the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# HUB: a method to model and extract the distribution of ice nucleation temperatures from drop-freezing experiments

### Ingrid de Almeida Ribeiro

### Konrad Meister

### Valeria Molinero

The heterogeneous nucleation of ice is an important
atmospheric process facilitated by a wide range of aerosols. Drop-freezing
experiments are key for the determination of the ice nucleation activity of
biotic and abiotic ice nucleators (INs). The results of these experiments
are reported as the fraction of frozen droplets *f*_{ice}(*T*) as a function
of decreasing temperature and the corresponding cumulative freezing spectra
*N*_{m}(*T*) computed using Gabor Vali's methodology. The differential freezing
spectrum *n*_{m}(*T*) is an approximant to the underlying distribution of
heterogeneous ice nucleation temperatures *P*_{u}(*T*) that represents the
characteristic freezing temperatures of all INs in the sample. However,
*N*_{m}(*T*) can be noisy, resulting in a differential form *n*_{m}(T) that is challenging to interpret. Furthermore, there is no rigorous
statistical analysis of how many droplets and dilutions are needed to obtain
a well-converged *n*_{m}(*T*) that represents the underlying distribution
*P*_{u}(*T*). Here, we present the HUB (heterogeneous
underlying-based) method and associated Python codes that
model (HUB-forward code) and interpret (HUB-backward code) the results of
drop-freezing experiments. HUB-forward predicts *f*_{ice}(*T*) and *N*_{m}(*T*)
from a proposed distribution *P*_{u}(*T*) of IN temperatures, allowing its
users to test hypotheses regarding the role of subpopulations of nuclei in
freezing spectra and providing a guide for a more efficient collection of
freezing data. HUB-backward uses a stochastic optimization method to compute
*n*_{m}(*T*) from either *N*_{m}(*T*) or *f*_{ice}(*T*). The differential spectrum
computed with HUB-backward is an analytical function that can be used to
reveal and characterize the underlying number of IN subpopulations of
complex biological samples (e.g., ice-nucleating bacteria, fungi, pollen)
and to quantify the dependence of these subpopulations on environmental
variables. By delivering a way to compute the differential spectrum from
drop-freezing data, and vice versa, the HUB-forward and HUB-backward codes
provide a hub to connect experiments and interpretative physical quantities
that can be analyzed with kinetic models and nucleation theory.

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Ice nucleators (INs) of biological and abiotic origins present in aerosols
are responsible for facilitating the heterogeneous freezing of atmospheric
water droplets above the homogeneous nucleation temperature (Murray et
al., 2012; DeMott et al., 2016, 2003). The potential of these
aerosols as ice nuclei has significant implications for cloud properties and
precipitation patterns (Gettelman et al., 2012; Mülmenstädt et
al., 2015; Froyd et al., 2022). Freezing experiments are key sources of
information to determine the range of temperatures over which INs promote
ice nucleation. The most common method to characterize INs is through
immersion freezing experiments, for which a wide range of assays and
instruments have been developed. A comprehensive report of various drop-freezing techniques can be found in Miller et al. (2021). The assays are typically performed by placing uniformly sized water
droplets with a known IN concentration or area on a substrate or in a
multiwall plate that is gradually cooled from a temperature above 0 ^{∘}C
until all droplets are frozen (Kunert et al., 2018; Budke and Koop, 2015).
Droplet freezing is detected visually or through the measurement of the latent
heat release (Stratmann et al., 2004; Budke and Koop, 2015; Kunert et
al., 2018; Reicher et al., 2018), allowing the assignment of a heterogeneous
nucleation temperature to each droplet. Drop-freezing experiments record the
fraction of frozen droplets, *f*_{ice}(T), as a function
of decreasing temperature; for soluble or dispersible INs *f*_{ice}(T) curves are typically collected at various 10-fold
dilutions of the IN sample.

Historically, there have been two interpretations of the dispersion of
nucleation temperatures in heterogeneous freezing experiments. The first
approach suggests that the stochastic nature of the nucleation process
dominates the variability in freezing temperatures (Bigg, 1953; Carte,
1956), while the second approach assumes that the dispersion in temperatures
mostly arises from a distribution of nucleation sites (Fletcher, 1969),
each with a deterministic, singular nucleation temperature (Levine, 1950;
Vali and Stansbury, 1966). Variability in the temperature, volume, and
amount of ice-nucleating particles per droplet can also contribute to the
dispersion of freezing temperatures (Vali, 2019; Knopf et al.,
2020). There is consensus now that both stochastic effects and sample
heterogeneities contribute to the distribution of freezing temperatures, and
both approaches are used for the modeling of drop-freezing experiments
(Vali, 1971; Marcolli et al., 2007; Niedermeier et al., 2011; Murray et
al., 2011; Broadley et al., 2012; Wright and Petters, 2013; Herbert et al.,
2014; Harrison et al., 2016; Alpert and Knopf, 2016; Vali, 2019; Fahy et
al., 2022b). Stochastic modeling of the freezing curves is based on
predicting the survival probability of liquid water containing INs as a
function of supercooling, and it requires a model for the temperature
dependence of the nucleation rate of the IN components. These models have
been solved numerically or evolved with Monte Carlo simulations to interpret
or resolve the distribution of ice nucleation properties of minerals
(Marcolli et al., 2007; Murray et al., 2011; Broadley et al., 2012;
Wright and Petters, 2013; Herbert et al., 2014; Harrison et al., 2016) and
organics (Zobrist et al., 2007; Alpert and Knopf, 2016) and to perform
parametric bootstrapping of experimental data (Wright and Petters, 2013;
Harrison et al., 2016). The advantage of the stochastic modeling approach
is that it enables a direct link to microscopic properties of the nuclei and
can account for the cooling rate dependence of the *f*_{ice}(T) data. These approaches require the use of analytical models for the
freezing rates and their distribution in the sample.

The modeling of freezing experiments based on the singular approach is
based on the framework proposed by Vali (1971). He assumed
that each particular IN has a characteristic ice nucleation temperature that
is independent of the cooling history. This implies that the IN with the
highest characteristic nucleation temperature in a droplet is responsible
for its freezing. Given a total number of droplets *N*_{0}, the number of
frozen droplets *N*_{F}(*T*) at a temperature *T* gives the range of
characteristic freezing temperatures that determines the ice nucleation
activity and is used to produce the cumulative freezing spectrum (Vali,
1971, 2014, 2019):

where ${N}_{L}\left(T\right)={N}_{\mathrm{0}}-{N}_{\mathrm{F}}\left(T\right)$ is the number of unfrozen
droplets; ${f}_{\mathrm{ice}}\left(T\right)={N}_{\mathrm{F}}\left(T\right)/{N}_{\mathrm{0}}$ is the fraction of
frozen droplets at temperature *T*; and *X* is a normalization factor per unit
volume of water, unit mass, or surface of the INs (Vali, 2019).
For soluble INs, the normalization factor is commonly defined by the mass of
the ice-nucleating material $X=\mathit{\rho}({V}_{\mathrm{drop}}\left(\right)open="/">\phantom{{V}_{\mathrm{drop}}d}d)$, where *ρ* is the density of the initial solution, *V*_{drop} is the droplet volume,
and *d* is the dilution factor (Kunert et al., 2018).
The IN surface area per drop, *X*=*A*_{drop}, is sometimes used as the
normalization factor for insoluble INs (e.g., dust, crystals), resulting in
a cumulative spectrum per area denoted as *N*_{s}(T). However,
it is challenging to measure the total IN surface area accurately (Knopf
et al., 2020). We note that Eq. (1a) can be used even when the
absolute concentrations or areas of the INs are unknown, provided that the
user knows the relative concentration of the dilution series derived from a
parent sample. The differential freezing spectrum *n*_{m}(T) is
obtained by the differentiation of the cumulative spectrum (Vali,
1971):

The differential spectrum identifies the density of IN active at each temperature and was identified by Vali as the central quantity that can be derived and interpreted from drop-freezing experiments (Vali, 1971, 2019).

The determination of the differential spectrum from the cumulative one by
finite differentiation is subject to significant noise, requiring a careful
selection of the temperature intervals and extensive sampling
(Vali, 2019). As stochastic effects are not considered in the
singular temperature formalism, the cumulative and differential spectra
should – in principle – depend on the cooling rate (Vali, 1994). The
stochastic nature of ice nucleation, combined with the uncertainties
associated with the experimental measurements (e.g., different droplet
volumes, inhomogeneous samples, different detection efficiencies), can
produce significant variations in the cumulative freezing spectra that
result in large uncertainties in *n*_{m}(T) and we provide its associated Python code and user manual (https://github.com/Molinero-Group/underlying-distribution, last access: 5 May 2023), published in de
Almeida Ribeiro et al., 2023). Parametric and
nonparametric bootstrapping based on the singular approximation and Monte
Carlo simulations have been used to estimate confidence intervals in
freezing spectrum measurements (Vali, 2019; Fahy et al., 2022a, b).

A central assumption of the singular freezing approximation is that the
freezing of a droplet containing multiple INs is promoted by the IN with the
highest nucleation temperature (Levine, 1950). The extreme value sampling
is apparent in the concentration dependence of *f*_{ice}(T) in
experiments (Marcolli et al., 2007; Budke and Koop, 2015; Kunert et al.,
2018; Lukas et al., 2022). Using probability theory, Joseph Levine demonstrated
that if the distribution of ice nucleation temperatures of the IN population
follows an exponential distribution, then the sampling of droplet freezing
temperatures corresponds to a Gumbel distribution, and the median freezing
temperature *T*_{MED} of the droplets scales with the logarithm of the number
(or total nucleating area) of IN per droplet (Levine, 1950). Richard Sear more
recently demonstrated that Levine's approach is a particular solution for a
generalized extreme value problem and used modern extreme value statistics
to derive the scaling of *T*_{MED} with the number of IN sites per droplet
for the three generalized extreme value (GEV) distributions: Gumbel that
would arise from an underlying IN distribution with exponential tails,
Frechet from those with power law tails, and Weibull from those with an
upper cutoff in the freezing temperature of the INs (Sear,
2013). However, there are limitations for the use of the analytical
approaches of Sear and Levine for the interpretation of actual drop-freezing
data. First, the extreme value sampling results in one of the three GEV distributions only
in the limit of an extremely large number of INs per droplet, while in
experiments the sampling is typically performed over dilutions down to a few
INs per droplet. There is no analytical formulation for the dependence of the
extreme value distribution in the low to intermediate concentration regime.
Second, the analytical theory assumes that the sampling is complete (i.e., the number of droplets is extremely large), while experiments are typically
performed with tens to hundreds of droplets. Third, Sear notes that there is
no general analytical theory to predict the GEV distributions from a mixture of
populations of nuclei with different temperature dependences
(Sear, 2013). In this study we overcome these three
limitations through a numerical implementation of extreme value statistics
for the modeling of drop-freezing experiments.

A consequence of extreme value sampling is that the differential spectrum
*n*_{m}(T) represents the underlying distribution of ice
nucleation temperatures of all INs in the sample, which we denote as
*P*_{u}(*T*), only when the sampling of INs in the drop-freezing experiments is
complete. The underlying distribution *P*_{u}(*T*) is akin to a hub that
connects the experimental freezing temperatures to physical analysis based
on nucleation theory or kinetic and equilibrium models that can elucidate
the mechanisms and origins of the distributions of INs (Fig. 1). We
here call the cumulative spectrum *N*_{m}(T) obtained through
Eq. (1a) in this complete sampling limit the intrinsic cumulative
spectrum of the system, *I*_{u}(T) (Fig. 1). While
there is consensus that the quality of the freezing spectrum increases with
the number of droplets, a rigorous analysis of how many droplets and IN
dilutions should be measured to provide accurate freezing spectra is still
lacking. The first goal of the present study is to provide a strategy to
optimize the sampling of drop-freezing experiments to derive interpretable
differential spectra that are a good approximant of the underlying
distribution of heterogeneous ice nucleation temperatures of the sample.

The existence of subpopulations or classes in the population of INs (e.g., different classes of bacterial INs, different ice-nucleating sites on
complex materials like dust) (Turner et al., 1990) is
common in atmospheric aerosols. While several studies have broadly defined
populations from the cumulative spectra by the range of nucleation
temperatures they encompass (Turner et al., 1990; Creamean et al., 2019)
or the origin of the sample (Steinke et al., 2020),
there is currently no simple procedure to identify and quantify
subpopulations or classes from cumulative freezing spectra *N*_{m}(T). The second aim of our study is to map the cumulative freezing
spectrum *N*_{m}(*T*) into the differential spectrum *n*_{m}(*T*) in terms of
subpopulations that may correspond to different physical nucleation sites in
the sample.

To reach the aims above, we develop a method we name HUB (for heterogeneous underlying-based) to model and interpret the results of drop-freezing experiments and provide its associated Python code and user manual (https://github.com/Molinero-Group/underlying-distribution, last access: 5 May 2023). Our method relies on the singular interpretation of freezing experiments: we assume that each individual IN has a characteristic nucleation temperature independent of its cooling history and that the freezing of a droplet containing multiple INs is promoted by the IN with the highest nucleation temperature. This second assumption allows the use of extreme value statistics (Castillo et al., 2005; David and Nagaraja, 2004; Gumbel, 2012; De Haan and Ferreira, 2006) to model and interpret the data.

We present two implementations of the HUB analysis code. The HUB-forward
code allows the user to postulate an underlying distribution of
heterogeneous nucleation temperatures *P*_{u}(*T*) in the system of interest.
The HUB-forward code uses the singular approximation and extreme value
statistics to generate an artificial IN dilution series similar to those
obtained in experiments, from which it computes the fraction of frozen
droplets *f*_{ice}(*T*) and from these derive *N*_{m}(*T*) using Vali's equation
(Fig. 1). The HUB-backward code works in reverse, extracting the
differential spectrum *n*_{m}(*T*) from a given cumulative *N*_{m}(*T*) using a
stochastic optimization procedure (Fig. 1). HUB-backward allows the
decomposition of the total population from *n*_{m}(*T*) into subpopulations. The
combination of HUB-forward and HUB-backward allows for an analysis of the
sensitivity of *N*_{m}(*T*) to the number of droplets and dilutions, as well as the
impact of the sampling on the closeness of the differential spectrum
*n*_{m}(*T*) to the underlying distribution *P*_{u}(*T*). The determination of
distributions obtained from the HUB-backward code could further enable the
interpretation of the experimental ice nucleation spectra with the size and
structure of INs using nucleation theory, kinetic models, and molecular
simulations. For example, Schwidetzky et al. (2023)
illustrate the use of the distribution of freezing temperatures obtained
with HUB-backward, together with classical nucleation theory for finite
surfaces, to interpret the size of the INs of *Fusarium acuminatum*.

This paper is organized as follows: Sect. 2 presents the
methodology, and Sect. 2.1 discusses the details on the
implementation of HUB-forward, while Sect. 2.2 describes the
HUB-backward procedure to find the differential spectrum *n*_{m}(*T*) and
discusses how to determine whether or not *n*_{m}(*T*) has converged to the
underlying distribution *P*_{u}(*T*). Section 3 presents examples of
applications of both HUB-forward and HUB-backward codes and their
capabilities. Section 3.1 analyses the effect of the number of
droplets sampled on the cumulative freezing spectrum *N*_{m}(T). Section 3.2 uses HUB-backward to compute the differential
spectra *n*_{m}(*T*) of various biological INs with increasing grades of
complexity in their cumulative freezing spectra. Section 3.3
demonstrates how to extract *n*_{m}(*T*) from the experimental fraction of ice
*f*_{ice}(*T*) and the impact of the cooling rate on
*n*_{m}(*T*). We end in Sect. 4 with a discussion of the main
conclusions and outlook.

## 2.1 HUB-forward method to compute the fraction of frozen droplets *f*_{ice}(*T*) and cumulative freezing spectrum *N*_{m}(*T*) from a known underlying distribution *P*_{u}(*T*)

In the HUB-forward analysis we know or assume an underlying distribution
*P*_{u}(*T*) of ice nucleation temperatures for the IN in the sample and generate from it an artificial IN dilution series similar to those obtained
in experiments, from which we compute the cumulative freezing spectrum
*N*_{m}(*T*) using Vali's equation (Eq. 1a). Using this approach,
we investigate the relationship between *N*_{m}(*T*) and *P*_{u}(*T*)
(Fig. 1) and the sensitivity of *N*_{m}(*T*) with respect to the
number of droplets and dilutions. For generality, we represent *P*_{u}(*T*) as
a linear combination of normalized continuous distributions *P*_{i}(T) that represent subpopulations of freezing temperatures:

where *p* is the total number of subpopulations, ${P}_{\mathrm{1}}\left(T\right),\phantom{\rule{0.125em}{0ex}}{P}_{\mathrm{2}}\left(T\right),\phantom{\rule{0.125em}{0ex}}\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{P}_{p}\left(T\right)$ are normalized
distribution functions, and *c*_{1}, *c*_{2}, …, *c*_{p} are their
weights such that $\sum _{i=\mathrm{1}}^{p}{c}_{i}=\mathrm{1}$. These subpopulations could
correspond to different chemical, topographical, or structural motifs in the
IN samples, although chemically distinct species could also produce
overlapped freezing signatures, and a single species could display a broad
freezing range. Our formalism does not require a mapping of subpopulations
of freezing temperatures to physical IN sites. The units of *P*_{u}(*T*) are the
same as for *n*_{m}(*T*), i.e., those of the cumulative spectrum divided by a
unit of temperature, but are generally omitted in what follows. Throughout
this work we assume that *P*_{i}(T) can be represented by
Gaussian (i.e., normal) distributions:

where each subpopulation *P*_{i}(T) is further characterized by
its most likely temperature of freezing *T*_{mode,i} and spread of
distribution of freezing temperatures *s*_{i}. We also provide in the HUB
code the option for the user to use the log-normal distribution, which has a
tail towards higher temperatures, or the left-tailed Gumbel distribution,
which has a tail towards lower temperatures. In our model, we assume that
the underlying distribution of ice-nucleating temperatures *P*_{u}(*T*) does
not change with the concentration of INs. This last condition is violated
when INs are involved in chemical, aggregation, or solubility equilibria that
alter the proportionality between their concentration and the dilution
factor of the sample, resulting in a lack of overlap of the pieces of the
cumulative spectra *N*_{m}(*T*) obtained from different dilutions
(Bogler and Borduas-Dedekind, 2020).

The number of INs in each droplet is given by the Poisson distribution:

where *n* is the actual number of INs in each droplet and *λ*
represents the average number of INs among all droplets of the corresponding
dilution. Figure 2a shows the probability mass function (PMF) for
*λ*=1, 5, and 10, computed according to Eq. (4) and
sampling over *N*_{0}=10^{4} droplets using the “SciPy Stats” Python
framework (Virtanen et al., 2020). As *λ* increases, the
probability that any droplet nucleates homogeneously rapidly approaches zero
(inset of Fig. 2a). When there is one IN on average per droplet
$(\mathit{\lambda}=\mathrm{1}),\sim \mathrm{37}$ % of the droplets do not have any
INs; i.e., they are “empty” droplets that would nucleate at the homogeneous
nucleation temperature. We note that by performing dilutions until a
sizeable fraction of droplets nucleate homogeneously, it is possible to
calibrate the absolute concentration of ice nuclei in the original,
undiluted sample.

To illustrate how the heterogeneous ice nucleation temperatures recorded in
drop-freezing experiments depend on the number of INs in the droplets, we
start from two examples with *P*_{u}(*T*) represented by one or two Gaussian
subpopulations, shown with dashed black lines in Fig. 2b and
c, respectively. We assign a temperature to each IN contained in
droplets from a 10-fold dilution series of five solutions with *λ*=1,
10, 10^{2}, 10^{3}, and 10^{4} average number of INs per
droplet. If the droplet volume is constant, *λ* is proportional to
the concentration of INs in the droplets. We sample *N*_{0}=10^{4} droplets for each concentration. This *N*_{0} is much higher than the
∼100 droplets usually sampled in laboratory experiments; we
address the effect of sampling in Sect. 3.1 below.

To sample independent random values for each IN, the number of random
variates, which are drawn from *P*_{u}(*T*), is the total number of INs among
*N*_{0} droplets. Thereby, each droplet has a set of temperatures
${T}_{j}^{\mathit{\lambda}}=({T}_{\mathrm{1}}^{\mathit{\lambda}},\phantom{\rule{0.25em}{0ex}}{T}_{\mathrm{2}}^{\mathit{\lambda}},\phantom{\rule{0.25em}{0ex}}\mathrm{\dots},\phantom{\rule{0.25em}{0ex}}{T}_{k}^{\mathit{\lambda}})$, where *j* is the droplet index and *k* is the IN index.
Since we assume that freezing occurs at the characteristic temperature of
the IN with the highest freezing temperature, the nucleation temperature for
each droplet is defined as the maximum, i.e., the extreme upper value, of
several independent freezing temperatures ${T}_{\mathrm{het},j}^{\mathit{\lambda}}=max({T}_{\mathrm{1}}^{\mathit{\lambda}},\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{2}}^{\mathit{\lambda}},\phantom{\rule{0.125em}{0ex}}\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{T}_{k}^{\mathit{\lambda}})$.
Figure 2b and c show the normalized distribution of ${T}_{\mathrm{het},j}^{\mathit{\lambda}}$ for different values of *λ*, namely ${P}_{\mathrm{max}}^{\mathit{\lambda}}\left(T\right)$.
Therefore, *P*_{u}(*T*) represents the underlying probability of heterogeneous
ice nucleation temperatures independent of the concentration of INs, while
${P}_{\mathrm{max}}^{\mathit{\lambda}}\left(T\right)$ represents the concentration-dependent distribution
and has the same units as *P*_{u}(*T*) and the differential spectrum.
According to the Fisher–Tippett–Gnedenko theorem, the distribution of
extreme upper values of the Gaussian distribution is the right-skewed Gumbel
distribution (Castillo et al., 2005; David and Nagaraja, 2004; Gumbel, 2012; De
Haan and Ferreira, 2006), which has a fatter tail on the high-temperature
side of its maximum. Indeed, the shift in ${P}_{\mathrm{max}}^{\mathit{\lambda}}\left(T\right)$ curves in
Fig. 1b and c evinces that as the number of INs in the droplet
increases, the probability of sampling the higher temperature tail of
*P*_{u}(*T*) increases significantly. This skew is the reason why several
dilutions are needed to sample the full population of ice nucleants.

HUB-forward computes the fraction of frozen droplets and cumulative spectra from a proposed underlying distribution of freezing temperatures, using extreme value statistics. The fraction of frozen droplets ${f}_{\mathrm{ice}}^{\mathit{\lambda}}\left(T\right)$ can be calculated as a function of the concentration-dependent distribution:

where the integration is from the ice melting temperature *T*_{m} to the
temperature $T,{N}_{\mathrm{F}}^{\mathit{\lambda}}$ is the total number of droplets that
freeze heterogeneously, and *N*_{0} is the total number of droplets. We note
that the approach taken in this work differs from that of previous studies
that start from a microscopic model for the nucleation sites and
nucleation theory to predict the fraction of frozen droplets using Monte
Carlo simulations, as well as from previous modeling using the singular
approximation, which do not account for the statistics of extreme sampling.

To use the HUB-forward code, the user must define the total number of
droplets “ndroplets” that serves as the total number of each concentration
and the number of subpopulations “nsubpop”. If “nsubpop” = 1, the user
must provide the temperature of maximum likelihood *T*_{mode,1} and the
spread *s*_{1}. If “nsubpop” = 2, the user must provide *T*_{mode,1},
*s*_{1}, *T*_{mode,2}, *s*_{2} and *c*_{2}. If “nsubpop” = 3, the user has
to provide *T*_{mode,1}, *s*_{1}, *T*_{mode,2}, *s*_{2}, *c*_{2}, and *T*_{mode,3},
*s*_{3}, *c*_{3}. To generate the cumulative freezing spectrum *N*_{m}(T), the user needs to define the total number of concentrations
“nconc”; the concentration of the parent suspension is defined in
“density” and the droplet volume in “volumedrop”. The output is
composed of different data plots and files: the normalized *P*_{u}(*T*) and
${P}_{\mathrm{max}}^{\mathit{\lambda}}\left(T\right)$, the artificially generated
${f}_{\mathrm{ice}}^{\mathit{\lambda}}\left(T\right)$, and *N*_{m}(T) built from the 10-fold
dilution series.

Figure 3a and b show the fraction of ice computed using
${P}_{\mathrm{max}}^{\mathit{\lambda}}\left(T\right)$ of Fig. 2b and c, which correspond
to *P*_{u}(*T*) with two subpopulations and one subpopulation, respectively. The
intermediate plateau in Fig. 3b indicates that no droplets freeze
at those temperatures. As discussed above, only 63 % of the droplets
freeze heterogeneously for *λ*=1. We assume droplets of uniform
volume *V*_{drop}=0.1 µL obtained through 10-fold dilution
of a parent suspension with *λ*=10^{4} INs per droplet
corresponding to a mass *m*=1 mg of IN in a volume *V*_{wash}=1 mL.
We use Eq. (5) and the *P*_{u}(*T*) of Fig. 2b and c to generate
${f}_{\mathrm{ice}}^{\mathit{\lambda}}\left(T\right)$ (Fig. 3a and b), sampling either 100 or
10^{4} droplets per dilution. We combine the ${f}_{\mathrm{ice}}^{\mathit{\lambda}}\left(T\right)$ using
Eq. (1a) to build the cumulative freezing spectra *N*_{m}(*T*) shown
in Fig. 3c and d (sampling 10^{4} droplets per dilution) and
Fig. 3e and f (sampling 10^{2} droplets per dilution).

The ability of HUB-forward to generate the cumulative freezing spectrum
*N*_{m}(*T*) from the underlying distribution *P*_{u}(*T*) allows for an
analysis of the sensitivity of *N*_{m}(*T*) and *P*_{u}(*T*) to the number of
droplets and dilutions, as seen in the comparison of *N*_{m}(T)
generated from the same underlying distributions using 100 and 10^{4}
droplets in Fig. 3. In Sect. 3.1 we show that the
sampling with 100 droplets for only four dilutions of a system with two
subpopulations of INs results in distortions of the distribution of freezing temperatures
and the proportions of these populations in the differential spectrum.

The knee point in *N*_{m}(*T*) corresponds to the point of maximum curvature
(Satopaa et al., 2011) and has been used to characterize the
nucleation temperature of a particular subpopulation
(Hartmann et al., 2022). Similar to Hartmann et al. (2022), we have
identified in Fig. 3c and d the knee points (dotted magenta line) of
the artificially generated *N*_{m}(*T*) by using a Python function named
“kneed”. The Python function “kneed” using *S*=1, curve = “concave”, and
direction = “decreasing”. The knee points *T*_{knee} are very close to the
temperatures of maximum likelihood *T*_{mode} (dashed black lines) of the
corresponding underlying distribution *P*_{u}(T), because under
these conditions the differential freezing spectrum *n*_{m}(*T*) is a very
good approximant for *P*_{u}(*T*). However, we find that the removal of the more
dilute solutions eliminates the plateau in *N*_{m}(*T*) and results in poor
estimation of the modes of *P*_{u}(*T*) from the knee of *N*_{m}(*T*).

## 2.2 HUB-backward method to recover the differential freezing spectrum *n*_{m}(*T*) from the cumulative freezing spectrum *N*_{m}(T) by a stochastic optimization procedure

The HUB-backward code implements a stochastic optimization procedure to
extract the differential spectrum *n*_{m}(*T*) from a given cumulative
spectrum *N*_{m}(*T*) or from an experimental *f*_{ice}(T) curve.
The latter is useful when data are available for a single concentration. One
possibility for obtaining *n*_{m}(*T*) from *N*_{m}(*T*) would be to follow the
following steps: (i) propose a trial function ${n}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$, (ii) use
HUB-forward to predict the concentration-dependent distributions
${P}_{\mathrm{max}}^{\mathit{\lambda},\mathrm{trial}}\left(T\right)$ for various IN concentrations,
(iii) use these in Eq. (5) to predict the freezing fractions
${f}_{\mathrm{ice}}^{\mathit{\lambda},\mathrm{trial}}\left(T\right)$, (iv) compute ${N}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$
from the freezing fractions using Eq. (1a), (v) evaluate the
difference between that trial and the target (experimental) value using

and then (vi) evolve the parameters that determine ${n}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$ until
the difference *δ*(T) is minimized. However, the use of
HUB-forward in steps (ii) and (iii) to generate and evaluate hundreds of
droplets containing up to tens of millions of INs would require significant
computations that render this optimization process inefficient.

The HUB-backward optimization procedure, sketched in Fig. 4, uses a
shortcut for steps (ii) and (iii) above to directly predict ${N}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$
from ${n}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$ with fast convergence. The shortcut is based on the
understanding that, in the asymptotic limit in which the sample is extremely
dilute (i.e., *λ*⟶0), each droplet that nucleates
heterogeneously contains a single IN. In such a case, sampling an infinitely
large number of droplets with ${P}_{\mathrm{max}}^{\mathit{\lambda}\u27f6\mathrm{0}}\left(T\right)$ is equivalent to sampling each and every IN, i.e., ${P}_{\mathrm{max}}^{\mathit{\lambda}\u27f6\mathrm{0}}\left(T\right)={P}_{\mathrm{u}}\left(T\right)$. In agreement
with this ansatz, Fig. 2b and c show that the underlying distribution
*P*_{u}(*T*) (dashed black line) and the concentration-dependent
${P}_{\mathrm{max}}^{\mathit{\lambda}=\mathrm{1}}\left(T\right)$ (blue squares) sampled with 10^{4}
droplets per dilution are already very close, i.e., ${P}_{\mathrm{u}}\left(T\right)\approx {P}_{\mathrm{max}}^{\mathit{\lambda}=\mathrm{1}}\left(T\right)$.

With this insight and considering the intrinsic cumulative spectrum, ${I}_{\mathrm{u}}\left(T\right)={\int}_{{T}_{\mathrm{m}}}^{T}{P}_{\mathrm{u}}\left({T}^{\prime}\right)\mathrm{d}{T}^{\prime}\times (\mathrm{1}-{e}^{-\mathrm{1}})$, we define the cumulative integral of the differential spectrum as

where the integration is from the ice melting temperature *T*_{m} to the
temperature *T* and *β* is an adjustable scaling factor to be
obtained from the optimization. Likewise, a similar estimate can be made for
a single fraction of ice curve ${f}_{\mathrm{ice}}^{\mathrm{trial}}\left(T\right)={I}_{\mathrm{u}}^{\mathrm{trial}}\left(T\right)$ using Eq. (7) and the mean squared error can be directly evaluated (Fig. 4). When the
target is a cumulative freezing spectrum, HUB-forward uses
${n}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$ to predict a trial cumulative freezing
spectrum (Fig. 4),

where $\mathrm{1}/X$ corresponds to the maximum of the cumulative spectrum
in the target distribution, $\mathrm{1}/X=max\left[{N}_{\mathrm{m}}^{\mathrm{target}}\left(T\right)\right]$. With Eq. (8) we obtain an ${N}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$
that we compare with the target using Eq. (6) (Fig. 4). To
do the comparison, HUB-backward uses a spline fit to interpolate the
experimental ${N}_{\mathrm{m}}^{\mathrm{target}}\left(T\right)$, in order to have equally spaced
temperature points to compare with the estimates in ${N}_{\mathrm{m}}^{\mathrm{trial}}\left(T\right)$.
We use the “interp1d” algorithm, which is available in the Python SciPy
library (Virtanen et al., 2020), with a linear interpolation to construct
new equally spaced data points within the range of the lowest and highest
temperature values in the freezing spectrum. The cost function for the
optimization is the mean squared error (MSE), computed from the difference
*δ*(T) in Eq. (6):

where *t* represents the total number of equally spaced points in *δ*(T).

We use a stochastic global optimization technique based on a simulated
annealing algorithm to find the set of parameters of ${n}_{\mathrm{m}}^{\mathrm{trial}}$
(Eqs. 2 and 3) and *β* (Eq. 7)
that globally minimize the MSE. We use the simulated annealing (SA)
algorithm “dual annealing” that is part of the SciPy minimize library
(Virtanen et al., 2020) with its default arguments predefined, except for
the parameters “maxfun” that sets the maximum number of evaluations of
the objective function (we select “maxfun” = 1 000 000 in the examples
below) and the seed for the generation of random numbers (a new random
integer is automatically generated every time the HUB-backward code is run).
We show below that the optimized differential spectra,
${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$, are quite insensitive to the value of
the seed.

The output of HUB-backward is an optimized differential spectrum
${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ or an optimized fraction of ice
${f}_{\mathrm{ice}}^{\mathrm{optimized}}\left(T\right)$. To quantify how much this optimized prediction
deviates from the known underlying distribution in the examples of
Fig. 5, where *P*_{u}(*T*) is known, we define the mean relative error
(MRE) for the set of parameters:

where *p* is the number of subpopulations.

We now turn our focus to how to select the input parameters required by
HUB-backward to start the search for the underlying distribution, using the
experimental ${N}_{\mathrm{m}}^{\mathrm{target}}\left(T\right)$ or ${f}_{\mathrm{ice}}^{\mathrm{target}}\left(T\right)$ as a guide. The
code requires the user to define the number of distinct Gaussian
subpopulations *P*_{i}(*T*) that comprise the underlying distribution
(Eq. 2) and to provide upper and lower bounds for the weights
*c*_{i}, their modes *T*_{mode,i}, and spreads *s*_{i} of each of these
populations. In general, we find that defining the minimum and maximum
values for the weights to ${c}_{i}^{\mathrm{max}}=\mathrm{1}$ and ${c}_{i}^{\mathrm{min}}=\mathrm{0}$ (see constraint in Eq. 2), for the modes ${T}_{\mathrm{mode},i}^{\mathrm{max}}$ and ${T}_{\mathrm{mode},i}^{\mathrm{min}}$ to between the homogeneous nucleation temperature (about −30 ^{∘}C) and
the melting temperature (0 ^{∘}C), and for the spreads to
${s}_{i}^{\mathrm{max}}=\mathrm{10}$ ^{∘}C and ${s}_{i}^{\mathrm{min}}=\mathrm{0.1}$ ^{∘}C works well.
However, these bounds can be tuned in order to better fit the data (as we
find for pollen in Sect. 3.2
below). If the existing experimental ${N}_{\mathrm{m}}^{\mathrm{target}}\left(T\right)$ data are very
noisy, they can be interpolated in HUB-backward using the method “interp1d”
with “npoints” = 100 and then smoothed with a Savitzky–Golay filter by
changing the parameters “window_length”, which is the length
of the filter window, and “polyorder”, which is the order of the polynomial
used to fit the samples (“filter” in Fig. 4). The default values
are 3 and 1, respectively. HUB-backward generates a plot that compares the
original and the interpolated target data.

To identify the minimum number of subpopulations needed to represent a given
freezing spectrum, we consider that every time a population is accumulated
in *N*_{m}(*T*) or *f*_{ice}(*T*), these functions display a sharp increase. We
note that assuming a large number of subpopulations may challenge the
interpretability of the optimized differential spectrum
${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$.

We apply the HUB-backward procedure to the *N*_{m}(T) obtained
in Fig. 3c and d by sampling four 10-fold dilutions with 100 droplets, i.e., only a total of 500 droplets. Figure 5 shows the
comparison between the predicted (solid magenta lines) and the target (dashed black
lines) *N*_{m}(T) (panels a and b) and
*n*_{m}(T) (panels c and d). Table 1
shows the predicted parameters and the precision of the optimization
procedure to recover the known underlying distribution *P*_{u}(*T*). The MRE
between the underlying distribution *P*_{u}(*T*) and the optimized
differential spectrum ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ is 2 % for the system with
one subpopulation and 13 % for the one with two despite the low number of
droplets used to sample the cumulative freezing spectra in the
computer-generated freezing experiments.

We conclude that the HUB-backward code gives a good estimate of the mode,
spread, and weights of the populations of INs in a sample and it can be
applied in a situation where *P*_{u}(*T*) is unknown. In Sect. 3.1
we discuss how the accuracy is of the underlying distribution recovered with
HUB-backward impacted by various schemes of the sampling of the number of droplets
and dilutions to construct *N*_{m}(T). In Sect. 3.2,
we apply the HUB-backward procedure to obtain ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ from
actual *N*_{m}(T) of experiments with various soluble
biological INs. In Sect. 3.3, we apply the HUB-backward procedure
to obtain ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ from *f*_{ice}(T) of
experiments of insoluble crystal INs.

## 3.1 Effect of the number of droplets and dilutions on the cumulative freezing spectrum *N*_{m}(*T*)

Figure 3d–f show *N*_{m}(*T*) generated with HUB-forward using five
dilutions from *λ*=10^{4} to 1 of a solution with *P*_{u}(*T*)
containing two populations in a ratio of 9 to 1. The *N*_{m}(*T*) are
different when the number of droplets per dilution is 100 (Fig. 3f)
or 10^{4} (Fig. 3d). As shown in the previous section, the
freezing spectrum obtained with 100 droplets and five dilutions has enough
sampling to recover this *P*_{u}(*T*) with good accuracy (Fig. 5c and d).
We test different number of droplets and concentrations, defined by the
average number of INs per droplet *λ*, to test the sensitivity of
*n*_{m}(*T*) to the number of droplets and dilutions when the underlying
distribution *P*_{u}(*T*) is known. We use HUB-forward to build *N*_{m}(*T*)
based on a combination of different numbers of droplets and concentrations,
similar to the case shown in Fig. 3f. Then, we use HUB-backward
to obtain ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$, compare it to *P*_{u}(T), and
test the accuracy of each prediction through its mean relative error (MRE) defined in Eq. (10).

The left panels of Fig. 6 show *N*_{m}(*T*) generated with
HUB-forward based on a combination of different concentrations using 100
droplets for each dilution. The magenta lines are based on the data provided by
the HUB-backward code. The right panels of Fig. 6 compare ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$
in magenta and the known underlying distribution *P*_{u}(T) in
black. In this example, *n*_{m}(*T*) is very close to *P*_{u}(T)
if both subpopulations are sampled enough. However, if the most dilute
solution with *λ*=1 is not included in *N*_{m}(*T*) (second panel),
the estimate of the underlying distribution is very poor. Thus, to improve
the sampling of the lower tail of *P*_{u}(T), we recommend
ending the dilution series always in the immediacy of *λ*=1, which
can be gleaned from the temperature range for which *N*_{m}(T)
becomes flat and a sizeable fraction of droplets of the more diluted sample
nucleates homogeneously (inset of Fig. 2a). The one-to-one correspondence between the fraction of droplets nucleated homogeneously and the average number of particles in the droplet in the highly diluted limit (inset of Fig. 2a) demonstrates that
reaching this limit allows for an *absolute* calibration of the number of INs in the
initial sample. Moreover, sampling to concentrations down to about one
nucleant per droplet is essential to recover a proper weight of the poorly
nucleating IN populations.

The relative weights of class A and C populations in *Pseudomonas syringae* is approximately 1 to
1000 (Sect. 3.2), while the ratio is 9 to 1 in the two-population
system example of Fig. 6. To understand the impact of highly
imbalanced populations on the sampling of the cumulative spectrum and
recovery of the underlying distribution, we show in Fig. 7 the
analysis of an example where the subpopulation of highly efficient INs is
3 orders of magnitude less likely to occur than the subpopulation at
lower temperatures, mimicking the one of *P. syringae*. Our analysis confirms that it is
important to end the dilution series in the immediacy of *λ*=1 to
fully represent the contribution of the poorer INs (Fig. 7b–f).
Furthermore, we find that it is important to sample a concentration high enough
to account for the rare INs that nucleate at the warmest temperatures
(Fig. 7d–h).

If only 25 droplets per dilution, instead of 100, are used to construct the
cumulative spectrum, the impact of insufficient sampling at the higher
concentrations is more pronounced: compare Fig. 8c and Fig. 7d obtained with the same underlying distribution *P*_{u}(T)
with 1000 to 1 subpopulation ratios and number of dilutions.

We conclude that an increase in the accuracy in the account of the subpopulations requires a higher number of dilutions and the checking of the predictions with the addition of each successive concentration to ensure convergence of ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$. Measuring fewer droplets or fewer dilutions leads to poor statistics and results in incompleteness or the misrepresentation of the underlying distribution in samples with multiple subpopulations. In principle, increasing the number of droplets of the most concentrated solutions or adding more 10-fold concentrated ones until there are no changes in the cumulative spectrum is recommended to ensure complete sampling. When that limiting scenario is not attainable, the use of HUB-forward to produce synthetic data from a proposed underlying distribution, followed by the recovery of the differential spectrum from these data sets, allows for an estimation of the errors that may be incurred for putative, proposed underlying distributions with the sampling scheme available in the laboratory.

## 3.2 Obtaining the differential freezing spectrum *n*_{m}(*T*) from the experimental cumulative freezing spectrum *N*_{m}(*T*) of biological INs using the HUB-backward code

In this section we use the HUB-backward code to obtain the differential
freezing spectrum *n*_{m}(*T*) from the cumulative freezing spectra *N*_{m}(*T*)
of the fungi *Fusarium acuminatum* strain 3–68 (Kunert et al., 2019),
the bacterium *P. syringae* (Schwidetzky et al., 2021), and birch
pollen (Dreischmeier, 2019). We select these systems because they
are important biological INs and show increasing complexity in terms of the
apparent number of underlying distributions that define their freezing
spectra.

The experimental *N*_{m}(*T*) obtained for *F. acuminatum* (black squares in Fig. 9a) was obtained by sampling six 10-fold dilutions, each with 96 droplets
(Kunert et al., 2019). Figure 9a shows the
cumulative spectra optimized assuming one (green curve) and two (cyan curve)
subpopulations; Fig. 9b shows the corresponding optimized
differential freezing spectra. The ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ with a single
subpopulation that peaks at −5.9 ^{∘}C is unable to represent the
cumulative density of the most potent nuclei and misses the inflection at
around −5.9 ^{∘}C in the experimental data, resulting in a mean squared
error MSE = 0.05. The ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ with two subpopulations has a
lower MSE = 0.003 and a better fit that suggests a population that peaks
at −7.3 ^{∘}C and another at −5.5 ^{∘}C, in comparable amounts
(Table 2). Most notably, the two subpopulations do not overlap in
the differential freezing spectrum, supporting that they may indeed
correspond to different physical entities. The improvement in the fit
becomes apparent in the inset of Fig. 9a, which shows *N*_{m}(*T*) on
a linear scale. The significant slope of *N*_{m}(*T*) even at the lowest
temperatures indicates that the sampling of more diluted solutions is needed to
capture the contribution of the less active INs. An attempt to represent *F. acuminatum*
nucleation data with three different subpopulations resulted in two of them
being almost identical. We conclude that adding a third subpopulation is
unnecessary to reproduce the experimental cumulative freezing spectrum of
*F. acuminatum*. We refer the reader to Schwidetzky et al. (2023) for an
interpretation of the size of the ice-nucleating surface of *F. acuminatum* based on its
differential spectrum and nucleation theory.

Next, we apply the HUB-backward code to analyze the experimental freezing
spectrum of Snomax^{®}, i.e., inactivated *P. syringae*. The cumulative spectrum
suggests the presence of two distinct subpopulations, usually called class A
(at warmer temperatures) and class C (at colder ones). We first assume the
differential freezing spectrum *n*_{m}(*T*) of *P. syringae* is a combination of two
Gaussian populations. The parameters of the optimized differential spectrum
with two subpopulations are listed in Table 2, and the curve is
shown in Fig. 9d with a cyan line. We use a
logarithmic scale to represent this ${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ because the
population corresponding to class A accounts for less than 0.1 % of the
total (Table 2). While the fit with two subpopulations results in a
good overall account of the target data, we note that there is some
difference in the region between classes A and C (Fig. 9c). The
fitting for *P. syringae* achieves an excellent agreement between optimized and target
cumulative spectra (Fig. 9c), through the prediction of an
additional peak located between classes A and C (the elusive class B), with
a population comparable to class A (Table 2 and red curve in
Fig. 9d). However, more measurements and analyses are needed to
establish whether this “class B” peak at −5.2 ^{∘}C is reproducible and
truly distinct from the one of class A at −3.7 ^{∘}C to warrant a physical
interpretation. Overall, both the analyses with two and three subpopulations
agree with previous ones (Govindarajan and Lindow, 1988; Warren, 1987)
that concluded that over 99 % of the INs active in *P. syringae* bacteria in
Snomax^{®} belongs to class C. The analysis presented here for
fungal and bacterial INs illustrates how HUB-backward can be used to reveal
and characterize the underlying number of IN subpopulations of complex
biological samples.

To further test the methodology, we model the cumulative freezing spectrum
of birch pollen. Given that the original *N*_{m}(*T*) data for pollen in Fig. 3.1 of Dreischmeier (2019) consist of multiple independent curves,
we took one of the many presented in this graph as the target
${N}_{\mathrm{m}}^{\mathrm{target}}\left(T\right)$ (black curve in Fig. 9e) and present some of
the additional data – not used in the optimization – with gray circles in
Fig. 9e. Section S4 in the Supplement shows that the differential
spectrum optimized from the whole data set and its sparse sampling are
almost identical because HUB-forward interpolates and smooths the input
data to produce an equispaced data set. The ${N}_{\mathrm{m}}^{\mathrm{target}}\left(T\right)$ seems to
contain three quite separated subpopulations, which is confirmed by the
accuracy of the optimized cumulative spectrum in Fig. 9e. The
parameters of the optimized differential freezing spectrum
${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ and the MSE are shown in Table 2. Our
analysis indicates that the two subpopulations that nucleate ice above
−16 ^{∘}C constitute less than 0.01 % of the active nucleating sites in
pollen (Fig. 9e), consistent with drop-freezing assays that only
measured solutions with low concentrations of birch pollen and did not
observe freezing at higher temperatures (Augustin et al., 2013; Pummer et
al., 2012; Felgitsch et al., 2018), while the more extensive data of
Dreischmeier (2019) reveal two more active subpopulations of INs.

To further illustrate the use of HUB-backward, Fig. 10 shows the
effect of pH in the modes, spread, and weights of the subpopulations that
contribute to the nucleation spectrum of *P. syringae* (Snomax^{®}), using
data from Lukas et al. (2020). Freezing in the temperature
range of class A drops about 3 orders of magnitude when the pH is lowered
from 6.2 to 4.4 (Fig. 10b). However, we note that the cumulative
number of INs is preserved in the experimental cumulative freezing spectrum
(Lukas et al., 2020), indicating that the change in pH did
not impact the number of nucleants. Figure 10c and d demonstrate that
the distributions associated with both subpopulations shift to lower
temperatures when the pH decreases, and the range of freezing temperatures
in class A becomes broader. An attempt to fit the cumulative spectra of
Snomax at different pH values with the same subpopulations, allowing only for
adjustment of their weights, resulted in a poor fit to the experimental
*N*_{m}(*T*), supporting the conclusions of Lukas et al. (2020) of a central role of electrostatic interactions in the assembly of the
bacterial ice-nucleating proteins and their ability to bind to ice. This
analysis exemplifies how HUB-backward can be applied to quantify the
dependence of IN on environmental variables.

## 3.3 Obtaining the differential freezing spectrum *n*_{m}(*T*) from the experimental fraction of ice *f*_{ice}(*T*) of insoluble ice nucleators using the HUB-backward code

Section 3.1 and 3.2 discuss how to obtain the
differential spectrum from a target cumulative one. However, there are many
cases where the results are presented as a fraction of frozen droplets as a
function of temperature, *f*_{ice}(*T*). In these cases, the HUB-backward code
can be used to obtain the optimized differential freezing spectrum
${n}_{\mathrm{m}}^{\mathrm{optimized}}\left(T\right)$ directly from ${f}_{\mathrm{ice}}^{\mathrm{target}}\left(T\right)$.
Section S5 illustrates this approach for the analysis of droplet freezing
data for a sample of lignin (Bogler and Borduas-Dedekind,
2020) in which the INs participate in aggregation equilibria. Here, we
exemplify the optimization of the differential spectrum of cholesterol from
experimental freezing data obtained at two cooling rates with droplets
sampled from a single dilution.

In the analysis of drop-freezing experiments, it is assumed that each IN has a
singular freezing temperature, independent of the cooling rate. However, ice
nucleation is a stochastic process, and the underlying distribution of
freezing temperatures *P*_{u}(T) strictly depends on both
temperature and cooling rate, as slower rates give more time for the system
to cross the nucleation barrier at warmer temperatures.

The triangles and squares in Fig. 11a display the experimental
*f*_{ice}(*T*) obtained by sampling the freezing of hundreds of 120 µL
droplets pipetted from a suspension of cholesterol monohydrate crystals in
contact with Teflon cooled at 0.18 K min^{−1} (triangles) and 0.06 K min^{−1}
(squares) (Zhang and Maeda, 2022). Our analysis of the freezing data of cholesterol
monohydrate shows that even a 3-fold change in the cooling rate can have a significant
effect on the differential spectrum (Fig. 11b).

As expected, the modes of the
three populations move towards warmer temperatures upon decreasing the
cooling rate. We note, however, that the shift in the peaks is not uniform;
the middle one seems to be more sensitive to the cooling rate. Different
sensitivity of the freezing rate of subpopulations has also been
reported in simulations of nucleation data of minerals using the
stochastic and modified singular frameworks (Herbert
et al., 2014; Murray et al., 2011). The modified singular model proposes an
empirical correction to the relation between *f*_{ice}(*T*) and *N*_{m}(*T*) to
account for the effect of the cooling rate on the shift in these quantities
(Vali, 1994). That analysis could be extended to the analysis of the
subpopulations of INs obtained with HUB-backward. Moreover, it would be
interesting in future studies to use the rate dependence of the mode of the
subpopulations to extract the steepness of the nucleation barrier with
temperature using nucleation theory (Budke and Koop, 2015)
and to investigate the relationship between the cooling rate dependence of
the differential spectrum obtained in the singular approximation with the
interpretation of the same data modeled with the stochastic framework, such
as in Wright et al. (2013) and Herbert et al. (2014).

In this study, we present the HUB method and associated Python codes that
model (HUB-forward code) and interpret (HUB-backward code) the results of
droplet freezing experiments under the assumptions that each ice-nucleating
site in the sample has a characteristic nucleation temperature that is
time-independent. The use of the singular approximation is the same as that used
by Vali (1971, 2014, 2019) in his derivation of the ice
nucleation spectra from data of fraction of frozen droplets. Different to
previous implementations of the singular model, HUB accounts for the
distribution of the number of INs in droplets at a given concentration and
uses extreme value statistics to represent the effect of dilutions in the
frozen fraction and freezing spectra. Our method and codes allow users to
obtain an analytical differential freezing spectrum *n*_{m}(*T*) from the
experimental distribution of freezing temperatures, and vice versa. The
differential freezing spectrum *n*_{m}(*T*) is an approximant to the
underlying distribution of ice-nucleating temperatures *P*_{u}(T), which provides a hub to connect the experimental freezing
temperatures with interpretative physical analyses using kinetic models or
nucleation theory that can be used to elucidate the mechanisms of nucleation
and origins of these distributions.

HUB-forward predicts the cumulative ice nucleation spectrum *N*_{m}(*T*) and
fractions of frozen droplets *f*_{ice}(*T*) from a known (or assumed)
underlying distribution *P*_{u}(*T*) of nucleation temperatures for the INs in the sample. The HUB-forward code can be used to investigate the effect of
the number of droplets and dilutions on the temperature range of the
cumulative freezing spectrum *N*_{m}(*T*). Our analysis shows that the
differential freezing spectrum *n*_{m}(*T*) is identical to the underlying
distribution of heterogeneous ice nucleation temperatures *P*_{u}(*T*) only
when sampling is complete. Measuring fewer droplets or fewer dilutions can
result in a biased representation of the differential and cumulative
spectra. HUB-forward predicts *f*_{ice}(*T*) and *N*_{m}(*T*) from a proposed
distribution of IN temperatures, allowing its users to test hypotheses
regarding the role of subpopulations of nuclei in the freezing spectra and
providing a guide for a more efficient collection of freezing data.

HUB-backward uses a non-linear optimization method to find the differential
freezing spectrum *n*_{m}(*T*) that best represents the experimental target
cumulative freezing spectrum *N*_{m}(*T*) or fraction of frozen droplets
*f*_{ice}(*T*) in the experiments. The analytical form of the differential
freezing spectrum *n*_{m}(*T*) obtained from HUB-backward offers an
interpretable physical basis. The interpretability of the results in terms
of subpopulations provides an advantage over polynomial fitting and
differentiation of *N*_{m}(*T*). Indeed, we show that the HUB-backward code
can be used to reveal and characterize the underlying number of IN
subpopulations of complex biological samples (Snomax^{®}, fungi
*Fusarium acuminatum*, and birch pollen) and quantify the dependence of their subpopulations on
environmental variables. Interestingly, our analysis evinces subpopulations
that are not obvious to the eye and have not previously been identified in
these samples. The robustness of the signals that correspond to these
populations and their physical nature require further investigation.

We illustrate the use of HUB-backward to obtain the differential freezing
spectrum *n*_{m}(*T*) from the fraction of frozen droplets *f*_{ice}(*T*)
collected at a single concentration. We apply that analysis to demonstrate
that *n*_{m}(*T*) depends on the cooling rate. The shift in the peaks of the
subpopulations to higher temperatures upon decreasing the cooling rate is
not unexpected, as longer waiting times allow for the surmounting of the same
nucleation barrier at warmer temperatures. By providing the temperature
dependence of the mode, spread, and weight of the subpopulation peaks,
HUB-backward can be combined with nucleation theory and other theoretical
analyses to extract the steepness, and maybe even the distribution, of
nucleation barriers that control the freezing process.

All codes, a user manual, and input files used in this project can be accessed at https://github.com/Molinero-Group/underlying-distribution (last access: 5 May 2023) and https://doi.org/10.5281/zenodo.7901549 (de Almeida Ribeiro et al., 2023).

All data used in this project can be accessed upon request to the authors.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-23-5623-2023-supplement.

VM, IdAR, and KM designed the project. IdAR developed the model code and performed the simulations. IdAR and VM prepared the manuscript with contributions from KM.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Ingrid de Almeida Ribeiro and Valeria Molinero gratefully acknowledge support by AFOSR through MURI award no. FA9550-20-1-0351. Konrad Meister acknowledges support by the National Science Foundation under grant no. NSF 2116528 and from the Institutional Development Awards (IDeA) from the National Institute of General Medical Sciences of the National Institutes of Health under grant nos. P20GM103408 and P20GM109095.

This research has been supported by the Air Force Office of Scientific Research (grant no. FA9550-20-1-0351), the National Institutes of Health (grant nos. P20GM103408 and P20GM109095), and the National Science Foundation (grant no. 2116528).

This paper was edited by Daniel Knopf and reviewed by Nadine Borduas-Dedekind and one anonymous referee.

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- Abstract
- Introduction
- Numerical modeling of drop-freezing experiments
- Using the HUB code to optimize and analyze drop-freezing experiments
- Conclusions
- Code availability
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

- Abstract
- Introduction
- Numerical modeling of drop-freezing experiments
- Using the HUB code to optimize and analyze drop-freezing experiments
- Conclusions
- Code availability
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement