Mathematical Modeling Predicts Enhanced Growth of X-Ray Irradiated Pigmented Fungi

Abstract

Ionizing radiation is known for its cytotoxic and mutagenic properties. However, recent evidence suggests that chronic sub-lethal irradiation stimulates the growth of melanin-pigmented (melanized) fungi, supporting the hypothesis that interactions between melanin and ionizing photons generate energy useful for fungal growth, and/or regulate growth-promoting genes. There are no quantitative models of how fungal proliferation is affected by ionizing photon energy, dose rate, and presence versus absence of melanin on the same genetic background. Here we present such a model, which we test using experimental data on melanin-modulated radiation-induced proliferation enhancement in the fungus Cryptococcus neoformans, exposed to two different peak energies (150 and 320 kVp) over a wide range of X-ray dose rates. Our analysis demonstrates that radiation-induced proliferation enhancement in C. neoformans behaves as a binary "on/off" phenomenon, which is triggered by dose rates 5000 mGy/h. Proliferation enhancement of irradiated cells compared with unirradiated controls occurs at both X-ray peak energies, but its magnitude is modulated by X-ray peak energy and cell melanization. At dose rates

Full text

Mathematical Modeling Predicts Enhanced Growth of X-
Ray Irradiated Pigmented Fungi
Igor Shuryak
1
*, Ruth A. Bryan
2
, Joshua D. Nosanchuk
3,4
, Ekaterina Dadachova
2,4
1Center for Radiological Research, Columbia University, New York, New York, United States of America, 2Department of Radiology, Albert Einstein College of Medicine,
Bronx, New York, United States of America, 3Department of Medicine, Albert Einstein College of Medicine, Bronx, New York, United States of America, 4Department of
Microbiology and Immunology, Albert Einstein College of Medicine, Bronx, New York, United States of America
Abstract
Ionizing radiation is known for its cytotoxic and mutagenic properties. However, recent evidence suggests that chronic sub-
lethal irradiation stimulates the growth of melanin-pigmented (melanized) fungi, supporting the hypothesis that
interactions between melanin and ionizing photons generate energy useful for fungal growth, and/or regulate growth-
promoting genes. There are no quantitative models of how fungal proliferation is affected by ionizing photon energy, dose
rate, and presence versus absence of melanin on the same genetic background. Here we present such a model, which we
test using experimental data on melanin-modulated radiation-induced proliferation enhancement in the fungus
Cryptococcus neoformans, exposed to two different peak energies (150 and 320 kVp) over a wide range of X-ray dose
rates. Our analysis demonstrates that radiation-induced proliferation enhancement in C. neoformans behaves as a binary
‘‘on/off’’ phenomenon, which is triggered by dose rates ,0.002 mGy/h, and stays in the ‘‘on’’ position. A competing dose
rate-dependent growth inhibition becomes apparent at dose rates .5000 mGy/h. Proliferation enhancement of irradiated
cells compared with unirradiated controls occurs at both X-ray peak energies, but its magnitude is modulated by X-ray peak
energy and cell melanization. At dose rates ,5000 mGy/h, both melanized and non-melanized cells exposed to 150 kVp X-
rays, and non-melanized cells exposed to 320 kVp X-rays, all exhibit the same proliferation enhancement: on average,
chronic irradiation stimulates each founder cell to produce 100 (95% CI: 83, 116) extra descendants over 48 hours.
Interactions between melanin and 320 kVp X-rays result in a significant (2-tailed p-value = 4.8610
25
) additional increase in
the number of radiation-induced descendants per founder cell: by 55 (95% CI: 29, 81). These results show that both melanin-
dependent and melanin-independent mechanisms are involved in radiation-induced fungal growth enhancement, and
implicate direct and/or indirect interactions of melanin with high energy ionizing photons as an important pro-proliferative
factor.
Citation: Shuryak I, Bryan RA, Nosanchuk JD, Dadachova E (2014) Mathematical Modeling Predicts Enhanced Growth of X-Ray Irradiated Pigmented Fungi. PLoS
ONE 9(1): e85561. doi:10.1371/journal.pone.0085561
Editor: Robin Charles May, University of Birmingham, United Kingdom
Received October 9, 2013; Accepted December 3, 2013; Published January 15, 2014
Copyright: ß2014 Shuryak et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported through NIAID grant U19-AI67773. The funders had no role in study design, data collection and analysis, decision to publish,
or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: is144@columbia.edu
Introduction
Survival (and sometimes relative predominance) of melanized
fungi in environments exposed to high levels of ionizing radiation
has been reported for several decades, for example from a nuclear
weapons test site in Nevada [1], and from a forest experimentally
exposed to chronic irradiation at the Brookhaven National
Laboratory [2]. Data from the zone contaminated by the
Chernobyl nuclear power plant accident [3–7] revealed that
certain melanized fungi survive chronic irradiation from multiple
radionuclides (even within the destroyed nuclear power plant
buildings), and that ionizing radiation can stimulate spore
germination and attract hyphal growth in some of these fungal
strains. Recent laboratory investigations confirm the conclusion
that some melanized fungi are not only radioresistant, but exhibit
enhanced proliferation during chronic sub-lethal irradiation [8–
10]. These findings suggest that the biological effects of ionizing
radiation are not limited to cell death, mutagenesis and
carcinogenesis [11], but can include growth stimulation of certain
life forms. Interactions of ionizing photons with fungal melanin
represent an important candidate mechanism for pro-proliferative
effects of radiation on melanized fungi [9,12–15].
To quantitatively analyze the phenomenon of fungal growth
modulation by chronic ionizing radiation exposure, we propose
the following mathematical model: Q = A(e, t, m) – [B(e, t, m) x
R]. Here Q is the predicted radiation effect on proliferation
relative to background conditions (for example, the excess number
of cells which descended from each founder cell due to radiation
effects on proliferation), and R is radiation dose rate. The
adjustable parameters (A and B) depend on X-ray peak energy (e),
duration of irradiation (t), and cell melanization status (m).
Parameter A represents radiation-induced growth enhancement,
which is assumed to be a ‘‘binary’’ qualitative response triggered
by very low dose rates, e.g. due to modulation of certain metabolic
and cell cycle-related pathways by interactions between melanin
and ionizing photons, by redox processes, and/or by responses to
low levels of DNA damage. Parameter B represents radiation-
induced growth inhibition, which is assumed to be caused by DNA
damage response pathways and hence proportional to dose rate.
This simple formalism provides a tractable tool for analyzing and
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mechanistically interpreting experimental data on the influence of
chronic sub-lethal irradiation on fungal growth. In particular, the
effects of cell melanization and other factors on the values of both
model parameters (A and B) can be analyzed quantitatively and
statistically.
Results
To generate experimental data for testing model predictions, we
selected the pathogenic fungus Cryptococcus neoformans (strain H99),
which becomes melanized (Figure 1a) if a melanin precursor (L-
DOPA in this case) is provided in the growth medium, and
remains non-melanized otherwise, while the genetic background
remains constant. Since melanin forms a contiguous layer with the
cell wall (Figure 1a), incoming X-rays unavoidably pass through
melanin before reaching the cytoplasm. We irradiated melanized
and non-melanized C. neoformans cells under identical conditions
with X-ray spectra of different peak energies (150 or 320 kVp) and
a range of dose rates (0.002 to 5500 mGy/h) using the X-RAD
320 biological irradiator (Figure 1b–c).
Analysis of Experimental Data
The effect of X-ray irradiation on the average number of
descendants (Q
e
) produced by each founder cell during the
exposure period was defined as follows: Q
e
=(X
r
(i) – X
c
)/X
0
,
where X
r
(i) is the number of colony-forming units per milliliter
(CFU/ml) for each irradiated culture, X
c
is the mean CFU/ml for
corresponding unirradiated controls, and X
0
is the mean CFU/ml
at the start of irradiation. As described in the Materials and
Methods section, Q
e
is less sensitive to random inter-experimental
variability than raw CFU/ml, because it represents the radiation-
induced change in CFU/ml, normalized by the initial cell
concentration. Therefore, Q
e
values are more convenient than
raw CFU/ml counts for detecting subtle effects of X-ray peak
energy and melanization on cell proliferation, which are of main
interest here.
Chronic exposure to X-ray dose rates below 5000 mGy/h had a
pro-proliferative effect on C. neoformans, resulting in extra
descendants being produced by each founder cell. This tendency
is shown by the positive mean Q
e
values and their corresponding
95% confidence intervals for C. neoformans cultures exposed to dose
rates in this range (Table 1 and Figure 2). Due to radiation-
Figure 1. Fungal model system and experimental setup. (a) Melanized cell of the fungus C. neoformans: immunofluorescent image shows the
melanin layer contiguous with the cell wall (left panel) and light microscopy image of the same cell (right panel). (b–c) External and internal views of
the X-RAD 320 biological irradiator, which was used for continuous X-ray irradiation of C. neoformans cells. The cells were suspended in liquid
medium in test tubes inserted into the sample rack. Radiation quality and dose rate were adjusted by varying the peak energy, current, filter, lead
shielding thickness, and distance from the source.
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induced alteration of cell proliferation, each founder cell gave rise
to an average of approximately 1–5 extra descendants over 24
hours of exposure, and to approximately 40–180 extra descen-
dants over 48 hours (see Table 1 and data points in Figures 3–4).
The modulation of radiation-induced growth enhancement by
melanin was more subtle, and depended on X-ray peak energy.
For 150 kVp X-ray exposures, there were no significant differ-
ences between the number of extra descendants per founder cell
produced by non-melanized and by melanized C. neoformans.On
average, each non-melanized founder cell produced 2.6 (range:
20.4, 5.5) extra descendants due to radiation effects over 24
hours, and 70 (range: 211, 153) extra descendants over 48 hours.
Table 1. Effects of continuous X-ray irradiation with different peak energies and dose rates on the proliferation of C. neoformans.
Duration of
irradiation (hours)
X-ray peak
energy (kVp)
Cell melani-
zation
Dose rate category
(mGy/h)
Mean dose rate
(mGy/h)
Radiation-induced change (Q
e
) in CFU/ml,
normalized by the initial cell concentration
Mean 95% CIs
24 150 21–1000 121 2.81 2.16, 3.45
2000–3000 2510 2.38 1.27, 3.49
+1–1000 121 3.45 2.57, 4.32
2000–3000 2510 2.45 1.74, 3.17
320 21–1000 383 0.78 21.93, 3.49
.5000 5490 25.03 27.67, 22.38
+1–1000 383 5.53 1.71, 9.34
.5000 5490 29.94 213.79, 26.09
48 150 21–1000 121 87.2 71.4, 103.0
2000–3000 2510 43.5 20.5, 66.4
+1–1000 121 79.2 63.6, 94.8
2000–3000 2510 36.4 15.5, 57.3
320 2,1 0.12 166.4 141.6, 191.1
1–1000 383 31.1 210.1, 72.4
.5000 5450 2137.0 2180.7, 293.2
+,1 0.12 178.9 147.1, 210.7
1–1000 383 125.4 84.7, 166.1
.5000 5450 2112.6 2145.3, 280.0
As discussed in the main text, radiation effects were quantified by calculating the radiation-induced change (Q
e
) in CFU/ml, normalized by the initial cell concentration,
as follows: Q
e
=(X
r
(i) – X
c
)/X
0
, where X
r
(i) is the CFU/ml for each irradiated culture, X
c
is the mean CFU/ml for corresponding unirradiated controls, and X
0
is the mean
CFU/ml at the start of irradiation. Q
e
represents the number of extra descendants produced by each founder cell due to the effects of irradiation on cell proliferation.
doi:10.1371/journal.pone.0085561.t001
Figure 2. Effects of 24 and 48 hours of continuous X-ray irradiation on the proliferation of
C. neoformans
.The symbols represent mean
values of the radiation-induced change (Q
e
) in CFU/ml (normalized by the initial cell concentration) for the same X-ray dose rate categories as in
Table 1, plotted as function of mean dose rate for each category. Error bars represent standard errors. The abbreviations ‘‘mel +’’ and ‘‘mel –’’
represent melanized and non-melanized cells, respectively.
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Each melanized founder cell produced 3.1 (range: 0.8, 5.8) extra
descendants over 24 hours, and 63 (range: 29, 150) extra
descendants over 48 hours.
However, for 320 kVp X-ray exposures, differences between
radiation responses of melanized and non-melanized cells began to
emerge after 24 hours of irradiation, and reached statistical
significance after 48 hours. On average, each non-melanized
founder cell exposed to X-ray dose rates ,5000 mGy/h produced
0.8 (range: 29.3, 6.6) extra descendants due to radiation effects
over 24 hours, and each melanized founder cell produced 5.5
(range: 24.6, 16) extra descendants under the same conditions.
Over a longer period of irradiation (48 hours), each non-melanized
founder cell exposed to X-ray dose rates ,5000 mGy/h produced
99 (range: 2100, 248) extra descendants, while each identically
exposed melanized founder cell produced a larger number of extra
descendants: 152 (range: 44, 278). This difference was statistically
significant (p = 0.038 using the Mann-Whitney U test and 0.030
using the 2-tailed Student’s t-test).
At sufficiently high dose rates (above 5000 mGy/h), the effect of
radiation was reversed from growth enhancement to growth
inhibition, and proliferation of C. neoformans was slowed down
(Table 1 and Figures 2–4).
Insight from Mathematical Modeling
Our quantitative model, Q = A(e, t, m) – [B(e, t, m) x R], was
fitted separately to each of 8 experimentally observed data sets,
which represent the 8 combinations of X-ray peak energy (150 or
320 kVp), exposure duration (24 or 48 hours), and cell melani-
zation status (melanized or non-melanized). The best-fit model
predictions corresponding to each of these data sets are shown as
lines in Figures 3–4. To assess how sensitive these predictions were
to random fluctuations in the data, the model was fitted to multiple
Monte Carlo simulated data sets generated from each experimen-
tally observed data set by nonparametric bootstrapping (as
described in Materials and Methods). The distributions of values
for parameters A and B, produced by fitting the model to
bootstrapped data, are shown as ‘‘clouds’’ of points in Figure 5.
The size and location of the region occupied by each ‘‘cloud’’
indicate the spread of model parameter values which are
consistent with the observed data set. The degree of separation
between regions which encompass model fits to different data sets
indicate how different the best-fit model parameter values
corresponding to one data set are from those corresponding to
the other data set, and how sensitive these differences are to
random variation in the data sets.
Our model had limited ability to describe the data from C.
neoformans cells irradiated for 24 hours. This conclusion was
suggested by: (1) the wide distributions of parameter values
produced by fitting the model to bootstrapped data (Figure 5), and
(2) the low adjusted R-squared (0.47), and the evidence against
normality of residuals (Shapiro-Wilk p-value ,10
24
), produced by
fitting the model to experimentally observed data using stepwise
Figure 3. Analysis of the proliferation of
C. neoformans
continuously X-irradiated for 24 hours. The symbols represent radiation-induced
change (Q
e
) in CFU/ml (normalized by the initial cell concentration) measured for melanized and non-melanized cells at each combination of X-ray
peak energy (150 or 320 kVp) and dose rate. Different symbol colors, shapes, and bar colors represent different experiments. Horizontal bars indicate
mean values from each experiment. The lines are best-fit predictions generated by the proposed model, Q = A(e, t, m) – [B(e, t, m) x R], which predicts
the radiation effect on proliferation relative to background conditions (Q, shown on the y-axis), based on radiation dose rate (R shown on the x-axis),
X-ray peak energy (e), duration of irradiation (t), and cell melanization status (m). Model parameter A represents radiation-induced growth
enhancement at low dose rates, and parameter B represents radiation-induced growth inhibition, proportional to dose rate.
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multiple linear regression (described in Materials and Methods).
The explanation is that variability in the duration of the lag phase
(i.e. of the time from the start of irradiation until cell proliferation
begins), which is known to occur in C. neoformans cultures with low
starting cell concentrations [16], caused large variations in CFU/
ml counts after 24 hours. This ‘‘noise’’, which was amplified
exponentially during the period of rapid cell division following the
lag phase, masked the relatively subtle effects of X-ray peak energy
and cell melanization. Consequently, the best-fit value for
parameter A, produced by fitting the model by stepwise multiple
linear regression, was common for both melanized and non-
melanized cells, because differences between their responses could
Figure 4. Analysis of the proliferation of
C. neoformans
continuously X-irradiated for 48 hours. Interpretations for the axes, symbols, and
curves are the same as in Figure 3.
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Figure 5. Analysis of best-fit model parameter values. The proposed model, Q =A(e, t, m) – [B(e, t, m) x R], which predicts the radiation effect
on proliferation relative to background conditions (Q), based on radiation dose rate (R), X-ray peak energy (e), duration of irradiation (t), and cell
melanization status (m), was fitted to 10,000 Monte Carlo simulated data sets, generated from each observed data set by nonparametric
bootstrapping. This procedure was repeated for 8 observed data sets, which represent combinations of 150 or 320 kVp X-ray peak energy, 24 or 48
hour exposure duration, and positive or negative cell melanization status (mel+or mel –). Each ‘‘cloud’’ of points represents the spread of values for
model parameters A and B which are consistent with the corresponding data set, randomly varied by bootstrapping. Black open symbols
superimposed on each ‘‘cloud’’ represent best-fit parameter values to the observed data (unperturbed by bootstrapping).
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not be discerned with statistical significance. However, this best-fit
value for parameter A was 3.2 (95% CI: 2.3, 4.1), significantly
different from zero (2-tailed p-value = 4.3610
210
), indicating that
the overall growth-enhancing effect of low dose rate irradiation on
C. neoformans became apparent as early as 24 hours after the start of
irradiation.
In contrast, the data set from C. neoformans cells irradiated for a
longer time (48 hours) was less affected by differences in lag phase
duration, because the lag phase constituted a smaller fraction of
the total irradiation time, compared with 24 hour exposures.
Consequently, applying the model to this data set was more
informative: the distributions of parameter values produced by
fitting the model to bootstrapped data were relatively narrow
(Figure 5), and fitting the model by stepwise multiple linear
regression produced a relatively high adjusted R-squared (0.70),
with no evidence against normality of residuals (Shapiro-Wilk p-
value = 0.81). The effects of X-ray peak energy and cell
melanization were detected. Growth enhancement at low dose
rates was significantly higher for melanized cells exposed to
320 kVp X-rays, than for all other cell and exposure groups. A
common value of parameter A (100, 95% CI: 83, 116) was
produced (by stepwise multiple linear regression) for melanized
and non-melanized cells exposed to 150 kVp X-rays, and for non-
melanized cells exposed to 320 kVp X-rays. However, for
melanized cells exposed to 320 kVp X-rays, parameter A was
significantly higher, by 55 (95% CI: 29, 81), with a 2-tailed p-value
of 4.8610
25
. Consequently, this analysis suggested that each
melanized founder cell exposed to 320 kVp X-rays produced (on
average) 55 more descendants over 48 hours of exposure, than an
identically irradiated non-melanized founder cell, or than a
founder cell exposed to less energetic 150 kVp X-rays.
Dependence on X-ray peak energy was also suggested for dose
rate dependent growth inhibition (parameter B), which, after 48
hours of irradiation, was 24.4 (95% CI: 11.6, 37.2) h/Gy for
150 kVp exposures, and 46.7 (95% CI: 41.3, 52.1) h/Gy for
320 kVp exposures. However, these differences in parameter B
should be regarded with caution because the range of tested dose
rates for 150 kVp X-rays did not include values .5000 mGy/h,
i.e. those dose rates at which C. neoformans growth inhibition was
strongest. We intend to investigate the dose rate and energy
dependences of growth inhibition in more detail in future studies.
These applications of the model again suggested that: (1)
proliferation of C. neoformans is enhanced by low dose rates of both
150 and 320 kVp X-rays, and this is most clearly visible after 48
hours of continuous irradiation, (2) the effect of melanin on the
radiation response of C. neoformans cells exposed to 150 kVp X-rays
is not discernible, (3) higher energy (320 kVp) X-rays, however,
are clearly more effective in stimulating the proliferation of
melanized C. neoformans cells, compared with non-melanized cells.
Discussion
By combining quantitative mathematical modeling with Monte
Carlo data simulation techniques, and applying them to experi-
mental data on the effects of X-ray dose rates spanning six orders
of magnitude, at two different X-ray peak energies, on genetically
identical fungi which differ only by melanization status, our study
provides new insight into the influence of chronic irradiation on
fungal proliferation. The finding that low dose rates of ionizing
photons (X-rays) stimulate the growth of certain fungi in a dose
rate independent manner (i.e. the magnitude of the growth
enhancement is constant over a wide range of dose rates) suggests
that chronic irradiation induces a qualitative (rather than a
quantitative) shift in cell homeostasis, e.g. due to activation (or
release from inhibition) of certain metabolic and cell cycle-related
pathways. Interestingly, the observed stimulation of fungal growth
by X-rays seems to be independent of the carbon source in the
growth medium, as in this study the growth enhancement was
observed with acetate as a carbon source while previous
observations were made when fungi were grown with sucrose [8].
Melanin plays a subtle, but important role in modulating the
phenomenon of radiation-induced growth enhancement. This
finding is consistent with the higher survival of melanized vs. non-
melanized fungi after high-dose acute irradiation [13], and also
with data from radioactively-contaminated environments, where
melanized fungal forms tend to be over-represented in comparison
with non-melanized ones [2,7,9], presumably because even a small
competitive advantage provided by melanin would be sufficient for
melanized forms to dominate over the long term.
Intriguingly, our study suggests that effects of melanin on fungal
proliferation are X-ray peak energy dependent, with more
energetic X-rays (320 kVp) producing a greater growth-stimula-
tory effect on melanized C. neoformans cells, than less energetic
(150 kVp) ones. The mechanisms for this energy dependence are
as yet unknown. Prior reports have shown that ionizing radiation
alters the oxidation-reduction behavior of melanin [15], and that
melanin affects ATP levels in irradiated cells [12]. Perhaps these
interactions between melanin and ionizing photons are photon
energy dependent, and we intend to investigate this in future
studies.
It will also be worthwhile to examine and quantitatively model
the effects of chronic irradiation and melanin on genetic and
epigenetic regulation in fungi, particularly under stressful condi-
tions of nutrient limitation and/or suboptimal temperature, which
are relevant for fungal ecology in radiation-contaminated areas.
Materials and Methods
C. neoformans Culturing and Irradiation
Cryptococcus neoformans (strain H99) was cultured in liquid
medium for 4 days. Cells grown in medium containing 1 mM
L-DOPA became melanized, while those grown without L-DOPA
remained non-melanized. Immunofluorescence of melanized cells
was performed using FITC-conjugated melanin-specific antibody
11B11, which only binds to pigmented cells.
Melanized and non-melanized cells, at a starting concentration
of approximately 1000 colony-forming units per milliliter (CFU/
ml), were irradiated in darkness at 25+/21uC for 24 and 48 hours
in minimal medium [14], with 1 mM acetate as a carbon source,
and plated for colony formation. Irradiations were performed
twice, with 4 or 6 samples for each condition in each exposure
time, using the X-RAD 320 Biological Irradiator (Precision X-ray,
Inc.). Al, Cu, and Sn filters and Pb shielding allowed delivery of
dose rates ranging from 0.002 milli-Gray/hour (mGy/h) to
5500 mGy/h of X-rays with peak energies of 150 and 320 peak
kilovoltage (kVp). The filtering and shielding remove the least
energetic photons from the original machine-generated X-ray
spectrum, thereby making the energy distribution of photons
which reach the cells narrower and closer to the peak energy. The
dose rates reaching the cells were calculated using measurements
from the ion chamber built into the X-RAD 320 unit (which
measures the dose rate after the beam has passed the filter), and
from a second ion chamber (clinical model PTW N30013
SN05188) inserted under the lead shielding.
The growth kinetics of C. neoformans cultures under experimental
conditions began with a lag phase of variable duration (up to 18
hours) during which the cells did not proliferate, followed by
exponential growth (most rapid between 24 and 48 hours). Cell
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concentrations began to saturate after around 72 hours, sometimes
reaching 1.0 million CFUs/ml.
Data Analysis and Mathematical Modeling
Because CFU/ml counts are sensitive to variations in the length
of the lag phase, which occur in C. neoformans cultures with low
starting cell concentrations [16], we quantified the effects of
radiation on C. neoformans proliferation by calculating the
radiation-induced change (Q
e
) in CFU/ml, normalized by the
initial cell concentration, as follows: Q
e
=(X
r
(i) – X
c
)/X
0
, where
X
r
(i) is the CFU/ml for each irradiated culture, X
c
is the mean
CFU/ml for corresponding unirradiated controls, and X
0
is the
mean CFU/ml at the start of irradiation. Q
e
represents the
average number of extra descendants produced by each founder
cell due to radiation effects on proliferation.
The experimentally-derived Q
e
values were analyzed by fitting
the following mathematical formalism: Q = A(e, t, m) – [B(e, t, m)
x R]. Here Q is the predicted radiation-induced change in CFU/
ml, normalized by the initial cell concentration, and R is radiation
dose rate. The adjustable parameters (A and B) depend on X-ray
peak energy (e), duration of irradiation (t), and cell melanization
status (m). Parameter A represents radiation-induced growth
enhancement (assumed to be independent of dose rate), and
parameter B represents radiation-induced growth inhibition
(assumed to be proportional to dose rate).
The experimental data were divided into 8 sets, which represent
the 8 possible combinations of X-ray peak energy (e), duration of
irradiation (t), and cell melanization status (m): (1) non-melanized
cells, 150 kVp X-rays, 24 hour exposure duration, (2) melanized
cells, 150 kVp X-rays, 24 hour exposure duration, (3) non-
melanized cells, 320 kVp X-rays, 24 hour exposure duration, (4)
melanized cells, 320 kVp X-rays, 24 hour exposure duration, (5)
non-melanized cells, 150 kVp X-rays, 48 hour exposure duration,
(6) melanized cells, 150 kVp X-rays, 48 hour exposure duration,
(7) non-melanized cells, 320 kVp X-rays, 48 hour exposure
duration, (8) melanized cells, 320 kVp X-rays, 48 hour exposure
duration. The model, Q =A(e, t, m) – [B(e, t, m) x R], was fitted to
each of these 8 data sets separately, generating 8 combinations of
best-fit values for parameters A and B.
The effects of possible non-linear dependences of Q on
radiation dose rate were tested by adding to the model terms
such as R
0.1
,R
0.5
,orR
2
. These terms did not increase the adjusted
R-squared values for the model fits to the majority of the 8 data
sets. Consequently, they were not included in the model.
The sensitivity of model predictions to perturbations of the
observed data were explored by nonparametric bootstrapping
[17], using a customized code written in FORTRAN 77. Each of
the 8 experimentally observed data sets contained a certain
number of ‘‘elements’’, where each element represented a
combination of X-ray dose rate (R) and its effect on cell
proliferation (Q
e
). A perturbed data set was created by randomly
selecting (with equal probability) elements from the observed data
set. Thus, in the perturbed data set some elements could be
repeated, while others could be absent, but the total number of
elements remained the same as in the original observed data set.
The model, Q = A(e, t, m) – [B(e, t, m) x R], was fitted to the
perturbed data set, producing a combination of parameters A and
B. This procedure was repeated 10,000 times on each of the 8
experimental data sets, generating distributions of parameters A
and B.
Stepwise multiple linear regression [18] was used to assess the
ability of the model to describe the data, and to investigate the
effects of individual predictors such as X-ray peak energy and cell
melanization status. Briefly, the procedure consisted of adding to
the model a new predictor only if it significantly improved the
model fit (i.e. forward selection, using a p-value threshold of
,0.05), and removing predictors from the model if doing so did
not reduce the fit quality significantly (i.e. backward elimination,
using a p-value threshold ,0.1). These steps were continued until
only those predictors which significantly improved the model fit
remained in the model. The regression was performed indepen-
dently on data from 24 and 48 hour irradiations.
Author Contributions
Conceived and designed the experiments: IS RAB JDN ED. Performed the
experiments: IS RAB JDN. Analyzed the data: IS. Contributed reagents/
materials/analysis tools: RAB ED. Wrote the paper: IS RAB ED.
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Modeling the Growth of X-Irradiated Fungi
PLOS ONE | www.plosone.org 7 January 2014 | Volume 9 | Issue 1 | e85561