world behavior of a filamentous fungus. The lesson learnt from the analysis of the second measurement series is that calculating medians are a better way of assessing central tendency measure when the data distribution is not known or is not normal.

Perhaps the most interesting measurement outcome in Table 3 is the one for the further generation hyphae sub-population, for which both mean and median are positive, compared to the central tendency values displayed for the parents and daughters. As the exploratory data analysis shows clearly that the highest number of the further generation hyphae is generated at the late stages of the colony growth, this would suggest that branching angles contribute significantly to the final geometrical fungus pattern formation. As both of the central tendency measure for the further generation hyphae (mean and median) are positive, it means that Neurospora crassa prefers to send further generation branches on the right side of the initial daughter branch. This observation can explain why the fungus biomass is shifted to the right with regards to the parent hypha (Figure 92).

Branching angle (with negative values) Ntotal Mean [deg] SD Min 1stQ Median [deg] 3rdQ Max Mode [deg] Parents 71 -1 61 -87 -57 -34 58 88 - Daughters 52 -18 83 -153 -86 -73 76 103 -82 Further 40 11.5 81 -113 -78 51 83 105 - All 163 -3 74 -153 -75 -34 70 105 -82

Table 5 Branching angles – a breakdown of total numbers of the measurements taken, means with standard deviations, and a 5-number data summary that consists of: the minimum value, 1st _{quartile (value at the end of the}

first 25% of the data of the whole data distribution), median (value in the middle of the distribution), 3rd _quartile

(value at the end of the 75% of the data of the whole data distribution), and the maximum value. The means, and standard deviations are part of the parametric statistical analysis while the 5-number summary is a non-parametric method and gives correct information even if the data is not distributed normally or when the data distribution is not known

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Figure 30 illustrates the distribution of branching angle data within different fungus sub- populations. In Figure 30 (A), one can notice a clear shift of the parent branching angles distribution to the left. Moreover, the range of the branching angles for the parents is from 33° to 88°, for the daughters from 26° to 153°, and for the further generation hyphae from 24° to 113°. The central tendency value (median) for parents is 58°, for daughters 82°, and for the further generation hyphae 78°. These results suggest different ranges of branching angles for parents versus other generations of hyphae.

Figure 30 Exploratory data analysis of branching angles in Neurospora crassa growing on the agar substrate (part II). The time of the observation varied from 32 minutes 46 seconds (Movies I-IV) up to 1 hour 05 minutes 31 seconds (Movies V and VI). (A) Overlay of 3 separate relative frequency counts plots (green-for the parent hyphae, blue-for daughters, and red – for the further generation hyphae. (B) Illustration of branching angle definitions for various sub-populations (green-parents, blue-daughters, and red-further generations). Arrows indicate BRANCHES: parent branch, daughter branch, and two further generation hyphae (C) Frequency counts plots for (a) parents, (b) daughters, (c) further generation hyphae

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