# Score-based diffusion priors for multi-target detection

Alon Zabatani  School of Electrical Engineering
Tel Aviv University
<EMAIL_ADDRESS>Shay Kreymer  School of Electrical Engineering
Tel Aviv University
<EMAIL_ADDRESS>Tamir Bendory  School of Electrical Engineering
Tel Aviv University
<EMAIL_ADDRESS>
###### Abstract

Multi-target detection (MTD) is the problem of estimating an image from a
large, noisy measurement that contains randomly translated and rotated copies
of the image. Motivated by the single-particle cryo-electron microscopy
technology, we design data-driven diffusion priors for the MTD problem,
derived from score-based stochastic differential equations models. We then
integrate the prior into the approximate expectation-maximization algorithm.
In particular, our method alternates between an expectation step that
approximates the expected log-likelihood and a maximization step that balances
the approximated log-likelihood with the learned log-prior. We show on two
datasets that adding the data-driven prior substantially reduces the
estimation error, in particular in high noise regimes.

###### Index Terms:

Diffusion models, multi-target detection, cryo-EM, score-SDE, expectation-
maximization

##  I Introduction

We study the two-dimensional multi-target detection (MTD) problem: estimating
an image $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ from a single, noisy
measurement that contains multiple copies of the image, each randomly rotated
and translated [1, 2]. In particular, we model the measurement
$M\in\mathbb{R}^{N\times N}$ as

$M[\vec{\ell}]=\sum_{i=1}^{p}F_{\phi_{i}}[\vec{\ell}-\vec{\ell}_{i}]+\varepsilon[\vec{\ell}],$
(1)

where $F_{\phi_{i}}[\vec{\ell}]=f_{\phi_{i}}(\vec{\ell}/n)$ is a discrete copy
of $f$, rotated by angle $\phi_{i}$ about the origin, $n$ is the known radius
of the image in pixels, $\\{\phi_{i}\\}_{i=1}^{p}$ and
$\\{\vec{\ell}_{i}\\}_{i=1}^{p}$ are the unknown rotations and translations,
respectively, and $\varepsilon[\vec{\ell}]$ is i.i.d. Gaussian noise with zero
mean and variance $\sigma^{2}$.

The goal is to estimate the image $f$ from measurement $M$, while the
rotations, translations, and the number of occurrences of $f$ in $M$, denoted
by $p$, are unknown. Importantly, since the rotations are unknown, it is
possible to reconstruct the target image only up to a rotation. For
simplicity, we assume that the translations are not densely packed, namely,

$|\vec{\ell}_{i}-\vec{\ell}_{j}|>4n,\quad\text{ for all }i\neq j.$ (2)

This assumption was also taken by previous works [2, 1] for simplification.
The algorithm presented in this paper can be readily extended to handle
arbitrarily spread signal occurrences, which violate (2), along with the
guidelines that were proposed in [3, 4]. Figure 1 illustrates measurements
with the digit three, taken from the MNIST dataset [5], at different signal-
to-noise ratios (SNRs). We define
$\text{SNR}\triangleq\frac{||F||^{2}_{\text{F}}}{A\sigma^{2}}$, where $A$ is
the area in pixels of $F$.

The MTD model is mainly studied as a mathematical abstraction of the single-
particle cryo-electron microscopy (cryo-EM) technology for determining
macromolecular structures [6, 7, 8]. In a cryo-EM experiment, individual
copies of a target biomolecule, rotated by unknown 3-D rotations, are
scattered across unknown 2-D locations within a thin layer of vitreous ice,
from which 2-D tomographic projection images are produced by an electron
microscope [9]. Maintaining a low electron dose is imperative to prevent
irreversible structural damage, leading to significantly noisy projection
images.

The conventional cryo-EM data processing pipeline [10, 11, 12, 13] involves
the initial detection and extraction of the 2-D projection images from the
measurement, followed by their rotational and translational alignment to
reconstruct the 3-D molecular structure. However, this approach fails for
small molecules, which are too difficult to detect and align [6, 14, 15, 10].
Consequently, the MTD model—that can be thought of as a simplification of the
cryo-EM model, where 2-D rotations replace the 3-D rotations, and the
tomographic projection operator is neglected—was devised in [15, 16], to study
the recovery of small molecular structures, without intermediate detection,
below the current detection limit [17].

(a) No noise

(b) $\text{SNR}=1$

(c) $\text{SNR}=0.1$

Figure 1: Three measurements at different SNRs: (a) no noise; (b)
$\text{SNR}=1$; (c) $\text{SNR}=0.1$. Each measurement contains multiple
rotated versions of the digit three, taken from the MNIST dataset. We are
primarily interested in low SNR regimes (e.g., panel (c)), in which it is
extremely challenging to estimate the locations and rotations associated with
the image occurrences.

Previous works proposed an expectation-maximization (EM) algorithm to solve
the MTD problem [3, 18], outperforming an autocorrelation analysis alternative
[2, 1, 4]. EM is an iterative algorithm to find maximum likelihood (or maximum
a posteriori) estimators, where the underlying model depends on unobserved
nuisance parameters (in our case, the rotations and translations associated
with the $p$ image occurrences). EM alternates between an expectation step
(E-step), which calculates the expected log-likelihood evaluated using the
current estimate of the parameters, and a maximization step (M-step), which
finds the parameters maximizing the function computed in the E-step [19].
While EM cannot be applied directly to the MTD problem, due to a large number
of possible translations that should be taken into account, the authors of [3]
suggested an approximation scheme that dramatically reduces the computational
burden, while providing high estimation quality; see Section III for details.

Motivated by recent attempts to utilize prior information to constitute
molecular structures in cryo-EM (see for example [20, 21]), we suggest
incorporating score-based diffusion priors into the approximate EM framework.
Diffusion models [22, 23, 24, 25] are a class of probabilistic generative
models that progressively perturb data by injecting noise, then learn to
reverse this process, thus capturing the implicit prior of the underlying data
distribution. Diffusion models have been successfully applied to a variety of
tasks [26, 27, 28, 29], such as image synthesis, where they have broken the
long-time dominance of generative adversarial networks (GANs) [30]. Score-
based stochastic differential equations (score-SDE) [25] is a popular
diffusion model, that makes use of the Stein score—the gradient of the log
probability density[31, 32]. The key idea behind score-SDE is to perturb the
data with a sequence of intensifying Gaussian noise governed by a closed-form
stochastic differential equation, and jointly estimate the score functions for
all noisy data distributions by training a deep neural network, conditioned on
the noise levels.

The main contribution of this paper is the integration of score-based
diffusion priors into the approximate EM framework, resulting in a flexible
algorithm that balances between the log-likelihood term (the data) and the
log-prior term using a regularized gradient ascent algorithm. We demonstrate
on two datasets that we can successfully apply it to the two-dimensional MTD
problem (1) and significantly improve image recovery quality, including in
highly noisy regimes.

The rest of the paper is organized as follows. Section II introduces the
score-SDE method. Section III presents the integration of score-SDE priors
into the approximate EM framework. Section IV provides numerical results on
two datasets (Gaussian images and MNIST), and Section V concludes the paper
and delineates future applications to cryo-EM.

##  II Score-based Diffusion Models

Score-based diffusion models learn the implicit prior of the underlying data
distribution by matching the gradient of the log probability density with
respect to data point $x$, i.e., Stein score function [31, 32]

$s(x)\triangleq\nabla_{x}\log p(x).$ (3)

We approximate the score function using a neural network $s_{\theta}(x)$,
where the network’s parameters $\theta$ are being learned by minimizing the
following score matching loss:

$\text{loss}(\theta)=\mathbb{E}_{x\sim p(x)}||s_{\theta}(x)-\nabla_{x}\log
p(x)||^{2}_{2}.$ (4)

However, in practice, the true gradients of the log probability density
functions are often unknown during training. In [32], it was shown that the
loss function (4) is equivalent to

$\text{loss}(\theta)=\mathbb{E}_{x\sim
p(x)}\left(\frac{1}{2}||s_{\theta}(x)||^{2}_{2}+\operatorname*{tr}\left(\nabla_{x}s_{\theta}(x)\right)\right),$
(5)

where $\text{tr}(\cdot)$ is the trace of the Jacobian matrix
$\nabla_{x}s_{\theta}(x)$. Unfortunately, learning the score function by
minimizing the loss function (5) tends to fail in regions where there are
fewer training samples (due to the expectation over the data density); this
can be mitigated by perturbing the data with various noise levels, as shown in
[24]. Specifically, score-SDE [25] perturbs the data with a sequence of
intensifying Gaussian noise, governed by a closed-form stochastic differential
equation, and jointly estimates the score functions for all noisy data
distributions. The SDE is defined as follows

$dx=-\frac{\beta(t)}{2}xdt+\sqrt{\beta(t)}dw,$ (6)

where $\beta(t):\mathbb{R}\rightarrow\mathbb{R}>0$ is the noise schedule of
the process, typically taken to be a monotonically increasing linear function
of $t$, and $w$ is the standard Wiener process. The data distribution is
defined when $t=0$, i.e., $x(0)\sim p_{data}$, and a simple, tractable
distribution (e.g., Gaussian) is assumed when $t=T$, i.e.,
$x(T)\sim\mathcal{N}(0,I)$.

With the evolution of the distribution along time $t$, the neural network
accepts both data point $x$ and time $t$ as inputs and produces an estimate of
the score function, i.e., $s_{\theta}(x,t)$. In this case, the denoising
score-matching loss function can be written as

$\text{loss}(\theta)=\mathbb{E}_{t\sim
U[0,T]}\mathbb{E}_{p(x(0))}\mathbb{E}_{p(x(t)|x(0))}\\\
\Big{(}\lambda(t)||s_{\theta}(x(t),t)-\nabla_{x(t)}\log
p(x(t)|x(0))||_{2}^{2}\Big{)},$ (7)

where $\lambda(t)$ is a positive weighting function and $p(x(t)|x(0))$ can be
analytically calculated using (6) to be:

$p(x(t)|x(0))\sim\mathcal{N}(\mu(t),\sigma^{2}(t)I),$ (8)

where

$\displaystyle\mu(t)=x(0)\exp\left(-\frac{1}{2}\int_{0}^{t}\beta(s)ds\right),$
$\displaystyle\sigma^{2}(t)=1-\exp\left(-\int_{0}^{t}\beta(s)ds\right).$

The expectations are approximated by averaging over the data points and the
integrals can be solved analytically for the specified functions (full
implementation details can be found in Table I).

##  III Approximate EM with score-based prior

The EM algorithm finds a (local) maximum of the posterior distribution (the
MAP estimator) by iteratively applying an expectation step (E-step), followed
by a maximization step (M-step). The E-step calculates the expectation of the
log-likelihood with respect to all admissible configurations of nuisance
parameters (translations and rotations in our case). The M-step then finds the
parameters maximizing the function found in the E-step, possibly with the
addition of a prior term [19].

Direct application of the EM algorithm to the MTD problem is computationally
intractable, due to the large number of possible translations that grows
quickly with the measurement size. Specifically, under the separation
constraint (2), the runtime of a single iteration grows as
$\mathcal{O}(N^{4})$, where $N^{2}$ is the size of the measurement M (1),
assuming the measurement is much larger than the target image. Approximate EM
that has been introduced in [3] runs in $\mathcal{O}(N^{2})$ per iteration, by
maximizing an approximation of the log-likelihood function. The approximate EM
algorithm has been applied to the two-dimensional MTD problem in [18] and to
the cryo-EM problem in [33]. However, in both cases, the prior on the signal
has been ignored, which is equivalent to considering an uninformative prior.
The main goal of this work is to introduce a data-driven prior into the
approximate EM algorithm and to pave the way to learn and implement such prior
for cryo-EM datasets.

Approximate EM begins with partitioning the measurement M into $D=(N/L)^{2}$
non-overlapping patches, each of size $L\times L$ (we assume for convenience
that $N/L$ is an integer). The separation condition implies that each patch
$M_{m},m=0\ldots,D-1$, can contain either no target image, a full target
image, or a part of the target image; overall there are $(2L)^{2}$
possibilities, excluding rotations. In particular, each patch can be modeled
by

$M_{m}=CT_{\vec{\ell}_{m}}ZF_{\phi_{m}}+\varepsilon_{m},\quad\varepsilon_{m}\sim\mathcal{N}(0,\sigma^{2}I),$
(9)

where $F$ is a square image of size $L\times L$, representing a discrete copy
of target image $f$, the operator $Z$ zero-pads $L$ entries to the right and
to the bottom of a rotated copy of $F$, and $T_{\vec{\ell}_{m}}$ circularly
shifts the zero-padded image by
$\vec{\ell}_{m}=({\ell_{m}}_{x},{\ell_{m}}_{y})\in\mathbb{L}\triangleq\\{0,1,\ldots,2L-1\\}^{2}$
positions, that is,

$(T_{\vec{\ell}_{m}}ZF_{\phi_{m}})\left[i,j\right]=\\\
(ZF_{\phi_{m}})\left[(i+{\ell_{m}}_{x})\bmod 2L,(j+{\ell_{m}}_{y})\bmod
2L\right].$ (10)

The operator $C$ then crops the image to size $L\times L$, i.e.,
$CT_{\vec{\ell}_{m}}ZF_{\phi_{m}}=T_{\vec{\ell}_{m}}ZF_{\phi_{m}}[0:L-1,0:L-1]$,
and the result is further corrupted by additive white Gaussian noise. Figure 2
provides an illustration of the model. In addition, since the EM algorithm
assigns probabilities to rotations (in the E-step), we uniformly discretize
the search space of rotations:

$\phi_{m}\in\Phi\triangleq\left\\{k\frac{2\pi}{K}\right\\},\quad
k=0,\ldots,K-1,$ (11)

where $K$ is a hyper-parameter for the algorithm.

(a)

(b)

(c)

(d)

(e)

Figure 2: A step-by-step illustration of the patch generation model described
in (9).

The patches $\\{M_{m}\\}_{m=0}^{D-1}$ are our observations, whereas the
rotations $\\{\vec{\ell}_{m}\\}_{m=0}^{D-1}$ and translations
$\\{\phi_{m}\\}_{m=0}^{D-1}$ are the unobserved nuisance parameters. Our goal
is thus to find the pixel intensity values of $F$, and we denote by
$F^{(\tau)}$ the estimated image at iteration $\tau$ of the EM procedure. In
the E-step, we calculate the expectation of log-likelihood function of the
model, with respect to the conditional distribution of the nuisance parameters
given the patches and the current estimate $F^{(\tau)}$:

$Q(F|F^{(\tau)})\triangleq\mathbb{E}\left[\log\mathfrak{L}\big{|}M_{0},\ldots,M_{D-1};F^{(\tau)}\right],$
(12)

where $\mathfrak{L}$ is the likelihood function, defined as

$\mathfrak{L}\triangleq
p(M_{0},\ldots,M_{D-1},\vec{\ell}_{0},\ldots,\vec{\ell}_{D-1},\phi_{0},\ldots,\phi_{D-1};F)\\\
\approx\prod_{m=0}^{D-1}p(M_{m},\vec{\ell}_{m},\phi_{m};F),$ (13)

where we neglect statistical dependencies between patches.

We note that Bayes’ rule implies

$p(\vec{\ell}_{m},\phi_{m}|M_{m},F^{(\tau)})\\\
=\frac{p(M_{m}|\vec{\ell}_{m},\phi_{m},F^{(\tau)})p(\vec{\ell}_{m},\phi_{m}|F^{(\tau)})}{\sum_{\vec{\ell^{\prime}}\in\mathbb{L}}\sum_{\phi^{\prime}\in\Phi}p(M_{m}|\vec{\ell^{\prime}},\phi^{\prime},F^{(\tau)})p(\vec{\ell^{\prime}},\phi^{\prime}|F^{(\tau)})},$
(14)

where from (9) we can write

$p(M_{m}|\vec{\ell}_{m},\phi_{m},F)\propto\exp\left(-\frac{\|M_{m}-CT_{\vec{\ell_{m}}}ZF_{\phi_{m}}\|_{\text{F}}^{2}}{2\sigma^{2}}\right),$
(15)

with the normalization
$\sum_{\vec{\ell}\in\mathbb{L}}\sum_{\phi\in\Phi}p(M_{m}|\vec{\ell}_{m},\phi_{m},F)=1$.
We assume that in each patch
$p(\vec{\ell}_{m},\phi_{m}|F)=p(\vec{\ell}_{m})p(\phi_{m})=\frac{1}{(2L)^{2}}\cdot\frac{1}{K}$,
namely, $\vec{\ell}_{m}$ and $\phi_{m}$ are independent of $F$ and of each
other, and are uniformly distributed. Finally, utilizing the approximation in
(13), we can write the expected log-likelihood function, up to a constant, as:

$Q(F|F^{(\tau)})=\sum_{m=0}^{D-1}\sum_{\vec{\ell}\in\mathbb{L}}\sum_{\phi\in\Phi}p(\vec{\ell}_{m},\phi_{m}|M_{m},F^{(\tau)})\\\
\times\log p(M_{m}|\vec{\ell}_{m},\phi_{m},F).$ (16)

The M-step updates the current image estimate $F^{(\tau)}$ by maximizing the
Q-function (16), with the addition of a regularized prior term:

$F^{(\tau+1)}=\operatorname*{argmax}_{F}\left(Q(F|F^{(\tau)})+\gamma^{(\tau)}\log
p(F)\right),$ (17)

where $\gamma^{(\tau)}$ is a weighting function, applied to the prior term at
iteration $\tau$. In our numerical experiments, the maximization process is
being conducted using a fixed number of gradient ascent steps with a fixed
learning rate $\mu$ and the initial guess of $F^{(\tau)}$. The gradient of the
Q-function can be analytically computed, whereas the gradient of the prior
term is approximated by our diffusion model $s_{\theta}(\cdot)$. Thus, a
single gradient ascent step can be written as:

$\begin{split}F^{(\tau+1)}&=F^{(\tau)}+\mu\nabla_{F}\left(Q(F|F^{(\tau)})+\gamma^{(\tau)}\log
p(F)\right)\Big{\rvert}_{F=F^{(\tau)}}\\\
&=F^{(\tau)}+\mu\left(\nabla_{F}Q(F|F^{(\tau)})\rvert_{F=F^{(\tau)}}+\gamma^{(\tau)}s_{\theta}(F^{(\tau)})\right).\end{split}$
(18)

The algorithm is summarized in Algorithm 1.

Input: measurement $M$; noise variance $\sigma^{2}$; initial guess $F^{(0)}$;
rotation range $K$ (11); max EM iterations $\mathcal{T}$; stopping parameter
$\epsilon$; learning rate $\mu$; regularization weights $\gamma$; prior model
$s_{\theta}(\cdot)$; gradient ascent steps $N$

Output: an estimate of $F$

1 for  _$\tau\leftarrow 0$ to $\mathcal{T}$_ do

2 calculate $\nabla_{F}Q(F|F^{(\tau)})$ according to (16)

3 set $F_{0}\leftarrow F^{(\tau)}$

4 for  _$i\leftarrow 0$ to $N$_ do

5 set $F_{i+1}\leftarrow
F_{i}+\mu\left(\nabla_{F}Q(F_{i}|F^{(\tau)})+\gamma^{(\tau)}s_{\theta}(F_{i})\right)$

6

7 set $F^{(\tau+1)}\leftarrow F_{N}$

8 if  _$||F^{(\tau+1)}-F^{(\tau)}||_{\text{F}} <\epsilon$_ then

9 break

10

return $F^{(\tau)}$

Algorithm 1 Approximate EM with score-based prior

##  IV Experimental Results

In this section, we implement our proposed algorithm on two different
datasets: (A) Gaussian $5\times 5$ images ($L=5$) and (B) MNIST $28\times 28$
images ($L=28$). As our baseline, we consider the plain approximate EM
framework (i.e., without diffusion priors) that has been proposed in [18]. We
examine both algorithms on a range of SNR values scaled for each dataset; for
example, the sparsity of the MNIST images requires lower SNR compared to
Gaussian images so that the target images are not easily detected (as in panel
(c) of Figure 1). Specifically, for the Gaussian case we test
$\text{SNRs}=\\{1,5,10\\}$, and for the MNIST case we test
$\text{SNRs}=\\{0.1,0.5,1\\}$. We generate the measurements according to the
MTD model (1), with $N=11L$ and density $\frac{pL^{2}}{N^{2}}=0.1$. The
rotations are uniformly sampled from the set
$\Phi\triangleq\\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\\}$, so that $K=4$ (11).
Finally, for each dataset we examine 10 target images (that are hidden from
the prior models during training) and report the mean estimation error,
defined by $E=\min_{\phi\in\Phi}\frac{||F^{*}-F_{\phi}||_{2}}{||F^{*}||_{2}}$,
where $F^{*}$ denotes the ground truth image and $F_{\phi}$ denotes the
estimated image, rotated by $\phi$. The error is minimized over the set of
rotations because of the intrinsic rotational symmetry of the problem.
Additional implementation details are provided in Table I. The code to
reproduce the experiments is available at
https://github.com/zabatani/2d-mtd_diffusion.

Figure 3: The mean estimation error of recovering target images $F$ drawn from
a Gaussian distribution, as a function of SNR, using approximate EM with and
without score-based prior. Evidently, the score-based prior addition is
beneficial in all SNRs, especially in low SNR regions.

###  IV-A Gaussian Images

The score network for this dataset consists of three fully-connected layers
with Softplus activation functions. The network has been trained on 10,000
samples drawn from a normal distribution with covariance identity and mean
$\mu$, where each entry of $\mu$ has been drawn i.i.d. from a uniform
distribution over $\\{0,1,2,3,4\\}$. The target images were drawn from the
same distribution. Training took approximately 10 minutes on an M1 Macbook
Air. The results are illustrated in Figure 3. Evidently, our algorithm is
superior in all SNRs and exhibits a significant improvement in recovery
accuracy, especially in low SNR levels.

Figure 4: The mean estimation error of recovering target images $F$ drawn from
the MNIST dataset, as a function of SNR, using approximate EM with and without
score-based prior. Evidently, the score-based prior addition is noticeably
beneficial in all SNRs.

###  IV-B MNIST

The score network for this dataset consists of a U-Net architecture [34], with
ten convolutional layers and LogSigmoid activation functions. The network has
been trained on the entire MNIST dataset (60,000 samples), excluding the 10
target images that have been tested. The training took approximately 10 hours
on a T4 GPU machine. In this case, we have noticed that proper weighting of
the prior term during gradient ascent plays a key role in our algorithm’s
success and that a monotonically increasing linear function of $\tau$ (the
iteration of the EM procedure) works best; see Table I for details. The
results are illustrated in Figure 4. Our algorithm is superior in all SNR
levels. In addition, we show reconstructions of all digits for a fixed
$\text{SNR}=1$, using both algorithms, in Figure 5. The improvement in
recovery quality is visible for all digits.

(a) Target image

(b) Approximate EM without prior

(c) Approximate EM with score-based prior

Figure 5: Recovery of all ten digits from the MNIST dataset of the MTD model
with $\text{SNR}=1$, using the approximate EM algorithm with and without
score-based prior. The recovery quality boost achieved by our algorithm (panel
(c)) is clearly visible. Besides the background denoising and digit
sharpening, it also successfully recovers the target image in cases where the
competitor deteriorates (e.g., digit six).

##  V Conclusion

In this paper, we proposed to integrate score-based diffusion priors into the
approximate EM framework in order to estimate images in the two-dimensional
MTD problem. Our method shows a superior image recovery quality compared to
[18], which considers an uninformative prior, in various SNRs and on two
datasets: a simple Gaussian case, and the more intricate data distribution of
MNIST.

This paper is part of an ongoing effort to recover small molecular structures
using cryo-EM. The MTD model (1) is a simplified model of the cryo-EM model
that includes 3-D rotations (rather than 2-D rotations in (1)) and tomographic
projections [10, 11]. The ultimate goal of this research is to learn a prior
of protein structures that were resolved using electron microscopes, and then
incorporate it into a computational method for recovering small molecular
structures [15, 33]. To learn the priors, we intend to devise 3-D steerable
CNN architectures [35] and train them in a denoising score-matching fashion
(7). This would require collecting enough data to train such neural networks.
To this end, we plan to use databases of resolved protein structures, such as
the electron microscopy data bank (EMDB) [36], as well to augment data
generated by computational tools, such as AlphaFold [37, 38]. Moreover, we
intend to explore additional ways to incorporate the score function into the
EM framework that might take advantage of the reverse-time SDE (6) (commonly
used for sample generation), for example, by a clever integration between the
SDE time evolution and the EM iteration progression. Finally, we intend to
integrate our prior into additional algorithms for MTD, such as
autocorrelation analysis [2, 1, 15, 39].

## Acknowledgment

The research is supported in part by the BSF grant no. 2020159, the NSF-BSF
grant no. 2019752, and the ISF grant no. 1924/21.

TABLE I: The parameters used for the numerical experiments. Parameter | Description | Gaussian | MNIST
---|---|---|---
$\mathcal{T}$ | EM iterations | 100 | 100
$N$ | Gradient ascent steps | 1 | 1
$\gamma^{(\tau)}$ | Regularization weighting function (18) | 1 | $\tau/\mathcal{T}$
$\mu$ | Learning rate (18) | $1e-3$ | $1e-3$
$\epsilon$ | Stopping parameter | $1e-5$ | $1e-5$
$F^{(0)}$ | Initialization | $U[0,1]$ | Average of ten digits
$\text{loss}(\theta)$ | Prior loss function | (5) | (7)
$\beta(t)$ | Noise schedule (6) | N/A | $0.1+(20-0.1)t$
$\lambda(t)$ | Weighting function (7) | N/A | $1-e^{\int_{0}^{t}\beta(s)ds}$

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# Nanoscale Water Behavior and Its Impact on Adsorption:
A case study with CNTs and Diclofenac

Patrick R. B. Côrtes  Departamento de Física, Instituto de Física e
Matemática, Universidade Federal de Pelotas, Caixa Postal 354, 96.001-970,
Pelotas, Brasil  Nicolás A. Loubet  Cintia A. Menéndez  Gustavo A. Appignanesi
INQUISUR, Departamento de Química, Universidad Nacional del Sur (UNS)-CONICET,
Avenida Alem 1253, 8000 Bahía Blanca, Argentina  Mateus H. Köhler
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Santa
Maria, 97105-900, Santa Maria, Brasil  José Rafael Bordin
<EMAIL_ADDRESS>Departamento de Física, Instituto de Física e
Matemática, Universidade Federal de Pelotas, Caixa Postal 354, Pelotas, Brasil

###### Abstract

Water is a fundamental component of life, playing a critical role in
regulating metabolic processes and facilitating the dissolution and transport
of essential molecules. However, the presence of emerging contaminants, such
as pharmaceuticals, poses significant challenges to water quality and safety.
Nanomaterials-based technologies arise as a promising tool to remove those
contaminants from water. Nevertheless, interfacial water plays a major role in
the adsorption of chemical compounds in the nanomaterials - as it plays in
biological processes such as protein folding, enzyme activity, and drug
delivery. To understand this role, in this study we employ Molecular Dynamics
(MD) simulations to explore the adsorption dynamics of potassium diclofenac
(K-DCF) on single-walled (SWCNT) and double-walled (DWCNT) carbon nanotubes,
considering both dry and wet conditions. Our findings reveal that the
structuring of water molecules around CNTs creates hydration layers that
significantly influence the accessibility of active sites and the interaction
strength between contaminants and adsorbents. Our analysis indicates higher
energy barriers for adsorption in DWCNTs compared to SWCNTs, which is
attributed to stronger water-surface interactions. This research highlights
the importance of understanding nanoscale water behavior for optimizing the
design and functionality of nanomaterials for water purification. These
findings can guide the development of more efficient and selective
nanomaterials, enhancing contaminant removal and ensuring safer water
resources, while also contributing to a deeper understanding of fundamental
biological interactions.

##  I Introduction

Water is indispensable for life, serving as a fundamental component in
regulating metabolic processes. It acts as a solvent, facilitating the
dissolution and transport of nutrients, gases, and waste products throughout
living systems Laage, Elsaesser, and Hynes (2017); Bordin  _et al._ (2023). In
biological systems, the relevance of water extends to the adsorption process,
which is vital for the proper function of cellular membranes and enzymes.
Water molecules form hydration shells around biomolecules, influencing their
structure, stability, and interactions Köhler, Barbosa, and Barbosa (2017);
Khodadadi and Sokolov (2017); Shikata  _et al._ (2024). This hydration layer
is crucial for the adsorption of substrates on enzyme active sites, affecting
the efficiency of metabolic reactions and nutrient absorption Mondal and
Bagchi (2022). Additionally, hydration water affects the adsorption of drugs
by biomolecules, influencing their bioavailability and therapeutic
effectiveness by contributing to the free-energy of the interaction between
the drug molecule and their targets Bagchi (2005); Fenter and Lee (2014).
Therefore, understanding the behavior of water in adsorption processes is
essential for advancing our knowledge of life Weng, Stott, and Toner (2019);
Pal and Chattopadhyay (2021).

Also, ensuring safe drinking water for everyone is a significant challenge. To
address this and other emerging social issues, in 2015, the United Nations
signed a pact with several countries to establish 17 Sustainable Development
Goals (SDGs) to be achieved by 2030 United Nations, Department of Economic and
Social Affairs - Sustainable Development (2015). Two of these goals are
directly linked to water quality: SDG 6, "Clean Water and Sanitation," and SDG
14, "Life Below Water." However, recent reports indicate that we are not on
track to meet these goals United Nations, Department of Economic and Social
Affairs - Sustainable Development (2022).

An even greater challenge is dealing with contaminants of emerging concern
Kumar  _et al._ (2022); Geissen  _et al._ (2015); Bell  _et al._ (2011);
Deblonde, Cossu-Leguille, and Hartemann (2011). Some pharmaceuticals, when
dispersed in water, can be categorized as pollutants that not only render
water unfit for consumption but also directly impact aquatic life Klimaszyk
and Rzymski (2018); Jørgensen and Halling-Sørensen (2000), as they are
designed to elicit physiological responses in living organisms Reichert  _et
al._ (2019); Wang  _et al._ (2021).

Non-steroidal anti-inflammatories are of particular interest because they are
widely available at a relatively low cost, which, combined with erroneous
disposal of medicines and packaging, leads to the pollution of canals and
river basins Svanfelt  _et al._ (2010); Lin  _et al._ (2023); Ellepola  _et
al._ (2022). One of the most consumed non-steroidal anti-inflammatories is
diclofenac (DCF). Moreover, it has been found that traditional water treatment
processes are ineffective in removing this compound from water. As a
consequence, it is often detected in the effluents from plants and in
receiving water bodies. Studies indicate that DCF is extremely toxic to many
animal species Ajima  _et al._ (2014); Galligan  _et al._ (2021); Herrero-
Villar  _et al._ (2021); Trombini, Blasco, and Hampel (2020); Swiacka  _et
al._ (2020); Joachim  _et al._ (2021); Wolf (2021). For instance, diclofenac
has been found to cause renal failure in vultures, leading to a significant
decline in their populations in regions where the drug is used in veterinary
medicine GREEN  _et al._ (2004); Oaks  _et al._ (2004). In aquatic
environments, DCF exposure has been linked to physiological and reproductive
harm in fish, including kidney damage and reduced fertility Schwaiger  _et
al._ (2004); Triebskorn  _et al._ (2004). Other aquatic organisms, such as
invertebrates and amphibians, are also vulnerable to the toxic effects of DCF,
which can disrupt their endocrine systems and negatively impact their
populations Mohd Zanuri, Bentley, and Caldwell (2017); Han  _et al._ (2010).

Nanomaterials-based technologies for water purification are promising
solutions to address the growing concern of water contamination by emerging
pollutants Leão, Bordin, and de Matos (2023); Bordin  _et al._ (2024); de
Azevedo  _et al._ (2023). Water molecules form hydration layers around
nanomaterials and pollutants, significantly influencing the adsorption
dynamics Schewe  _et al._ (2021); Suzuki (1997). These hydration layers affect
the accessibility of active sites on nanomaterials, the diffusion of
pollutants, and the overall interaction strength between pollutants and
adsorbents. Therefore, a detailed understanding of water behavior at the
nanoscale is crucial for optimizing the design and functionality of
nanomaterials for water purification. By studying these interactions,
researchers can develop more efficient and selective nanomaterials, ultimately
improving the removal of contaminants and ensuring safer water resources for
various applications.

Carbon-based nanomaterials, such as carbon nanotubes (CNTs) and graphene,
stand out as promising candidates for developing new technologies for water
decontamination, particularly in the removal of emerging contaminants Yin  _et
al._ (2019). These nanomaterials exhibit exceptional properties, including
high surface area, chemical stability, and tunable surface functionalities,
which make them highly effective adsorbents for a wide range of pollutants
Bordin  _et al._ (2024); Thakur, Kumar, and Singh (2024); Hsu  _et al._
(2023). The unique structure of carbon-based nanomaterials allows for strong
interactions with various contaminants, including pharmaceuticals, pesticides,
and heavy metals, thereby enhancing their removal efficiency from water
systems Manimegalai  _et al._ (2023). Then, studying the adsorption of
diclofenac in carbon nanotubes is particularly relevant as it helps to
understand the mechanisms and factors influencing the adsorption efficiency
and capacity of CNTs, guiding the optimization and design of more effective
CNT-based nanomaterials Mishra and Sundaram (2023).

Computer simulations emerge in this scenario as an alternative to access
information at an atomic level and for the rational design of nanomaterials
with specific properties for water treatment Li, Zhang, and Wang (2015).
Specifically, Molecular Dynamics (MD) simulations have been widely used to
study the adsorption of many emerging contaminants on carbon-based materials
Côrtes  _et al._ (2024); Ibrahim and Gharbia (2024); Yang  _et al._ (2021); S.
Araújo  _et al._ (2024); Leão  _et al._ (2023). More recently, the influence
of the solvent in processes taking place in aqueous solutions has been
investigated Moscato  _et al._ (2024); Chow  _et al._ (2023); Franco  _et al._
(2023); Bux and Moin (2020). This is even more remarkable at the solid-liquid
interface, especially considering the unique behavior of confined water, whose
structure is quite different from that of bulk water Köhler  _et al._ (2019);
Ho and Striolo (2014); Köhler and da Silva (2016).

In this direction, we use MD simulations to study how the molecular structure
of water near carbon nanotubes (CNTs) impacts the adsorption of emerging
contaminants. The potassium diclofenac (K-DCF) molecule was chosen to
represent the contaminant. We compared the adsorption efficiency of single-
walled (SWCNT) and double-walled (DWCNT) carbon nanotubes in two scenarios:
dry, when the CNT interior is not filled with water, and wet, when the CNT
interior is filled with water. Our findings indicate that the combination of
water structuration and the number of CNT layers plays an important role,
explaining the differences between both systems. Our paper is organized as
follows: in Section 2, we present the simulation details and methods. In
Section 3, the results are discussed, and in Section 4, we highlight the main
conclusions and remarks.

##  II The Model and Simulation Details

The simulated systems, as illustrated in Figure 1, consist of a single- or
double-walled nanotube fixed at the center of the box, a potassium diclofenac
molecule, and water. The simulation setup was built using the Moltemplate
Jewett  _et al._ (2021) software. We use a 5 nm long zigzag (24,0) nanotube
for the SWCNT system and a (24,0) enclosing another (16,0) nanotube for the
DWCNT system, both also with 5 nm in length. The chiralities were chosen to
reproduce the inter-wall nanotube gap found in the literature Lu (1997). The
CNT Lennard-Jones (LJ) parameters were obtained from the work by Saito et al.
Saito  _et al._ (2001)

The potassium diclofenac structure was downloaded from the Protein Data Bank
Berman  _et al._ (2000) (PDB) and kept rigid during the simulation. The LJ
parameters and charges were based on the work of Levina et al. Levina  _et
al._ (2018) for the diclofenac and the sodium parameters as parametrized by
Fuentes-Azcatl and Barbosa Fuentes-Azcatl and Barbosa (2018). The water was
modeled as SPC/E water Berendsen, Grigera, and Straatsma (1987), and Lorentz-
Berthelot mixing rules were employed. 4100 water molecules were accommodated
in a $5\times 5\times 5$ nm simulation box to achieve the bulk water density 1
g/cm3, and the temperature was kept constant at 300 K using the Nosé-Hoover
thermostat Nosé (1984); Hoover (1985).

All simulations were performed using the LAMMPS package Thompson  _et al._
(2022). They all started with an energy minimization step, followed by 1 ns of
thermalization and then 10 nanoseconds of data accumulation. Periodic boundary
conditions were applied. Molecules were maintained rigid with the RATTLE
algorithm Andersen (1983). Three sets of independent simulations, with
different initial positions and velocity distributions, were performed for
each configuration (SWCNT and DWCNT with or without water inside).

Figure 1: Depictions of simulated systems, diclofenac molecule, and CNTs
orientation. The color scheme for atoms follows: red is Oxygen, white is
Hydrogen, gray is Carbon, green is Chloride, blue is Nitrogen, and purple is
the Potassium ion.

To analyze the trajectories of the DCF molecules, we chose to map out the
Nitrogen coordinates through the simulation time. With the radial distance
distributions between the DCF and the outer walls of the CNT we could
establish adsorption criteria. The water radial density profile from the
nanotube was also computed to assess a possible route involving the water-
mediated DCF adsorption in CNTs. Additionally, the Potential of Mean Force
(PMF) Abal  _et al._ (2021); Köhler, Bordin, and Barbosa (2018); Bordin  _et
al._ (2012) was calculated for all systems.

To quantify the hydrophobicity level of the nanotubes, we calculate the
probability $P(N)$ of finding N water molecules inside a sphere of radius 3.3
Å tangent to the CNT. Larger values of $P(N=0)$ indicate higher
hydrophobicity, which is inversely proportional to the work necessary to
completely remove water molecules from the observation volume. In this
scenario, the quantity $-k_{B}T\ln{P(N=0)}$ can be understood as the energy
spent to create a cavity tangent to the nanotube Loubet, Verde, and
Appignanesi (2024) and can be related to the energy barrier confronted by the
DCF molecule to occupy that same cavity.

Recently, the $V_{4S}$ index was proposed Loubet, Verde, and Appignanesi
(2024, 2023). It accounts for energetic information at the atomic level in an
MD simulation being calculated by establishing four interaction sites in the
water molecule and estimating the potential energy due to each of them. Once
we are using the SPC/E water model, the first two sites are located at the
positions of hydrogen atoms. They are 1 Å away from the oxygen, forming an
H-O-H angle of $109.47^{\circ}$, the tetrahedral angle. The last two sites are
determined to complete a perfect tetrahedron, i.e., 1 Å away from the oxygen
on the opposite side, producing the same angle as H-O-H but in an orthogonal
plane to the first plane formed by the atoms of the water molecule. Four
tetrahedral interaction sites are established this way. Having determined the
4 points, we calculate the potential energy between the water molecule and
each heavy atom surrounding it and then attribute the contributions to the
nearest site. After adding up all contributions at each site, we have four
energy interaction values that we organize from the highest in absolute value,
that is, the most negative one ($V_{1S}$) to the lowest contribution
($V_{4S}$). More details can be found in the very recent works Loubet, Verde,
and Appignanesi (2024, 2023); Montes de Oca, Sciortino, and Appignanesi
(2020), while a programming code to implement this indicator can be found in:
https://github.com/nicolas-loubet/V4S.

##  III Results and Discussions

Figure 2: Radial position of Diclofenac molecule over time in the four cases
simulated. Each color line represents one round of simulation.

To estimate the adsorption of the compound in the CNT we evaluate the radial
distance $r=\sqrt{x^{2}+y^{2}}$ of the k-DCF molecule relative to the center
of the CNT, as shown in Figure 2. The dashed line indicates the position of
the nanotube wall, and each line represents the radial trajectories of the
molecule from three distinct simulations. The trajectories in Figure 2(a) and
(b) indicate that for single-walled CNTs, the k-DCF molecule performs a random
walk, with both shorter (less than 1 ns) and longer walks (approximately 8
ns), until it encounters the CNT wall. It was observed that once diclofenac
reaches this minimum distance from the CNT, it remains at this radial distance
until the end of the simulation, indicating adsorption. However, for DWCTN
simulations, adsorption appears more challenging.

For wet DWCNTs, three distinct cases were observed: in the first case (blue
curve in Figure 2(c)), the molecule is adsorbed at the beginning of the
simulation (approximately 1 ns). In the second simulation (green curve),
adsorption occurs at the end of the simulation (approximately 1 ns), and in
the third simulation (orange curve), the k-DCF molecule does not reach the
contact distance during the simulation time. On the other hand, adsorption in
dry DWCNTs was observed in one of the simulations, as shown in Figure 2(d).

Figure 3: Probability density to find an Oxygen atom of a water molecule at a
certain distance from CNTs for SWCNT (a) and (b) DWCNT.

Figure 4: (a) Qualitative interpolation of return point of k-DCF molecule in
DWCNT dry system and b) Probability Density to find an atom of a water
molecule at a given distance. The pink dashed lines are only guides pointing
to the pink circles interpolation in a) and the end of the structuration in
(b).

It is unclear whether longer simulations would lead to all drug molecules
being adsorbed by the CNTs and how this would happen regardless of the number
of walls or whether they are filled with water. However, it is clear that
adsorption is more challenging in DWCNTs. Interestingly, when molecules are
not adsorbed, they appear to encounter a barrier and bounce back. To clarify
this, let’s take a closer look at the dry DWCNT case.

As seen in Figure 4(a), even in cases where the molecule was eventually
adsorbed, it initially reaches a minimal distance of $r\approx 18$ Å and
bounces back before overcoming this barrier and adsorbing. The magenta circles
indicate points where the blue trajectory curve reaches the location of this
barrier, marked by the dashed magenta line in Figure 4(a).

To understand the origin of this barrier, we can analyze the water structure
around the nanotube in this case. Figure 4(b) shows the probability density of
water molecules in the radial direction, highlighting the two layers structure
of water around the CNT. By comparing Figures 4(a) and (b), we can relate the
barrier near 18 Å to the water’s two layer structure. In other words, the
k-DCF molecule must overcome this barrier created by the water structure
around the CNT. Since adsorption was observed in all cases with SWCNTs but
only in half of the DWCNT simulations, it suggests that the water structural
barrier is higher near DWCNTs than SWCNTs.

However, by comparing the probability densities of finding water molecules
near SWCNTs and DWCNTs, as shown in Figure 3, no clear differences in water
distribution around the distinct nanotubes are observed. In all cases, two
layers of water molecules are observed, indicating that this parameter alone
is insufficient to distinguish the water structuring around different
nanotubes. To better understand these differences, it is essential to analyze
the energy and tetrahedrality of the water molecules. By examining the energy
interactions, we can gain insights into the stability and binding strength of
the water molecules with the nanotube surfaces. Additionally, evaluating the
tetrahedrality of water molecules can help us understand their local
structural organization. Tetrahedrality measures the degree to which water
molecules form tetrahedral arrangements, a hallmark of their hydrogen-bonding
network. By combining these analyses, we can better understand the water
molecule structure and behavior around SWCNTs and DWCNTs, providing a clearer
picture of how different nanotube structures influence water molecule
arrangement and interactions.

Figure 5: Potential of Mean Force around for a) dry and b) hydrated CNT
nanotubes, c) Probability of finding N water molecules inside the observation
sphere, comparations for SWCNT and DWCNT systems and, d) Distribution of
$V_{4S}$.

The energy barrier felt by the k-DCF molecule, created by the water structure,
is shown in Figure 5(a) and (b) by the radial PMF around the nanotube. It
provides insight into the most favorable radial positions for a solute
molecule, indicated by the valleys. It’s interesting to notice that for the
dry CNTs there is a significant difference for the SWCNT and DWCNT PMFs, see
Figure 5(a), while for completely hydrated CNTs the number of walls does not
play a significant role, as Figure 5(b) indicates. It indicates that, for the
dry case, the carbons from the inner wall affect the water structure on the
outside, while when the CNT is fully hydrated, this effect vanishes. This
result indicates that, within the analyzed systems, a molecule will face more
difficulty approaching the double-walled, dry nanotube.

This behavior is explained by the relationship of the $V_{4S}$ indices
presented in Figure 5(c). Here, we explore only the dry case to see if any
difference can be observed. The $V_{4S}$ index is determined as the smallest,
in magnitude, of the potential energies of the four interaction points related
to the tetrahedral arrangement of the water molecule. This means that, since
water molecules have a lower average $V_{4S}$ around DWCNTs than SWCNTs (from
$-7.17$ to $-6.81$), the interactions of water molecules are stronger in the
former case. This result is consistent with the findings shown in Figure 5(c),
which quantifies the hydrophobicity of the nanotube by calculating the
probability of finding N water molecules inside a small observation sphere
tangent to the nanotube wall. The differences in $\ln(P(N=0))$, with a
slightly lower value for the DWCNT, indicate greater work required to create a
vacuum cavity at the water-nanotube interface. Indeed, we note that this 5%
difference in ln(P(N=0) translates itself into a rougly 50% increase in the
probability of cavity creation, P(N=0), at the SWCNT as compared to the DWCNT,
pointing to the fact that it is easier to remove hydration water from the
SWCNT. Consistently, the difference in the mean value of V4s for the two
nanotubes also amounts to around 5%. Such a difference arises from the fact
that a water molecule close to the DWCNT interacts with a greater number of C
atoms as compared to the SWCNT (even when the interaction with the inner wall
of the DWCNT is much atenuated by its distance). In turn, since $V_{4S}$
senses the water-wall interaction (which should be overcome in order to remove
the water molecule), it also implies that the SWCNT presents around 50% higher
dehydration propensity.

In summary, the differences in the behavior observed in Figure 2 are explained
by subtle differences in the structuring of water around the nanotubes, where
the number of walls and the existence or not of water inside the CNTs can
affect the adsorption process. The return point in the DWCNT systems is
explained both by the energy barriers for transitioning between the first
solvation layers, denoted by the PMF peaks in Figure 5a, and by the greater
difficulty in in removing water molecules around the double-walled nanotubes
compared to the SWCNT, allowing the diclofenac molecule to be adsorbed.

##  IV Conclusion

In this study, we utilized MD simulations to investigate the impact of water
molecular structure near CNTs on the adsorption of K-DCF, a prominent example
of emerging contaminants. Our results reveal that the structuring of water
molecules and the number of CNT walls significantly influence the adsorption
dynamics. We observed that water molecules form hydration layers around the
CNTs, which affect the accessibility of active sites and the strength of
interaction between pollutants and adsorbents. In single-walled nanotubes,
K-DCF molecules exhibited a random walk until adsorption occurred, while in
the double-walled case the adsorption process was more complex and often
hindered by an energy barrier created by the structured water layers.

The energy barrier for adsorption in DWCNTs, highlighted by the Potential of
Mean Force (PMF) analysis, was higher compared to SWCNTs. This difference is
attributed to the stronger interaction of water molecules with DWCNTs, as
indicated by the lower $V_{4S}$ indices and higher work required to create a
vacuum cavity at the water-nanotube interface. These subtle differences in
water structuring and energetics explain the observed adsorption behaviors and
highlight the importance of considering nanoscale water behavior in the design
of nanomaterials for water purification.

Our findings emphasize that both the structural characteristics of CNTs and
the presence of confined water play crucial roles in the adsorption efficiency
of contaminants. This knowledge can guide the development of more effective
and selective nanomaterials, ultimately enhancing the removal of pollutants
and ensuring safer water resources. Future research should continue to explore
the interplay between water structuring and nanomaterial properties to
optimize water treatment technologies. Additionally, the approach employed in
this work has broader implications for understanding the role of water in
biological processes. For instance, hydration layers are critical in protein
folding, where water influences the structural stability and function of
proteins. Similarly, water molecules around active sites affect substrate
binding and catalytic efficiency in enzyme activity. By studying water
behavior in these contexts, we could gain insights into fundamental biological
processes such as signal transduction, where water mediates interactions
between biomolecules, and drug delivery, where hydration layers influence the
bioavailability and efficacy of therapeutic agents. Thus, the insights from
this study would not only be helpful in advancing water purification
technologies but might also enhance our understanding of the central role of
water in various biological processes.

## Acknowledgement

Without public funding this research would be impossible. J. R. B. thanks the
National Council for Scientific and Technological Development (CNPq), under
grant numbers 405479/2023-9, 441728/2023-5, and 304958/2022-0, as well as from
the Rio Grande do Sul Research Foundation (FAPERGS), grant number
21/2551-0002024-5. M. H. K. thanks CNPq Universal grant number 306709/2021-0
and FAPERGS PqG Grant number 21/2551-0002023-7. P. R. B. C. acknowledge
support from the Coordination for the Improvement of Higher Education
Personnel Brasil (CAPES), Financing Code 001. C.A.M., N.A.L. and G.A.A.
gratefully acknowledge funds from UNS and CONICET.

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