Theodore Aouad
email: theodore.aouad@centralesupelec.fr

Hugues Talbot
email: hugues.tablot@centralesupelec.fr

A foundation for exact binarized morphological neural networks

Training and running deep neural networks (NNs) often demands a lot of computation and energy-intensive specialized hardware (e.g. GPU, TPU...). One way to reduce the computation and power cost is to use binary weight NNs, but these are hard to train because the sign function has a non-smooth gradient. We present a model based on Mathematical Morphology (MM), which can binarize ConvNets without losing performance under certain conditions, but these conditions may not be easy to satisfy in real-world scenarios. To solve this, we propose two new approximation methods and develop a robust theoretical framework for ConvNets binarization using MM. We propose as well regularization losses to improve the optimization. We empirically show that our model can learn a complex morphological network, and explore its performance on a classification task.

Introduction

Binary weight neural networks (BWNNs) are attractive because they can provide powerful machine learning solutions with much less storage, computation, and energy consumption than conventional networks [START_REF] Rastegari | Xnor-net: Imagenet classification using binary convolutional neural networks[END_REF]. Several methods, such as BinaryConnect [5], DoReFa-Net [27], and XNOR-Net [START_REF] Rastegari | Xnor-net: Imagenet classification using binary convolutional neural networks[END_REF], have shown good to excellent results in a variety of applications. These networks usually use the sign function to binarize the weights in the forward pass. They must use a special gradient function, like the Straight-Through Estimator (STE) [START_REF] Bengio | Estimating or propagating gradients through stochastic neurons for conditional computation[END_REF], to overcome the zero gradient problem of the sign function during the backward pass. However, this approach lacks a solid theoretical basis, suggesting the need for a different approach. Some methods avoid using the STE by first training a floating-point neural network and then binarizing it afterwards [12,8]. However, this approach may lead to approximate binarization and a drop in performance. In this paper, we present a new approach that uses the concepts of Mathematical Morphology (MM) [23] to overcome the drawbacks of existing methods. MM, based on modern set theory and complete lattices, offers a non-linear mathematical framework for image processing. Its basic operators, erosion and dilation, are equivalent to thresholded convolutions [14], creating a link between MM and deep learning. Combining these fields can improve the efficiency and results of morphological operations while enhancing our knowledge of deep learning [START_REF] Angulo | Some open questions on morphological operators and representations in the deep learning era[END_REF]. Recent works on morphological neural networks have explored learning operators and structuring elements using various approaches, such as the max-plus definition [19,[START_REF] Franchi | Deep morphological networks[END_REF] and differentiable approximations [START_REF] Shen | Deep morphological neural networks[END_REF][START_REF] Masci | A learning framework for morphological operators using counter-harmonic mean[END_REF]9]. However, these methods primarily focus on learning gray-scale MM operators and have not focused on NN binarization.

The Binary Morphological Neural Network (BiMoNN) [2] proposed a well-defined mathematical method that can binarize NN weights without losing performance under certain conditions. However, it is limited to binary inputs and can learn only one filter per layer, limiting the design of modern architectures. In this paper, we improve the Bi-MoNN framework to overcome these limitations and introduce a new model that can learn any sequence of morphological operators while achieving complete binarization in all cases. We also introduce novel regularization techniques to encourage our model to become morphological. By combining MM concepts with deep learning, our work establishes the basis for a more robust and theoretically sound framework for NN binarization. Our contributions in this work are as follows:

• In §2, we refine the BiMoNN theoretical framework and generalize it to any kind of gray-scale / RGB inputs. We introduce two new layers analogous to the dense and convolutional layers, enabling the transposition of modern architectures. • In §3, we present a well-defined mathematical binarization method based on MM that works seamlessly with standard frameworks and tools. We also propose two new approximate binarization techniques to deal with the cases where exact binarization is not possible. • In §4, we introduce three applicable regularization losses, and a fourth one that is only theoretical due to its long computation time. • In §5, we evaluate the capacity of the binarized Bi-MoNN to learn complex morphological pipelines without performance loss. Additionally, we investigate its behavior on the MNIST [16] classification task, and the behavior of the introduced regularization techniques. Our code is publicly available at https://github. com/TheodoreAouad/LBQNN2023.

General Binary Morphological Neural Network

We build on the BiMoNN framework [2] and propose a new Binary Structuring Element (BiSE) neuron. We show how it is morphologically equivalent to binary images by using the notion of almost binary images. We also propose the BiSE Layer (BiSEL) that can learn multiple filters per layer, which is similar to a convolutional layer; and the DenseLUI, a dense layer that can be binarized. In §2.1 and §2.2, we consider binary inputs only. We generalize to any types of inputs in §2.3.

Binary Structuring Element neuron

Let D be the dimension of the image (usually D = 2 or 3). We denote Ω I ⊂ Z D the support of the images, and Ω the support of the weights kernel. For the remainder of the paper, we assume S ⊂ Ω and S ̸ = ∅. For a set X ⊂ Z D , we denote its indicator function 1 X :

Z D → R such that 1 X (i) = 1 if i ∈ X, else 1 X (i) = 0.
We denote the set of its subsets P(X). We denote the convolution by ⊛. We denote [•] + := max(•, 0) and [•] -:= min(•, 0). Definition 2.1 (BiSE neuron). Let (W, B) be a set of reparametrization functions and (ω, β, p) a set of learnable parameters. Let ξ be a smooth threshold activation (e.g. normalized tanh: 1 2 tanh(•) + 1 2 ). Then the Binary Structuring Element neuron (BiSE) is:

χ : x ∈ [0, 1] Ω I → ξ p x ⊛ W (ω) -B(β) ∈ [0, 1] Ω I (1)
The reparametrization functions (W, B) are hyperparameters. The BiSE neuron is a convolution operator with weights and biases that are reparametrized (see §2.4), a smooth threshold activation and a scaling factor p that can invert the output if negative. We introduce the following almost binary image representation, to handle images that are not exactly binary and to be able to apply gradient descent optimization.

Definition 2.2 (Almost Binary Image). The set of almost binary images of parameter δ is denoted I(δ). We say an image I ∈ [0, 1] Ω I is almost binary and define X I its associated binary image if:

∃δ ∈ 0, 1 2 , I(Ω I ) ∩ 1 2 -δ, 1 2 + δ = ∅ (2)
X I := I > 1 2 (3)

Morphological Equivalence

We now express the conditions under which a BiSE neuron can be seen as a morphological operator.

Theorem 2.3 (Dilation -Erosion Equivalence). For a given structuring element (SE) S ⊂ Ω, and an almost binary parameter δ ∈]0, 1 2 ], a set of reparametrized weights W ∈ R Ω and bias b ∈ R, we define:

L ⊕ (W, S) := k∈Ω\S [W k ] + + 1 2 -δ k∈S [W k ] + (4) U ⊕ (W, S) := 1 2 + δ min k∈S W k + k∈Ω [W k ] - (5)
U ⊖ (W, S) := k∈Ω W k -L ⊕ (W, S) (6)
(7)

L ⊖ (W, S) := k∈Ω W k -U ⊕ (W, S) (8)
Let ψ ∈ {⊕, ⊖} be a dilation or erosion. Then:

∀I ∈ I(δ) , ψ S I > 1 2 = I ⊛ W > b ⇔ L ψ (W, S) ≤ b < U ψ (W, S) (9)
In this case, ∀s ∈ S, W s ≥ 0 and b ≥ 0 and we say that a BiSE χ with weights W (ω) = W and B(β

) = b is activated. If ψ = ⊕, then B(β) ≤ 1 2 k∈Ω W (ω) k . If ψ = ⊖, then B(β) ≥ 1 2 k∈Ω W (ω) k
For any almost binary image I ∈ I(δ), χ(I) ∈ I(δ out ) is almost binary with known parameter δ out . Finally

∀I ∈ I(δ), χ(I) > 1 2 = ψ S I > 1 2 (10) I χ(I) X I X χ(I) χ •> 1 2 •> 1 2 ψ S
This theorem states that if a BiSE is activated, it transforms an almost binary inputs into an almost binary outputs with known δ out . Moreover, if we threshold the input and output with respect to 1 2 , it is equivalent to applying the corresponding morphological operation, exhibiting an almost commutativity property between thresholding and convolution. Further, equation (10) shows that if a BiSE is activated for operation ψ, performing the BiSE operation in I(δ) is equivalent to performing the binary morphological operation ψ in the binary space P(S). This presents a natural framework for binarization (see §3).

Binary Morphological Neural Network

Our objective is to build a binarizable neural network. We now define binarizable neural layers based on the BiSE neuron. By combining these layers, we can create flexible architectures tailored to the desired task.

As mentioned earlier, the BiSE neuron resembles a convolution operation. However, a single convolution is insufficient to create a morphological layer. In this context, we explain how we handle multiple channels. We observe that the BiSE neuron can be used to define a layer that learns the intersection or union of multiple binary images x 1 , ..., x n ∈ I(δ). For example, their union can be expressed as the dilation of the 3D image x := (x 1 , ..., x n ) ∈ I(δ) ⊂ [0, 1] Ω I n with a tubular SE applied solely across the dimension of depth. Therefore, we define the Layer Union Intersection (LUI) as a special case of the BiSE layer, with weights restricted to deep-wise shape. It is analogous to a 1 × 1 convolution unit. A LUI layer can learn any intersection or union of any number of almost binary inputs. By combining BiSE neurons and LUI layers, we can learn morphological operators and aggregate them as unions or intersections.

Definition 2.4 (BiSEL). A BiSEL (BiSE Layer) is the combination of multiple BiSE and multiple LUI. Let (χ n,k ) n,k be N * K BiSE and (LUI k ) k be K LUI. Then we define a BiSEL as:

ϕ : x ∈ [0, 1] Ω I N → LUI k χ n,k (x n ) n k (11)
The BiSEL mimics a convolutional layer. In conventional ConvNets, to process multiple input channels, a separate filter is applied to each channel, and their outputs are summed to create one output channel. In the case of BiSEL, instead of summing the results of each filter, we perform a union or intersection operation (see Figure 1).

DenseLUI the LUI layer is similar to a 1 × 1 convolution, which is equivalent to a fully connected layer. Given an input vector x ∈ R n , we can apply the LUI layer to the reshaped input x ∈ R n×1×1 treating it as a 2D image with width and length of 1 and n channels, respectively. Therefore, we can utilize the BiSEL to create binarizable fully connected layers.

Gray-Scale / RGB Inputs Up until now, all inputs were assumed binary. We extend these definitions to gray-scale and RGB images. The main idea is to separate an input channel I c ∈ R Ω I into its set of upper level-sets to come back to binary inputs:

{(I c ≥ τ ) | τ ∈ R} (12)
Considering all possible values of τ from a continuous image would result in an excessive number of level-sets to process. Alternatively, we can define a finite set of values for τ in advance. Subsequently, each channel of an image I ∈ R c×w×l is separated into its corresponding level-sets, and these level-sets are provided as additional channels. If we have N values for level-set, the resulting input is a binary image We have introduced two types of binarizable layers: the DenseLUI, which is similar to a fully connected layer, and the BiSEL, which resembles a convolutional layer. By combining these layers, we can create a Binary Morphological Neural Network (BiMoNN), which encompasses various architectures suited for different tasks.

I B ∈ {0, 1} (N •c)×w×l .

Training considerations

The BiMoNN (Γ R ) is fully differentiable. If L is a differentiable loss function, given a dataset of N labeled images {(x i , y i )}, we minimize the error 1 N N i=1 L(Γ R (x i ), y i ) using a gradient descent algorithm (like Adam [13]). The gradients are computed with the backpropagation algorithm [START_REF] Rumelhart | Learning Representations by Back-propagating Errors[END_REF]. The binarization scheme, which is defined in the next section, is applied post-training or during training to measure the model's evolution.

To facilitate the convergence of our networks towards morphological operators, certain constraints are beneficial. In order to avoid dealing with a multitude of constraints, we reparametrize some variables to ensure that the constraints are always satisfied. We introduce two reparametrization functions for the weights, and three for the bias.

Positivity Our objective is to reach the set of activable weights and bias. Theorem 2.3 indicates that we only have to look at positive parameters. We can enforce them to be positive by setting W and B as the softplus function:

B pos (•) = W pos (•) := f + (•) := log(1 + exp(•)) (13)
Dual reparametrization For the weights, we introduce the dual reparametrization as follows:

W dual (ω) := K • f + (ω) W (ω) w (14)
with K := 2 • ξ -1 (0.95). We can show that this dual reparametrization ensures that the training process is similar for both erosion and dilation.

Other reparametrization for the bias can be defined to keep it into coherent range values. If the bias is smaller than min(W ), then ∀X ⊂ Ω, χ(1 X ) < 0.5: no values will be close to 1. On the other side, if the bias is higher than W w, then χ(1 X ) > 0.5. Therefore, we want min(W ) < B < W w. Let {W 1 < ... < W K } be the ordered values taken by the weights W . The previous inequality is ensured if B belongs to the closed convex set

C b := [ W1+W2 2 , W w -W1 2 ] = [l c (W ), u c (W )].
There are two ways of ensuring that B ∈ C b .

Projected First, we can project the bias after the gradient iteration: this comes down to a projected gradient algorithm. We call this approach "Projected" reparametrization. We apply B p (β) := f + (β), and we project B p (β) onto C b after the gradient update iteration.

Projected Reparam

The second way is to redefine B before the end of the iteration, instead of after. We call this approach "Projected Reparam" reparametrization:

B pr (β) :=    l c (W ) if f + (β) < l c (W ) u c (W ) if f + (β) > u c (W ) f + (β) else (15)
Initialization The BiSE layer performs convolutions with a smooth threshold function as the activation, resulting in values in the range of [0, 1]. Since we have reparametrized the weights to be positive, the classical initialization proposed in [7], which is tailored for ReLU activation and includes negative weights, needs to be adapted. We ensure that the gradients do not vanish during initialization, especially when stacking BiSE neurons in a deep network. Inspired by [7], we sample uniform weights with an adjusted distribution that maintains a constant variance. Additional details can be found in Appendix A.

Binarization

Binarization of a neural network involves converting the real-valued weights and activations, typically stored as 32-bit floats, into binary variables represented as 1-bit booleans. While the conventional approach results in variables in {-1, 1} [26], which is not suitable for learning morphological operations, we instead utilize {0, 1}.

BiMoNNs inherently correspond to binarized networks, making the BiSE neuron a natural framework for binarization when dealing with binary inputs. If a specialized hardware tailored for morphological operations is available, it can offer significant improvements in efficiency and performance, facilitating the binarization process. Alternatively, dilation and erosion can be expressed using binaryweighted thresholded convolutions.

In our approach, binarization occurs after the training phase. We present two types of binarization for the BiSE neuron: the exact method (as introduced in [2]), when the BiSE neuron is activated, and two novel approximated methods. Then, we sequentially binarize the entire network.

Exact BiSE Binarization

The real-value operations performed by an activated Binary Structuring Element (BiSE) in the almost binary space can be replaced with binary morphological operations on the binary space after thresholding by 0.5, without sacrificing performance (as per Theorem 2.3). To determine if a BiSE is activated and which operation it corresponds to, we introduce Proposition 3.1, which provides a linear complexity method for extraction.

τ ⊕ := 1 1 2 + δ B(β) - w∈W (ω) [w] - (16)
τ ⊖ := 1 1 2 + δ w∈W (ω) [w] + -B(β) (17)
Given a BiSE neuron χ and an almost binary output in I(δ), we check if χ is activated for S ⊕ or S ⊖ , where S ⊕ = (W (ω) > τ ⊕ ) and S ⊖ = (W (ω) > τ ⊖ ). If yes, we binarize by replacing χ with the corresponding morphological operator. If no, we use proposition 3.1 to confirm that χ is not activated, and we approximate the binarization using the methods in section 3.2. The exact method requires only the computation of L ⊕ , U ⊕ , L ⊖ , U ⊖ and at most O(|Ω S |) operations.

Approximate BiSE Binarization

In practice, BiSE are rarely activated, necessitating an approximate binarization method. Let ( W, b, p) := (W(ω), B( β), p) be the learned reparametrized parameters.

Projection onto activable parameters

To find the closest morphological operation, we minimize the Euclidean distance d for a given ψ ∈ {⊕, ⊖} and the set A ψ,S of activable parameters:

A ψ,S := (w, b) ∈ R Ω × R L ψ (w, S) < b < U ψ (w, S) (18) minimize S⊂Ω,ψ∈{⊕,⊖} d ( W, b), A ψ,S (19)
For each set A ψ (S) corresponding to a possible morphological operation, we find the smallest distance for each (S, ψ) and apply complementation if p < 0. The following proposition allows linear search instead of exponential.

Proposition 3.2. If S * , ψ * minimize d ( W, b), A ψ,S , then S * = ( W ≥ min S * W s )
Proposition 3.1 guarantees that the SE is a set of thresholded weights when the weights are activated. Proposition 3.2 ensures that this property is preserved for the optimal SE even when the weights are not activated. Thus, we only need to compute the distance to all possible sets of thresholded weights, denoted as S k := ( W ≥ W k ), and select the smallest distance. To compute the distance for ψ and S fixed, we solve the following optimization problem.

minimize (w,b)∈R Ω + ×R 1 2 i∈Ω (W i -W i ) 2 + 1 2 (b -b) 2 subject to L ψ (w, S) -b ≤ 0 b -U ψ (w, S) ≤ 0 (20)
When the initial weights W are positive, the projected weights are also positive. Consequently, the constraints can be rewritten for dilation and erosion as follows:

⊕          k∈Ω\S W k -b ≤ 0 (21) ∀s ∈ S, b ≤ W s (22) ∀k ∈ Ω \ S, -W k ≤ 0 (23) ⊖              b - s∈S W s ≤ 0 (24) ∀s ∈ S, i∈Ω W i -W s ≤ b (25)
∀k ∈ Ω \ S, -W k ≤ 0 (26) These new constraints are then linear. By utilizing the positive reparametrization technique (as described in §2.4), we can compute the distance to any activable parameter set A ψ,S by solving a QP problem, using the OSQP solver [START_REF] Stellato | OSQP: an operator splitting solver for quadratic programs[END_REF]. This needs to be done for all possible sets of thresholds S k , of which there are |Ω|. For a single layer with 4096 input and output neurons, the computation would take up to 28 days (on a Intel(R) Xeon(R) W-2265 CPU, 3.50GHz). Future work may involve finding an analytically computable form for efficient distributed computing. Alternatively, we introduce the following approximation technique.

Projection onto constant weights

We use a similar technique to [17]. First, we define A S the set of constant weights over S, and replace d ( W, b), A ψ,S with d( W, A S ).

A S := {θ • 1 S | θ > 0} (27) d( W, A S ) = i∈Ω W 2 i - 1 |S| s∈S W s 2 (28)
Similarly to proposition 3.2, we can prove that the optimal S * is a set of thresholded weights. The analytical form in 28 allows for efficient computation of S * in distributed systems, significantly faster than the first method. For a layer with 500 input and output channels, this step takes less than 3 seconds. Once S * is obtained, the bias term helps determine ψ ∈ {⊕, ⊖}, based on Theorem 2.3. If b > W w/2, the operation is an erosion; otherwise, it is a dilation.

BiMoNN binarization

The core of the successive binarization is Theorem 2.3. To simplify, we suppose that a BiMoNN is a succession of BiSE. If the first BiSE is activated, then with a binary input (e.g. almost binary with δ 0 = 1 2 ), the output is almost binary of known parameter δ 1 . If the next BiSE is activated for the parameter δ 1 , its output is also activated for parameter δ 2 , and so on. Hence, every BiSE operation on the almost binary space is equivalent to the morphological operation on the binary space. In case one BiSE is not activated, its output is not on the almost binary space, breaking the exact equivalence between BiSE and morphology. We apply an approximate method (projection §3.2.1 if small network, otherwise fast projection §3.2.2), and suppose that the input for the next BiSE is binary.

Morphological Regularization

The binarization schema, as described in §3, is separate from the training process. During standard loss optimization with classical gradient descent, there is no explicit constraint that makes the network behave morphologically. Consequently, the network may not tend to a morphological operation: BiSE operators may not be activated and the weights may stay far from their respective projection space, resulting in potential errors in the approximate binarization process compared to the floating-point operator. To encourage the network to exhibit a more morphological behavior, we propose the inclusion of a regularization term in the loss, denoted as L morpho . The loss function now becomes:

L = L data + c • L morpho ( 29
)
L data represents the loss used for the data-driven task (e.g. Cross-Entropy Loss for classification), and c is the hyperparameter controlling the strength of the regularization term. To define the regularization term, we reuse the two approximate binarization techniques defined in section 3.2.

Regularization onto activable parameters

A first idea is to reduce the distance to the closest set A ψ,S defined in [START_REF] Masci | A learning framework for morphological operators using counter-harmonic mean[END_REF] for each BiSE neuron. Let (W, b) := (W (ω), B(β)) be the weights in a given iteration. Then:

L morpho = L acti := min S,ψ d (W, b), A ψ,S (30)
To do this, we must compute this distance in a differentiable fashion. We proceed in two steps: first, we compute the optimal (S * , ψ * ) as in §3.1, by checking all possible thresholded set of weights and solving the corresponding QP with OSQP, with the Lagrangian dual method. This yields the best (W * , b * ) as well as the Lagrangian dual values, from which we deduce the differentiable form of the distance. If ψ * = ⊕, let λ * be the dual value for the constraint [START_REF] Rastegari | Xnor-net: Imagenet classification using binary convolutional neural networks[END_REF] and

T := {t ∈ S * | W t ≤ b * } (31) K := {k ∈ Ω \ S * | W k ≤ λ * } (32) K := (Ω \ S * ) \ K (33) D := |K|(|T| + 1) + 1, (34)
then we can show that we obtain the following differentiable expressions for (W * , b * ).

b * = 1 D j∈K W j + |K| t∈T W t + b (35) ∀j ∈ K, w * j = W j + 1 D t∈T W t + b -(|T| + 1) i∈K W i (36) ∀k ∈ K , W * k = 0 (37) ∀t ∈ T , W * t = b * (38) ∀s ∈ S * \T , W * s = W s . ( 39
)
Then, we have a differentiable expression for the loss.

L acti = i∈Ω (W i -w * i ) 2 -(b -b * ) 2 (40)
However, the computational burden described in §3.1 persists: the first step of computing S * is too long in practice.

Regularization onto constant set

Instead of trying to enforce a fully morphological network, we can encourage the weights to stay in the set of constant weights A S defined in (27). We proceed in two steps: we find the best S * in the same way as in §3.2.2, i.e by computing the distance d( A S , W) defined in (28) for all set of thresholded weights, and selecting the smallest distance. Then, a differentiable expression for the distance is:

L morpho = L exact := i∈Ω W i - 1 |S * | s∈S * W s 2 (41)
Computing the distance for all set of thresholded weights still takes significant time, and in our experimentation, this slows the training by up to 80×. Instead, we propose to use the technique introduced in [17]. If we assume that the weights follow a uniform or normal distribution, we can approximate the optimal S * ≃ (W > τ ) with the following thresholds:

Uniform τ u := 2 3 1 |Ω| i∈Ω W i S u := (W > τ u ) (42) Normal τ n := 3 4 1 |Ω| i∈Ω W i S n := (W > τ n ) (43)
From this, we can define two regularization loss which can be computed without slowing the training down.

L morpho = L unif := i∈Ω W i - 1 S u s∈Su W s 2 (44)
L morpho = L normal := i∈Ω W i - 1 S n s∈Sn W s 2 (45)

Experiments

In this section, we empirically validate the capabilities of BiMoNNs in learning a binarized morphological pipeline through a denoising task, without the need for regularization. We also evaluate the model and regularization techniques on the MNIST classification task.

Binary Denoising

We generate a second dataset to evaluate the denoising capacity of BiMoNNs. The target images in this dataset consist of randomly-oriented segments with width h, with added Bernoulli noise. To filter these images, an MM expert would use a union of opening operations, where the SEs are segments with width 1 and angle one of (0 • , 90 • , -45 • ). The SEs should be longer than the noise and shorter than the smallest stick in the target image (usually a length of 5). Examples are given in figure 3a.

Our architecture uses two consecutive BiSEL layer of kernel size 5 and 3 hidden channels (see figure 3b). We optimize the MSE Loss, with an initial learning rate of 0.01. We train over 6000 iterations and halve the learning rate after 700 iterations with non-diminishing loss. We stop once the loss does not decrease after 2100 iterations. We employ a positive reparametrization for the bias and a dual reparametrization for the weights, and binarize with the projection onto activable parameters.

The network achieves excellent denoising performance, with a DICE score of 97.5%. The remaining 2.5% discrepancy is due to artifacts between the sticks that cannot be denoised using an opening operation. The binarized network (Figure 3b) accurately learns the intersection of three openings (which remains an opening) the same way a human expert would combine such operators to achieve the denoising task optimally. Additionally, 4 out of the 6 BiSE are activated during the process. This experiment shows that our network can learn accurate and interpretable composition of morphological operators.

Classification

We conduct classification experiments on the MNIST dataset [16]. All images are thresholded at 128. Our Bi-MoNN model comprises one hidden DenseLUI layer with 4096 neurons. To handle the large number of parameters, we adopt the fast projection defined in §3.2. We compare the classification accuracy of our float and binarized models against the SOTA and baseline models with fully connected layers and 1D batch normalization [11], employing 1 2 (tanh(•) + 1) as the activation layer. The accuracy results are summarized in Table 1.

In our framework, binarizing the weights also entails binarizing the activations. Consequently, binarizing the last layer would yield binary decision outputs for each output neuron, possibly leading to multiple labels with a score of 1.

To overcome this issue, we refrain from binarizing the last layer, thus retaining the real-valued activations. This decision affects a negligible proportion of parameters (≃0.1%).

In traditional classification neural networks, the softmax activation is commonly used at the end of the last layer to produce the final probability distribution over the classes. However, in the BiMoNN architecture, we utilize the same activation function as the hidden layers, which is the normalized tanh. Additionally, we compare the performance of our BiMoNN model when replacing the last normalized tanh activation with a softmax layer. When using the normalized tanh, L data is the Binary Cross-Entropy loss. When using the softmax, we use the Cross-Entropy loss. Applying the softplus reparametrization to the weights led to a slight increase in the floating-point error (2.2% vs. 2.8%). Similar findings were observed in [20] for nonnegative neural networks in a different task. Generally, positive neural networks exhibit lower accuracy but offer enhanced robustness and interpretability [4]. In our case, it significantly improved the binarized results, along with a substantial increase in the rate of activated BiSE neurons, rising from a median of 1.5% to 10% with softplus. Without imposing positivity, the binarized network performed randomly.

We analyze the impact of regularization on the performance of the binarized model, which improves as expected. The float accuracy also increases, given that we select the model with the best binary accuracy on validation. Surprisingly, L unif and L normal outperform L exact , despite being designed as approximations. This discrepancy might be due to the number of searches performed: L exact performed only 42 searches, while other configurations went through 100 searches. However, we have not yet achieved parity with the baseline for the float model or reached the state-ofthe-art for the binarized model. With 4.5% error compared to 2.8% for the same number of parameters, this emphasizes the need for improved architecture, better regularization techniques, or exploration of alternative optimization methods.

Discussion

The state-of-the-art BWNN methods commonly rely on the XNOR operator to emulate multiplications. However, this algebraic framework proves unsuitable for morphological operators, as it contradicts the set-theoretical principles of morphological operations. In our experiments, we observed that the BNN operator [10] failed to learn even the simplest dilation operator. Furthermore, the performance of the state-of-the-art method on the denoising task was un-satisfactory, with a DICE coefficient of only approximately 0.3, indicating the need for improved approaches in handling morphological operations.

In contrast, our findings reveal that the float BiMoNN exhibits enhanced binarization capabilities when trained to closely approximate a set of morphological operators. As a result, the float BiMoNN naturally acquires morphological properties, leading to a more effective subsequent binarization process. However, when applied to the classification of the MNIST dataset, the resulting float BiMoNN does not retain morphological characteristics, causing a noticeable performance degradation after binarization. To address this issue, we emphasize the importance of positive reparametrization and applying morphological regularization. By incorporating these techniques, we significantly improved the model's overall performance and mitigate the accuracy loss upon binarization. This shows the potential of our proposed BiMoNN framework in leveraging morphological properties and offers insights into the development of more effective BiMoNN models for many and diverse tasks.

Conclusion

In this paper, we have presented a novel, mathematically justified approach to binarize neural networks using the Binary Morphological Neural Network (BiMoNN), achieved by leveraging mathematical morphology. Our proposed method establishes a direct link between deep learning and mathematical morphology, enabling binarization of a wider set of architectures, without performance loss under specific activation conditions and providing an approximate solution when these conditions are not met.

Through our experiments, we have demonstrated the effectiveness of our approach in learning morphological operations and achieving high accuracy in denoising tasks, surpassing state-of-the-art techniques that rely on the straightthrough estimator (STE). Furthermore, we proposed and evaluated three practical regularization techniques that aid in converging to a morphological network, showcasing their efficacy in a classification task. Additionally, we introduced a fourth regularization technique that, though promising in theory, currently faces computational challenges. We will adress these shortcoming in future work.

Despite promising results, there is still room for enhancing both the floating point and binary modes of our network. In future work, diverse architectures, such as incorporating convolution layers, could be explored to further improve the performance and applicability of BiMoNN. Overall, our research lays the foundation for advancing the field of binarized neural networks with a morphological perspective, offering valuable insights into developing more powerful and efficient models for a wide range of tasks.

Figure 1 :

 1 Figure 1: BiSEL vs Conv Layer. Input x with 3 channels. Output ϕ(x) with 2 channels.

Figure 2 :

 2 Figure 2: Gray to level-set for 5 different values, generating 5 input channels.

Proposition 3 . 1 (

 31 Linear Check). Let us assume the BiSE of weights W (ω) and B(β) is activated for ψ with SE S ⊂ Ω for almost binary images I(δ). Then S = (W (ω) > τ ψ ) with

  (a) Input-output example. (b) Binarization of the network

L

  BCE ( ŷi , y * i ) := 10 c=1 y * i log( ŷi ) + (1 -y * i ) log(1 -ŷi ) (46)L CE ( ŷi , y * i ) := 10 c=1 y * i log( ŷi )(47)We conduct a comprehensive random search to identify the optimal hyperparameter configuration for the Binary Morphological Neural Network (BiMoNN). The hyperparameters explored include the learning rate, last activation function (Softmax layer vs. normalized tanh), positive vs no reparametrization for weights, and several bias reparametrization schemas (no reparametrization, positive, positive reparam, and positive projected). Additionally, we investigate regularization losses, such as no regularization, L exact , L uni , and L nor . If regularization is applied, only positive weight reparametrization is considered. We vary the coefficient c in the regularization loss and explore different batch value starting time for when we start applying regularization during training. For each regularization schema, we select the model with the best binary validation accuracy, and the corresponding results are displayed in Table1

Table 1 :

 1 . Detailed hyperparameter configurations and hy-perparameters study are provided in appendix B. Accuracy error on test set for MNIST classification, with float error R and binarized error B.

		Architecture	Params R	B
		DLUI (W = Id)	3.3 M	2.2% 90.2%
	Ours	DLUI (No Regu) 3.3 M DLUI L exact 3.3 M	4.6% 10.1% 4.0% 7.3%
		DLUI L unif	3.3 M	3.6% 4.5%
		DLUI L normal	3.3 M	2.8% 4.6%
		EP 1fc [15]	3.3 M	-	2.8%
	SOTA	BinConnect [5]	10 M	-	1.3%
		BNN [10]	10 M	-	1.4%
	Float	FC (4096) FC (2048x3) [10] 10 M 3.3 M	1.5 % -1.3% -

A. Initialization

A. 1. Initialization In this section, we investigate the initialization of the BiSE neuron. Let us suppose that we stack L ∈ N * BiSE neurons one after the other.

Notations Let l ∈ [ [1, L]] be the layer number. Let W l ∈ R Ω S be the random set of weights of this layer. Let b l ∈ R be the bias (not random). Let µ l := E(W l ) and σ l := V ar(W l ). Let x 0 ∈ {0, 1} Ω I be the input image, and for l ∈ [ [1, K]], let x l be the output of the l th layer: x l := ξ(y l ) ∈ [0, 1] Ω I with y l := x l-1 ⊛ W l -b l ∈ R Ω I such that for j ∈ Ω I , y l (j) = i∈Ω S W l (i)x l-1 (j -i) -b l . Let n l = |W l |. Assumptions 1. For l ∈ [ [1, L]], the W l (i) i∈Ω l S are independent, identically distributed (IID) 2. For l ∈ [ [1, L]], the x l (i) i∈Ω I are IID 3. For l ∈ [ [1, L]] , E[y l ] = 0 and p(y l ) is symmetrical around 0 4. For (l, i, j) ∈ ([ [1, L]] × Ω S × Ω I ) , W l (i) and x l-1 (j -i)) are independent Under these assumptions, we can drop the i and j index and rewrite y l = W l x l-1 -b l , with implicit indexing. The same goes for x l .

According to Fubini-Lebesgue theorem, and using assumption 3, we have

We use the unbiased estimator to compute the mean, by summing all the weights of the layer: µ l := 1 n l w∈W l w. Therefore, with E(x 0 ) the mean of the inputs:

In some cases, we initialize the scaling factor p = 0 to avoid bias towards applying complementation or not. Then, this initialization leads to a point with zero grad (see annexe A.2 for details). Therefore, we add some noise to the bias:

ϵ bias depends on the length of the network. It is set between [10 -4 , 10 -2 ].

Variance computation We find a recurrence relation between V ar(y l ) and V ar(y l-1 ). To simplify the computations, let p ′ = dξ(p•x) dx (0) be the tangent at 0, and let us assume that p ′ > 0. We approximate the smooth threshold by a piece-wise linear function:

with

By independence, we have

with

We unroll the recursive relation. Let

Then

The right hand side of the inequality comes from the fact that

If the variance is equal to 0, then the output is constant: therefore, the gradient is 0 and we cannot learn the parameters. On the other hand, if the variance is too high, the model is poorly conditioned: we can expect the learning to be unstable.

If the number of layers is too big, the product term V (p ′ ) can either vanish or explode. To avoid variance exploding when G(p ′ ) ≥ 0, and to avoid variance vanishing when G(p ′ ) ≤ 0, we choose to ensure that the product is equal to 1 by imposing each term to be 1.

The variance needs to be positive, which gives

We have a range of possible means and variances. Let us choose a uniform distribution for W l . Then W k ∼ U(µ l -√ 3σ l , µ l + √ 3σ l ). We want positive weights (see theorem 2.3), therefore we want µ l > √ 3σ l , which gives

Using equations 71 and 72, we choose µ l as the mean of the bounds:

Role of the scaling factor p Finally, p ′ is related to the value of p: p ′ = d(ξ(p•x) dx (0) = pξ ′ (0). Ideally, to avoid bias towards applying complementation or not, p should be set at 0, therefore p ′ = 0. We consider that after the first few iterations, p ̸ = 0, and our computations become valid.

How to set p ′ in practice ? We want the BiSE output to be in the full range [0, 1]. Let us suppose that after a few iterations, p = 1. With B l = w∈W l w, the lowest output value is given by an image full of 0, giving ξ(-B l ). The highest output is given by an image full of 1 and gives ξ(B l ). Let h ∈]0, 1[ (e.g. h = 0.95). We want ξ(B) > h, which is the same as ξ(-B l ) < 1 -h thanks to the binary-odd property. By using the fact that µ l n l ∼ w∈W l w, we have:

Summary With h = 0.95, ϵ bias ∈ [10 -4 , 10 -2 ] and p ′ l :=

We can approximate E(x 0 ) with the mean of a few samples, for example the first batch.

Reparametrization We computed the weights and biases of the convolution. In the BiSE definition 2.2, they correspond to W (ω) and B(β). The true initialization must be done on ω and β. If W and B are invertible, the problem is solved:

If the functions are not invertible, the study must be done case by case in order to find generative distributions for ω l and β l that respects the resulting distributions for W (ω l ) and B(β l ).

Dual reparametrization For the dual reparametrization, we need to adapt the parameter K.

be a sequence of independent, identically distributed random variables, S

Proof.

E[|X

Then, according to the strong law of large numbers,

Finally if we take a = µ k -√ 3σ k the initialization becomes:

Remark on bias initialization Theorem 2.3 expresses the operation approximated by the BiSE depending on the bias.

From the second to the last layer, the bias is initialized at the middle of both operation. Therefore, the BiSE are unbiased to learn either dilation or erosion. However, for the first BiSE layer, the bias is initialized differently. If E(x 0 ) < 1 2 , then the bias indicates a dilation. We bias the BiSE to learn a dilation. This is explained by the assumption that the output must mean at 1 2 . If the input is a lower than 1 2 , we must increase its value. The same goes for the erosion: if the value of the input is too high, we reduce it by applying an erosion.

A.2. Gradient Computation

Let Γ = χ L • ... • χ 1 be a sequence of BiSE neurons, with parameters p l , b l and W l for all layers l. We compute L(Γ(X), Y ) for one input sample. Let

We re-denote W l as the linear matrix such that

If for all l, p l = 0, then dL db l = dL dW l = 0. Moreover, by initialization of the bias b l , we have χ l-1 W l -b l = 0, leading to dL dp l = 0. With our current initialization, all the gradients are equal to 0. Therefore, in practice, we add a uniform noise to the bias:

This will lead to p l moving away from 0, which unblocks the other gradients as well. We introduce a small bias towards either dilation or erosion. However, its significance is negligible, and does not prevent from learning one operation or the other. We perform a random search across a range of hyperparameters to identify the optimal configuration. The hyperparameters explored in the search are as follows:

B. Experiments

• Learning rate between 10 -1 and 10 -3 .

• Last activation: Softmax layer vs Normalized tanh. If Softmax Layer, we use L CE , else we use L BCE • Applying the softplus reparametrization to the weights or not.

• The bias reparametrization schema between no reparametrization, positive, projeceted and projected reparam.

• Regularization loss: either no regularization, or the projection onto constant set (choose between L exact , L uni and L nor ).

If we apply regularization, then we also apply softplus reparametrization. • If regularization: the coefficient c in the loss, either 0.01 or 0.001.

• If regularization: the number of batches to wait before applying the regularization, in [0, 5000, 10000, 15000, 20000].

For each regularization schema, we select the model with the best binary validation accuracy, and the corresponding results are displayed in Table 1. Detailed hyperparameter configurations are provided in Table 2 for reference.

We investigate the effects of each hyperparameter.

Bias Reparametrization For the choice of bias reparametrization function, we observed that not applying any reparametrization to the bias resulted in less robustness, and the network occasionally failed to learn. Applying projected reparametrization improved robustness, but it did not increase the proportion of activated neurons. Conversely, applying the positive projected and projected reparametrization functions increased the ratio of activated BiSE neurons from 0.9% to around 20%. Ensuring that the bias falls within the correct range of values enhances the interpretability of the network.

Regularization Delay when activating the regularization loss at the beginning of training, the results were notably worse, especially in binary accuracy. Our hypothesis is that the network does not have enough time to explore the right morphological operations and is prematurely drawn to a non-optimal operator, getting stuck in its vicinity. We observed that the accuracy improved when increasing the waiting time before activating the regularization loss. Future work may explore more advanced policies for applying the loss, such as using the loss at a given frequency instead of every batch, increasing the coefficient of the loss at each step, or waiting for the network to converge before applying regularization.

Regularization coefficient and last activation The coefficient of the regularization loss did not have a significant impact, as well as replacing the normalized tanh by a softmax layer.


Dan-Dan Su

Karim Aissou

Yan Zhang

Virginie Gervais

Sebastien Ulrich
email: sebastien.ulrich@cnrs.fr

Mihail Barboiu
email: mihail-dumitru.barboiu@umontpellier.fr

Squalene-polyethyleneimine-dynamic constitutional frameworks enhancing the enzymatic activity of carbonic anhydrase †

Carbonic anhydrase is an essential enzyme that catalyzes the hydration/dehydration of carbon dioxide, which is highly relevant to carbon capture processes. However, its efficient encapsulation in host materials is of utmost importance for the system performance (durability, stability, and efficiency). In this paper we demonstrate that polyethyleneimine-polyethylene glycol squalene constitutional nanoparticles PEI-DCFs, synthesized via reversible imine/amino-carbonyl chemistry, are efficient host matrixes for bovine carbonic anhydrase (bCA) encapsulation. This system showed an impressive one-order-of-magnitude-improved catalytic proficiency (k cat /K m = 7396 M -1 s -1 ) as compared to bCA alone (k cat /K m = 504 M -1 s -1 ). This performance rivals the current state-of-the-art systems with equivalent amounts of the enzyme, even after heating for a prolonged period at 80 °C, translating into its direct application for enhancing carbon dioxide capture and conversion.

Introduction

The significant growth of carbon dioxide (CO 2 ) emissions is a serious environmental and public health issue causing atmospheric pollution and the green-house effect. Many attempts have been made using electrocatalytic reduction, photosynthesis, chemical absorption and membrane transport methods to capture and store CO 2 , 1,2 but so far the high costs and energy requirements restrict their effective application. 3 On the other hand, a modern alternative rests on dynamic and adaptive materials which can be obtained by cheaper and more versatile covalent/supramolecular selfassembly processesa proof-of-concept having already reported simultaneous CO 2 capture and metal purification. 4 During the past decades, the use of carbonic anhydrases (CAs) for CO 2 hydration and storage has attracted attention due to its high turnover rate, its environmentally benign properties and wide sources. 5 CAs are already exploited in artificial systems for applications in CO 2 hydration/ dehydration, 6,7 CO 2 hydrogenation 8 or CO 2 conversion to methanol. 9 A wide range of materials have been used for CA immobilization in order to improve its tolerance to hazardous conditions such as high temperature (up to 80-100 °C), high acidity, etc., but also to maintain its efficiency when embedded in artificial matrix hosts. 10 Nanotechnology has the potential to address this problem. Indeed, multivalent binders can lead to interfacial interactions/ stabilization with the enzyme while also promoting its encapsulation in nano-bioreactors or biomimetic protocells. The high surface-to-volume ratio and the controllable display of binding/reactive groups have been used in artificial 11 and CA-based catalytic nanoparticles. 12,13 The catalytic center of CAs contains a zinc-binding pocket and a hydrophobic binding pocket. In the catalytic cycle of CA, hydroxyl ions are produced under neutral pH conditions to generate bicarbonate upon reaction with CO 2 . Protons are therefore byproducts which are expelled out of the catalytic center in a rate-limiting step. 14,15 While most of the host materials developed hitherto inhibit the catalytic activity of CAs either by altering the enzyme conformation or by hindering the access of the substrates to the catalytic center, a few recent systems showed an enhancement of catalytic activity instead. This peculiar behavior is best understood by the host facilitating the proton shuttle out of the active site. 16 We previously reported interesting results in that direction using dynamic constitutional frameworks (DCFs) which adaptively stabilize a microenvironment through confining H-bonding or electrostatic interactions on the outer enzyme surface and play the role of multi-proton sponges, both essential for an effective catalytic activation mechanism. 17,18 In this case, CA can be activated by simple addition of building blocks in aqueous solution, which are easily and modularly self-assembled via reversible covalent bonds on the external enzyme surface. 19 On the other hand, solid absorbents, made with polyethyleneimine (PEI) loaded on different porous substrates, are promising materials for carbon dioxide capture. 20 The gas absorption/transport mechanism is based on the same hydrolytic reaction as performed by CA, which may further convert CO 2 into carbamic acid and/or bicarbonate at the secondary amine sites of PEI in the presence of water. This makes PEI one of the highly performing nanomaterials for CO 2 capture from flue gas or air.

Herein, we report a novel strategy, aimed to be as simple and broadly applicable as possible relating to the encapsulation of bovine carbonic anhydrase (bCA) within a polyethyleneimine (PEI)-DCF nanoparticles with enhanced catalytic activity. This results in the production of scalable biomimetic DCF-bCA nanoparticles that remarkably outperform the classical bCA activity in solution and combine the extraordinary absorption capacity of PEI materials. 20 This study leads to a greater fundamental understanding of how CA incorporation can be optimized at the nanoscale to simultaneously stabilize and activate the overall catalytic cycles of a DCF-PEI-bCA system operated under harsh conditions, as required by industrial CO 2 capture processes. The relatively straightforward development of DCF-PEI-bCA nanoparticles is a very important and previously unreported strategy. They are operating synergistically to give enhanced catalytic turnovers, higher that those resulted from the sum of the former components that are relevant to their subsequent use in larger scale applications as absorbent materials or membranes.

Results and discussion

Design and synthesis

The preparation of DCFs was carried by self-assembly of the following components: (1) 1.3.5-benzenetrialdehyde, BTA as a core center, PEGylated-1,1′,2-tris-norsqualene constructed from (2) hydrophobic squalene aldehyde, SQ-CHO with a remarkable folded conformation in water, 21 (3) hydrophilic poly(ethylene glycol)-bis-(3-aminopropyl)-terminated (Mn ∼ 1500) PEG1500, with a favorable solubility in water and used as an artificial biocompatible chaperone for bCA refolding and limiting aggregation 22 and (4) different polyethyleneimines-PEIs: a) branched polyethyleneimine, bPEI800 (Mn ∼ 800) (12 PEI units), b) pentaethylenehexamine, PEH (5 PEI units), and c) triethylenetetramine, TAA (3 PEI units) used as charged H-bonding heads interacting with the bCA surface and presumed to act as catalytic multivalent proton shuttles for bCA activation (Fig. 1). DCF-PEI, DCF-PEH and DCF-TAA were prepared via the aminocarbonyl/imine reversible chemistry between (1), (2), (3) and positively-charged bPEI800, PEH and TAA, respectively in order to quantify their role in the enhanced catalytic effects of DCF-bCA systems at room temperature and 80 °C. The 1 H-NMR spectra are in accordance with the proposed structures (Fig. S1-S5 †).

Binding to bCA

The binding behaviors of DCFs or PEI monomers to bCA were first evaluated by monitoring the intrinsic fluorescence emission of bCA, at its maximum of emission λ em = 343 nm, following the excitation at λ ex = 280 nm. 23 An obvious decrease in fluorescence emission was observed, upon addition of the DCFs (DCF-PEI, DCF-PEH and DCF-TAA) onto an aqueous solution of bCA (Fig. 1b andS6 †). From the Stern-Volmer analysis 24 the observed Stern-Volmer constants of DCF-PEI, DCF-PEH and DCF-TAA with bCA were determined to be 15 052 M -1 , 8400 M -1 and 7912 M -1 respectively (Fig. 1c). The superiority of DCF-PEI compared to DCF-PEH and DCF-TAA reveals that an increasing number of cationic groups results in a more effective quenching of the fluorescence emission of bCA, which can be a result of either a quenching triggered by the DCFs or by enforced structural changes of the enzyme upon binding to the DCFs. Compared with the DCFs, monomeric bPEI800, PEH and TAA show two orders of magnitude weaker effects, with Stern-Volmer constants of 201 M -1 , 121 M -1 and 196 M -1 , respectively, demonstrating the importance of multivalent presentation of positive PEIs on DCF surfaces.

Isothermal titration calorimetry (ITC) experiments were carried out in parallel (either by titrating the enzyme with ligands or by titrating ligands with bCA) and revealed marked differences (Table 1). While DCF-PEI, DCF-PEH, and bPEI800 were found to be similarly potent binders of bCA with association constants in a narrow range of 0.93-1.02 × 10 6 M -1 , notable differences in thermodynamic behavior were observed. DCF-PEI with its cationic superiority leads to the most pronounced enthalpic contribution compared to DCF-PEH. DCF-TAA with the smallest cationic group gives the largest entropy change including contributions of the solvent entropy and the protein conformational entropy. The difference with the above fluorescence analysis indicates that binding (measured by ITC) is not correlated with the structural changes (monitored by intrinsic fluorescence measurements) of the enzyme. Altogether, these results show that the multivalency displayed by the DCF with the largest cationic group PEI (DCF-PEI) triggers the most drastic structural changes in bCA, through enthalpy-and entropydriven binding, which may affect the catalytic efficiency of the enzyme. On the other hand, DCF-TAA with the smallest cationic group was found to be the most potent binder, with

Catalysis Science & Technology Communication

an association constant of 7.14 × 10 6 M -1 and a reaction mainly driven by entropy, but it does not trigger structural changes within the bCA enzyme according to the fluorescence assay.

Nano-scale observation of DCFs-bCA conjugates

Self-assembly at the nanoscale of DCFs-bCA conjugates. Dynamic light scattering (DLS) was used to estimate the average diameter of nanoparticles formed after 12 h at room temperature in solutions of bCA in the presence of DCF-PEI, DCF-PEH and DCF-TAA. First, it is worth noting that there is no fluctuation in size for the DCFs alone, within a concentration range of 0.05 mM to 0.2 mM, with average sizes of 173.5 ± 0.2 nm, 127.0 ± 0.5 nm and 171.1 ± 1.6 nm for DCF-PEI, DCF-PEH and DCF-TAA, respectively (Table S3 †). In the presence of bCA, the DLS results showed the formation of dimensionally similar nanoparticles (Fig. 2a-c). Thus, the addition of bCA does not affect the overall selfassembly in solution, occurring with the total inclusion of enzyme within bCA/DCF nanoparticles without affecting their size and shape. In contrast, the polyamine building blocks bPEI800, PEH, and TAA lead to the formation of much larger (>300 nm) and more polydisperse nanoparticles in the presence of bCA (Table S3 †).

In order to further elucidate their aggregation behaviors, the bCA/DCF bioconjugates were imaged by atomic force microscopy (AFM) (Fig. 2d-f and S8 †). The AFM topography images of DCF-PEI with bCA (Fig. 2d), DCF-PEH with bCA (Fig. 2e) and DCF-TAA with bCA (Fig. 2f) showed homogeneous particles over the entire samples, with no sign of aggregation. The apparent sizes of all samples were found to be below 200 nm, which is in good agreement with the DLS results, revealing the uniform particle distribution.

The transmission electronic microscopy TEM showed that the DCF-PEI (Fig. 2g) and DCF-PEI with bCA (Fig. 2h) had an aggregated morphology with an average overall size of 200 nm for all formulations. Similar results were obtained in previous studies on PEI decorated nanoparticles for gene delivery. 25,26 Our results are in accordance with the DLS data, confirming the narrow particle size distribution of the aggregates, which was observed by TEM. DCF-PEI is obtained via aggregation of several discrete brighter and darker nanoparticles, most probably formed by segregation of squalene and PEG-PEI components, respectively, selfassembled via reversible covalent interactions at the interfaces between nanoparticles. Interestingly DCF-PEI exhibited a more complex core and shell morphology when bCA is added. The aggregates were composed of a darker central core of ∼50 nm and of a surrounding corona, in which well-defined and homogeneously distributed brighter nanoparticles can be identified (Fig. 2h). These white grey spots of ∼5 nm are hypothesized to be mostly composed of bCA for which the crystallographic data suggest an overall ellipsoid shape with the dimensions of 4 × 4 × 5 nm. 27 The dark color of the bPEI800-bCA samples suggested that strong electrostatic aggregation occurred between bPEI800 and bCA, homogeneously distributed within dense nanoparticles. In contrast the DCF-PEI with the bCA sample showed that larger voids and bright less dense spots can be observed inside these hybrid samples when compared with the pristine PEI sample micrographs, where very few or no bright spots were seen.

The nano-structural distribution of dense and light regions within the DCF-bCA aggregates is essential to control the catalytic enzymatic performance, since they determine the pathways for the substrate flow and diffusion across the active nanoparticle.

Enzymatic activation activity of bCA

The esterase catalytic activity of cationic monomers alone/ DCFs in the presence (Fig. S9 †) or in the absence (Fig. S10 †) of bCA was tested by monitoring the hydrolysis reaction of p-nitrophenyl acetate (NPA) in PBS buffer (pH 7.4) by UV-vis spectroscopy. The esterase activity through the hydrolysis reaction of NPA is directly related to CO 2 hydration reaction catalyzed by both bCA and PEI systems. 28 The time course of the absorbance at λ = 400 nm was monitored to estimate the catalytic activity relative to the reference activity of free bCA in solution at different concentrations of PEI monomers. Unless the concentration exceeds 0.1 mM, bPEI800 and DCF-PEI present negligible catalytic activity in the absence of bCA (Fig. 3a), which is in line with only a few examples in the literature reporting the catalytic activity of PEI itself 28 on the CA stabilization within a PEI/dopamine protecting matrix. 29 It is also similar to the literature data where only up to 43% of the primary amine of PEI is acetylated using acetic anhydride at 60 °C in anhydrous methanol. 30 For further support, the 1 H NMR analysis of aqueous solutions of PEI and NPA indeed showed the exclusive formation of p-nitrophenol (NP) and acetic acid which completely exclude the acetylation of PEI in aqueous solution at room temperature (Fig. S19 †).

We then hypothesized that combining an increasing number of PEI with bCA in hybrid bioconjugates would lead to strong synergistic improvement of the catalytic turnover and enzyme protection within a multicharged environment. When combined with bCA, important enhancements of catalytic activity were observed for DCF-PEI and bPEI800. In the presence of bCA, the multivalent DCF-PEI system reaches an important ∼5.8-fold increase of catalytic activity at 0.2 mM (Fig. 3a). Compared with that, bPEI800 reached a ∼2.4fold increase in catalytic activity in the presence of bCA. In contrast, the enhancements of activity are very moderate when PEH, DCF-PEH, (Fig. 3b) and TAA, DCF-TAA (Fig. 3c) have been added to bCA, with a maximum of ∼1.8-fold increase in catalytic activity observed for DCF-PEH at 0.2 mM of PEH.

Synergistic activity was calculated by using the following equation: N = A 1 /(A 0 + A 2 ), where A 1 is the absorbance measured with DCF or a cationic monomer with bCA at 4 min, A 0 is the absorbance measured with CA alone at 4 min, and A 2 is the absorbance measured with DCF or a monomer alone at 4 min (Fig. 3d). It is worthy to note that the synergistic activity of DCF-PEI and DCF-PEH is strongly dependent on the nature and the concentration of bPEI800 and PEH monomers and shows that the multivalent effect of enzyme encapsulation by DCFs and enhanced enzyme activity play a role in the overall catalytic process.

The catalytic activity was then quantified by determining K m , equal to the substrate concentration at half of the maximum rate V max , which were both determined by fitting the data with the Michaelis-Menten model. The observed ∼2-fold decrease of K m for DCF-PEI with bCA (K m = 2.08 mM) or bPEI800 with bCA (K m = 1.68 mM) as compared to bCA alone (K m = 3.13 mM) reveals the stronger affinity with the bCA enzyme under DCF confinement. In contrast, DCF-PEH, DCF-TAA, PEH, and TAA show much greater K m values (Table 2), meaning lower binding affinity which is in line with all results presented above. A proficiency constant (k cat / K m ) is used to measure the catalytic efficiency of the enzymatic biohybrids. Since the proficiency constant reflects the affinity and catalytic ability of the enzyme to the substrate at the same time, it can be used to compare the catalytic efficiency of different enzymes for a specific substrate (Fig. 4a, Table 2). With k cat /K m values of 7396 and 6630 M -1 s -1 for DCF-PEI with bCA and bPEI800 with bCA, respectively, these two systems clearly stand out, being one order of magnitude more powerful catalysts than all others tested, including bCA alone. It was obvious that DCF/cationic monomer with bCA presents higher catalytic efficiency, with an estimated k cat range of 2.49 s -1 to 15.36 s -1 compared to the reaction rate k of DCF/monomer without bCA (Table S2 †), emphasizing a significant catalytic effect. These results are in agreement with previous results, when the enzymatic activity of hCAII is slightly inhibited by its immobilization of PEI/ polydopamine layers 31 or by the absorption on positively charged polystyrene nanoparticles that do not inhibit the enzyme. 32 The exceptional activation effects observed with DCF-PEI or bPEI800 with bCA strongly demonstrate the importance of a critical number of ethylene imine units and more importantly shed light on their multivalent presentation within DCFs. They can strongly increase the activity of bCA up to one order of magnitude under confinement within a dynamic microenvironment. Moreover, while direct interactions of monomeric PEI groups with the internal enzyme active site can lead to enzyme activation, the activation mechanism observed with nanometric hybrids is mostly related to the external binding of larger DCF-PEI. The multivalent network of PEI substituents mostly accelerates outwardly the proton-transfer step via amino groups and are equally conferring a dynamic crowding microenvironment for the enzyme activation.

The bCA aggregation driven by intermolecular hydrophobic interactions is one of the problems that hinder its use in real applications. The confinement within a protecting matrix may enhance its stability and disfavor its denaturation via aggregation which might occur at high temperature for example. 33,34 The activation effect of bCA at 80 °C was tested by UV experiments with the time-course analysis at pH 7.4. The final concentration of DCF-PEI was 0.2 mM. As shown in Fig. 4b andc, the hydrolysis reaction of p-nitrophenyl acetate was promoted at 25 °C (black line) as compared to the reaction with bCA at 80 °C (grey line).

Interestingly, catalytic activation effects can be obtained by simple addition of bPEI800 (light blue dot) and DCF-PEI (light blue) in the absence of bCA. The addition of bPEI800 (blue dot) and DCF-PEI (blue line) components to a solution of bCA induce a 3-or 5-fold, increased enzymatic activity at 80 °C, respectively. The same activation effect was observed with the addition of DCF-PEI in an aqueous solution of bCA after 180 s reaction time at 80 °C (red line).

The particle size of bPEI800 with bCA and DCF-PEI with bCA evaluated through DLS increases to 1000-1200 nm when heated in aqueous solution at 80 °C (Fig. S16 †). The same aggregation effects were observed by TEM. Heterogeneous dark and bright nanoparticles of DCF-PEI with bCA with an overall diameter of 800-1000 nm (Fig. S17 †) are formed. Larger fractal aggregates were observed for bPEI800 with the bCA samples when heated at 80 °C (Fig. S18 †). The multivalent hydrophilic bPEI800 and DCF-PEI cross-linked nanoparticles enable to activate the catalytic properties of the native bCA, through accelerated proton sponge effects of the external PEI groups. Moreover, at high temperature they strengthened significantly the structural stability, and effectively inhibited the formation of protein aggregate occurred to free bCA.

Conclusions

In conclusion, we have established that multivalent DCFs provide a possibility to enhance the bCA enzyme catalytic activity by changing the enzyme microenvironment. Moreover, those DCFs can dynamically interact and generate the optimal enzyme-DCF structure through adaptation, leading to the formation of nanoparticles that optimally encapsulate the enzyme, thereby 8-fold accelerating the overall catalytic activity of DCF-bCA hybrids compared with bCA alone. The results obtained in this work showed that multivalent presentation of bPEI800 groups within DCF-PEI platforms presents outstanding binding behaviors triggering structural changes within bCA, which results in an impressive improved catalytic proficiency of k cat /K m of 7396 M -1 s -1 , more than one order of magnitude higher than that for bCA alone. The DCF-PEI with bCA aggregates was notably robust and able to provide catalytic activation effects after being exposed to relatively harsh temperatures. The observations implied their feasible deployment in real systems for CO 2 capture where they would undergo cycles requiring materials to maintain high performance at 80-100 °C. This rationalization corroborates the results discussed above, suggesting that a direct perspective is to immobilize encapsulated and activated enzymes in a solid substrate for enhancing CO 2 capture and conversion. The expected performances must be greater than simple enzyme immobilization under such confined conditions within hydrophobic-hydrophilic nanoparticles. Their stability must be preserved under wet conditions, so they must be used as solid sorbents or entrapped with a high loading content in the thin layers of membranes. The "facilitated transport" with the CO 2 enzyme carriers would compensate for the low CO 2 partial pressure difference over the membrane and increase the driving force. This leads to HCO 3 -transport under wet conditions in applications where water vapors are present: flue gas, space exploration, breathing, and fermentation. [START_REF] Sandru | Composite hollow fiber membranes for CO2 capture[END_REF] Experimental Materials and methods, Fig. S1 to S19, Tables S1 to S3, and supplementary references are provided in the ESI. †

Fig. 1 a

 1 Fig. 1 a) Schematic representation of molecular components used to construct multivalent dynamic constitutional frameworks (DCFs) for bovine carbonic anhydrase (bCA) immobilization in hybrid nanoparticles. The four-component DCFs are synthesized step-wise through imine/aminocarbonyl chemistry, and different cationic headgroups (bPEI800, PEH, and TAA), shown in orange, were tested. b) Fluorescence spectra showing the quenching of bCA upon addition of increasing amounts of DCF-PEI (top) or bPEI800 (bottom). c) Stern-Volmer plots 23 of relative fluorescence intensity versus cationic monomer concentration present in free and DCF bonded forms in solution, used for the determination of the Stern-Volmer constant (K a ) between bCA and DCFs/PEI monomer.

Fig. 2

 2 Fig. 2 DLS experiments showing the size distribution of colloidal aggregates in aqueous solutions of a) DCF-PEI with bCA, b) DCF-PEH with bCA and c) DCF-TAA with bCA. Representative AFM scans of d) DCF-PEI with bCA (1.5 × 1.5 μm; scale bar: 200 nm), e) DCF-PEH with bCA (1.5 × 1.5 μm; scale bar: 200 nm) and f) DCF-TAA with bCA (1.5 × 1.5 μm; scale bar: 200 nm). Representative TEM images of g) DCF-PEI, h) DCF-PEI with bCA and i) bPEI800 with bCA. The dense PEI-PEG regions are darker while the less dense grey brighter spots correspond to the SQ regions. The 5 nm spots in h) correspond to the bCA dispersed in the hydrophilic corona of PEG-PEI components.

Fig. 3

 3 Fig. 3 Hydrolysis of p-NPA, monitored by UV-vis absorption spectroscopy (Aabsorbance measured at 4 min) with the addition of a) DCF-PEI, bPEI800, b) DCF-PEH, PEH, c) DCF-TAA, TAA to the solutions of bCA, used as a reference. d) Synergistic activity of all monomers/DCFs.

Fig. 4

 4 Fig. 4 Estimation of K m of bCA enzymatic reaction based on the Michaelis-Menten equation for DCF and monomers in the a) presence of bCA. The concentration of p-NPA varied from 0.1 to 10 mM, and the concentration of DCFs/monomer was fixed to 0.2 mM/0.4 mM. b) and c) Kinetic hydrolysis experiments at 80 °C: blank = p-NPA in water at 25 °C (light magenta) and 80 °C (black), bCA at 25 °C (magenta) and 80 °C (grey); TAA (yellow dot), DCF-TAA (yellow), PEH (green dot), DCF-PEH (green), bPEI800 (light blue dot), DCF-PEI (light blue) at 80 °C and bCA at 80 °C with the addition of DCF-PEI at 180 s (red).

Table 1

 1 Thermodynamic parameters derived from isothermal titration calorimetry binding studies for the binding of DCFs and PEI to bCA

	Samples	K d (μM)	K a (10 6 M -1 )	ΔH (kcal mol -1 )	-TΔS (kcal mol -1 )
	DCF-PEI	1.04 ± 0.66	0.96	-3.39 ± 0.32	-4.77
	DCF-PEH	1.07 ± 0.82	0.93	-0.86 ± 0.10	-7.28
	DCF-TAA	0.14 ± 0.44	7.14	-0.15 ± 0.04	-9.22
	bPEI800	0.98 ± 0.84	1.02	-0.22 ± 0.04	-7.97

Table 2

 2 The K m values, k cat values and the proficiency constant (k cat /K m ) of enzymatic reaction with the addition of DCF/monomer

		K m (mM)	k cat (s -1 )	k cat /K m (M -1 s -1 )
	Samples	Enzyme DCF	Enzyme DCF	Enzyme DCF
	DCF-PEI	2.08	15.36	7396
	DCF-PEH	4.20	3.79	902
	DCF-TAA	3.71	2.49	672
	Samples Enzyme monomer Enzyme monomer Enzyme monomer
	bPEI800 1.68	11.12	6630
	PEH	3.36	2.77	826
	TAA	3.31	3.24	978
	bCA	3.13	1.58	504

Catal. Sci. Technol., 2022, 12, 3094-3101 | 3099 This journal is © The Royal Society of Chemistry 2022

This journal is © The Royal Society of Chemistry 2022

Acknowledgements DS appreciates the scholarship as support from China Scholarship Council at the University of Montpellier, France. This work was supported by the M-ERA NET 2019 grant to MB via Agence Nationale de la Recherche ANR-20-MERA-0001-01, SMARTMATTER. YZ thanks financial support from the National Natural Science Foundation of China (NSFC) (22008090) and the Natural Science Foundation of Jiangsu Province (BK20180625). The ITC experiments were performed on the PIM Platform of I2BC (Platform for measurements of Interactions of Macromolecules).

Notes and references

Conflicts of interest

There are no conflicts to declare.