The light dispersion becomes an intrinsic phenomenon associated to the colloidal systems, in our case, associated to gold nano-particles. The optical and spectroscopic properties of gold nano-particles have been extensively exploited for chemical, bioanalytical, and biomedical applications, in which the size controllability and surface tunability are essential (Njoki et al.). if the particles are smaller than the wavelength of the colliding haze, Rayleigh dispersion phenomena occurs. The particles are able to disperse in an easier way the blue corresponding wavelength and consequently, blue coloration is observable. These phenomena explain the blue coloration of the sky, triggered by tiny particles that are dispersing the blue light radiation.

On the other hand, it must be taken in account that as tinier the size of these particles gets, their nature tents to be metallic, the dispersion phenomena previously introduced could recoil and become predominant the radiation absorption. Surface Plasmon resonance phenomena arise. It’s a matter of fact that the dispersion processes occur frequently on big size nano-particles, whereas the absorption by surface plasmon resonance occurs when the nano-particles are tinier. The change of the size in particles at nano-scale, triggers changes in the coloration very useful to build a sensor.

The main basic conjugation strategies are, covalent conjugation and unspecific adsorption.

Unspecific adsorption: AuNPs Surface unspecific covering is fast and direct, non-covalent interactions are involved. The proteins will adhere forming the so-called protein corona at random orientations. A determined range of orientations could lead to gluten peptide and HLA union.

Covalent conjugation: Covalent conjugation between a linker (N-hydroxysuccinimide), previously funcionalized with the AuNPs and the amides present in proteins (HLA).

In network science, most specifically in random graphs theory, a simple model about network formation show light over the previous questions. The Erdös-Rényi (ER) Graph expose the basic tools to deal with the process of network formation from a complete disordered system. The foundations of the ER graph lie on the presence of nodes and exist a probability p of finding a linker joining two of them, randomly (in a full mixed state). Then the algorithm chooses a pair of agents and generates a random number between 0 and 1, η = (0,1), then if η is lower than p, a link appears in between. This model, apparently too simple it is possible to expand a model no more complex, that could explain our sensor dynamics from the basics but robust at system level. An anti-intuitively property exhibit by these network forming models is that there exists a threshold in the dynamics, that leads the system from a disconnected network (where the system remains disconnected and the connected hole doesn’t exist) to a connected one (where the system remains mostly connected and the hole contains the most). The transition from one state to the other follow a specific dynamic, that change exactly in a determined critical point that turns of the previous dynamic in order to begin the new one.

These two states diverge from having disconnected nodes o tiny clusters of them to summarizing into a network mostly connected, or even fully connected. Retrieve information from these two states in the process of network formation could be of great value when the design parameters of the sensor have to be chosen. Conceptually addressing questions such the existing relationship between the number of nodes or beads and linkers in order to acquire the desired network state, or the relationship between the HLA deposited over the beads and the connectivity resulting at a given number of linkers, the role of the affinity epitope-HLA and the dynamic of the aggregations.

The necessity of creating a theoretical framework to support the preliminary designing process of the UV-vis sensor is obvious due the complexity underlying the network formation, before starting with the assays in the wetlab and designing DOE (Design of Experiences) to rationally perform approaches to choose the parameters and prove the functionality of the layout.

An ER like graph model have been developed, beginning by analytical and probabilistic approximation and subsequently followed by and, where the intrinsic dynamics that emerge from it aligns exactly with the results of the analytical approximation as it is going to be shown below.

Has been simulated within stochasticity. The dynamics obtained from the mathematical approach arise. Furthermore, reliable features useful for a future design emerge.

Stochastic Erdös-Rényi model

A Stochastic, non-spatial and time discrete model has been created into NetLogo environment (García & Sancho). The stochastic models involve noise as an intrinsic characteristic of the complex reality of the biological systems and introduce the effects of the second principle of the thermodynamics. The time step is dynamic and discrete. This stochastic model doesn’t take in account the space adopting the simplest scenario that could reproduce a robust basic behavior.

The same principle as in Erdös-Rényi has been followed to built up the stochastic model. Taking two random nodes at each time stem and connecting them using an existing linker available. The giant component of the system is pursued by the red colour and will determine how connected the network at increasing the connections per node as the linkers are running out. It is important to notice that que time scale is not important in this model, the basis falls on increase the connections per node by adding linkers and follow the formation of the giant component and the general system itself. The resulting graph would be exactly the same that the one predicted analytically, and the noise of the system will be observable, but the tendency as robust as before.

*Stochastic adimensional ER model. The tunable parameters as p (affinity HLA – epitope) assumed maximal, num-node as the number of beads computed, the linker available quantity. Real time dynamics of network formation while the average connections per node are increased.

Probability of union is a tunable parameter, representing the affinity of the HLA and the gluten epitope or the linker. It is important to notice that changing the union rate it would not change the intrinsic dynamics of network formation. It will affect, naturally, the fastness of the connection process between nodes but will not change the biphasic reality of the network formation or the potential number of edges per node required to change between phases and subsequently to weave the connected net, or alternatively, the fragmented one. The number of nodes or nanoparticles is, of course, a tunable parameter. Yet this aspect could not change the dynamics either. Adding more nodes will affect directly on the accuracy of the results without changing the resulting dynamics. As more nodes we could simulate the closer will be the resulting data to the pure mathematical approach and the interference related to noise will be softer.

As a result, the following 3 graphs represent how the simulation using different node amount will not affects the underlying dynamic predicted mathematically above. At the same time, clear view of the biphasic nature of the connectivity of the network can be taken. As well as, could be identified the critical point that separates two well defined states. The same lecture could be extracted from the stochastically model in comparation to the mathematical approach. The connectivity of the network will not increase linearly with the increase of the average connections per node, which means that, increasing the number of HLA per node the connectivity of the hole will change the dynamic like a switch when a critical number of an average connections per node is reached. As a matter of fact, when the average connections per node is below the unit, the system tents to become fragmented, alternatively when the average connections per node overpass the criticality of the unit, the connectivity emerges for free.

*The growth of the giant component while the average connections per node changes. Computed at different gold nanoparticles amount.

These corroborations of the dynamics could give the designer valuable information related to the sensor nature while arise important technical considerations relevant for the practical implementation.

First, is crucial to observe that an increasing on the linkers proportion over nodes will not lead to a better connectivity of the network, because the biphasic dynamics is not ruled by the number of linkers. Nevertheless, the average connections per node will affect directly to the resulting network. Its true that a lack of linkers will preclude the connectivity of the network no matter how potential connectivity has the hole system. For this reason, the average linkers/nanoparticles must be around 1:1 at least if the full connectivity is on the search or increase this average in order to improve the availability of linkers to join the nanoparticles and solve a matter of crossing each other into the space. But it is important to remark, that adding exorbitant quantities of linker is senseless. Following the same idea, in the connected phase, as more connections the network makes, less is the improvement in connectivity terms, because they are redundant.

As a matter of fact, another important lesson could be obtained, tuning the proportion of HLA per nanoparticle, the transition between phases could be driven through dynamics. In other words, a distant average HLA per node from the critical point, will lead to a fully connected network in all possible scenarios except one, when all HLA conjugated on to the nanoparticles will be blocked. The intermediate scenario where the presence of reactive epitopes is a fact but not blocks all the connectivity potential must be taken in account. That is why working with proportions of HLA per node near the criticality, will lead to a detection more accurate and the triggering eventuality between fragmented network and fully connected will be more dependent on the presence of blocking epitopes and less dependent on the free HLA.

It is interesting also, that the requirements in order to achieve a full connected scenario, in terms of HLA per node or average connections per node is no superior to 3.5 – 4 units of HLA per nanoparticle. Allowing the designer to focus on this numerical framework. When we have the scenario of a disconnected network, with a final average less than 1 connections per node. The tiny clusters predominate the scenario and the giant component is no different in size to other clusters in the mixture.

Take information of how the formation of clusters relates the connectivity in the network and the output of the sensor is a logical straightforward question that needs to be answered. Subsequently the construction of stochastic models taking in account the space where developed.

3D stochastic spatial model of network connectivity

Tridimensional (3D) model has been designed as cellular automata, using discrete time and asynchronous iteration algorithm. An initial distribution of nanoparticles or nodes are distributed all along the space randomly. Each node diffuses trough the space as a random walker in isotropic tridimensional space (there is no preferential directionalities) (Railsback and Koppel). The collision between nodes trigger their chance of connectivity, just as in Erdös-Rényi theory, assuming definite affinity rate value. It is assumed full accessibility to linkers when two nodes collide, in order to simplify the model. Normal distribution is used to assign the HLA per bead average. Consequently, each connection consumes one free HLA each bead. The number of nodes is also a tunable parameter just as in the adimensional model.

*Stochastic 3D ER model. The tunable parameters as p (affinity HLA – epitope) assumed maximal, num-node as the number of beads computed, the linker available quantity could be adjusted. Real time dynamics of network formation while the average connections per node are increased. The first animation corresponds to a fragmented system, where the critical value of average connections per node has not been achieved. The second one corresponds to a connected system, where the critical value of average connections per node has been overcome.

Biphasic dynamic is expected from tridimensional model, yet the dynamic of the network evolves with time. The HLA per bead distribution is prefixed and will determine the final connectivity of the system. The next graph constitutes the final state of connectivity at different HLA per bead concentration (average connections per node). The final state has been achieved running the code until 5000 iterations, enough time steps to achieve the maximum connectivity at every HLA per bead distribution. Computing 600 nodes or beads in a tridimensional space 60x60x60 units, being the diameter of a bead 1.5 folds the unit. It is important to notice, again, that a variation in the number of nodes and the tridimensional space computed will accelerate o deaccelerate the connectivity process but the intrinsic dynamics will remain unaltered. The critical point has been set at one average connection per node, just as in the adimensional model. However, the real critical point when 3D space is taken in account, could be displaced. In fact, the criticality is displaced between 1 and 1.2 HLA per bead. This change is common when space plays a role in the simulation. Yet the dynamics remains invariant in comparison to the previous ER models.

*Each value corresponds to the final state of a given set of parameters in 3D environment. The affinity and the number of nodes remain invariant (maximum affinity and 600 nodes). tridimensional space 60x60x60 units, being the diameter of a bead 1.5 folds the unit. Varying the average connections per node. The final state has been achieved in each simulation and recorded, finally joined in the graph.

The 3D approach shows important features that must be taken in account to understand closely, how the nanobeads could aggregate and which could be their special conformation. The color change will be triggered by a virtual increase of the nanoparticle’s diameter. In other words, virtual increase of the diameter could be understood under the simplest approximation. Two linked beads lead to a virtual single bead of doubled diameter, and so on (Njoki et al).

The increase in core diameter of gold nanoparticles introduced dramatic and continuous increase in the extinction coefficients. From a core diameter of 4-35 nm, the extinction coefficient increased three orders of magnitude. It has been demonstrated (Liu et al), good linear relationship could be applied and could be expressed as:

lnε = klnD + α

Where, ε is extinction coefficient in M-1cm-1, D is the core diameter of the nanoparticles in nm, and k = 3.32111, a = 10.80505 experimental values. The linear fitting of the experimental data gives a correlation coefficient of 0.99754 and standard deviation of 0.21983. In accordance wit (El-Sayed et al).

This correlation has been implemented in our 3D model, being able to obtain and approximation of the coefficient of extinction at each point of the simulation, directly related to the nature of the network connectivity at each time. Departing from well-defined and characterized gold nanoparticles from Liu et al. work (see table).

*The extinction coefficient of gold nanoparticles with different core sizes and capping ligands determined in Liu et al. study.

*Extinction coefficient at different core diameters. Starting from individual beads of different core diameter. 3.76, 8.55, 20.6 and 34.45 respectively from top left to bottom right.

It is important to remark that the data and information obtained from all these models and mathematical approaches doesn’t allow the designer to use directly the data in a quantitative manner directly. The qualitative uptake that could be obtained from the data in valuable enough to build a theoretical basis to understand the underlying dynamics of our sensor and obtain quantitative starting points to allow further on steps in the practical design of the UV-vis HLA mediated sensor.

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