# On morphological and functional complexity of proteinoid microspheres

Saksham Sharma  Unconventional Computing Laboratory, UWE Bristol; Cambridge
Centre for Physical Biology, Cambridge, UK  Adnan Mahmud  Department of
Chemical Engineering, Cambridge, Philippa Fawcett Drive, Cambridge CB3 0AS, UK
Giuseppe Tarabella  Institute of Materials for Electronic and Magnetism,
National Research Council (IMEM-CNR), Parma (Italy)  Panagiotis Mougoyannis
Andrew Adamatzky  Unconventional Computing Laboratory, UWE Bristol, UK

###### Abstract

Proteinoids are solidified gels made from poly(amino acids) based polymers
that exhibit oscillatory electrical activity. It has been proposed that
proteinoids are capable of performing analog computing as their electrical
activity can be converted into a series of Boolean gates. The current article
focuses on decrypting the morphological and functional complexity of the
ensembles of proteinoid microspheres prepared in the laboratory. We identify
two different protocols (one with and one without SEM) to prepare and
visualize proteinoid microspheres. To quantify the complexity of proteinoid
ensembles, we measure nine complexity metrics (to name a few: average degrees
$\textrm{Deg}_{av}$, maximum number of independent cycles $u$, average
connections per node $\textrm{Conn}_{av}$, resistance
$\textrm{res}_{\textrm{eff}}$, percolation threshold
$\textrm{perc}_{\textrm{t}}$) which shine light on the morphological,
functional complexity of the proteinoids, and the information transmission
that happens across the undirected graph abstraction of the proteinoid
microspheres ensembles. We identify the complexity metrics that can
distinguish two different protocols of preparation and also the most dense,
complex, and less power consuming proteinoid network among all tested. With
this work, we hope to provide a complexity toolkit for hardware designers of
analog computers to design their systems with the right set of complexity
ingredients guided one-to-one by the protocol chosen at the first place. On a
more fundamental note, this study also sets forth the need to treat gels,
microspheres, and fluidic systems as fundamentally information-theoretic in
nature, rather than continuum mechanical, a perspective emerging out from
recent program by Tao to treat fluids as potentially Turing-complete and thus,
programmable.

††preprint: APS/123-QED

Proteinoids are the gels in the shape of microspheres which are the copolymers
made out of amino acids of various kinds [1, 2]. Applications of such
microparticles range from producing protein-like molecules [1], adding
fluorescent probes to the proteinoid nanoparticles (PNPs) [3], drug delivery
[3, 4, 5], producing anti-fog additive (surbitan monooleate encapsulated into
PNPs) [6] to using micro/nano particles in potentially delivering drug for
targeted applications [7] and using as model systems for acoustomanipulation
in sealed microfluidic platform in microfluidic platform [8]. There are
plentiful of protocols available in the literature to produce proteinoids,
each suited to the specific application A comprehensive list of the most
widely used protocols published in the literature was introduced by us
previously here [9]. There we two protocols for the production of proteinoid
microspheres: (i) one based out of Fox & Waeheldt (1968) [10]; and (ii)
another one based out of Przybylski & Fox (1983) [11]. While the former
protocol focused more upon the careful choice of the combination of several
amino acids which are mixed together and heated to form proteinoids (detailed
protocol in the Section I below), the latter protocol ensures that the
proteinoid microspheres have vesicles on their periphery with controlled
permeability, such that the periphery can encapsulate substances such as
$\alpha-$chlorophyll which correlates to high membrane potential.

In [12] we outline pathways for computing with proteinoids. We proposed that
electrical spiking [13, 14] of proteinoid microspheres can be use to construct
neuromorphic devices [15] and implement reservoir computing [16, 17] and
mining for Boolean circuit [18]. In future computing devices made of
proteinoid microspheres an information will need to be transferred along the
microspheres’ ensembles. This is why it is important to analyse the morphology
of the proteinoid ensembles.

Despite plethora of work on tuning the specific details of the protocol for a
desired outcome, the morphological and functional complexity of proteinoid
microspheres, and the information transmission inside it, has not been studied
in a greater depth till date. The most closest to this is about 50-years old
study by Brooke and Fox (1971) [19]. In this work, the authors found that the
thermal proteinoids synthesised artificially have similar morphology as the
microfossils. This morphology is influenced by the presence of salts outside
the proteinoids: salt water or 5% calcium chloride induce larger size to the
proteinoids than the distilled water. Acetic acid was found to induce
dissolution of proteinoids: at 20% concentration, the dissolution was
initiated and at 50% concentration, microspheres were completely dissolved.

The morphology of the bulk of microspheres is another line of enquiry that has
received less attention in the literature. Young (1965) investigated the
chains which are formed by the proteinoid microspheres by inducing slight
pressures on the microscopic slide, cooling the solution, altering the pH of
the medium [20]. However, the morphological and functional complexity of
proteinoid chains, once they are formed, was not investigated. This is because
the focus of the research in 1950s was to allude the possibility that
proteinoid microspheres (or coacervates) can be used as a pre-cell model to
enhance our understanding of the pathways which might be responsible for the
origin of cellular life.

##  I Protocol for the preparation of proteinoids

The materials/apparatus required to prepare proteinoids [21] include: amino
acids in powdered form; Tri-block heater; 3-hole round bottom flasks; nitrogen
gas; magnetic stirrer; cellophane membrane for dialysis; a water bath. To
start, 1.5 grams each of aspartic acid and glutamic acid is mixed and heated
in the 3-hole round bottom flask at 290∘C initially. The temperature is step-
wise increased by 10∘ count, until fumes start to emerge out. The exhaust in
the fumehood (with N2 gas as the inlet) is started to release the fumes out.
Firstly, the powder changes from white colour to green colour, and then
changes its morphology into a series of boiling microspheres. When the heating
is turned off, the residue becomes solid and is extracted out and allowed to
cool for half an hour. The extracted residue is put inside the Slide-A-Lyzer
mini dialysis device, of 10,000 molecular weight cut-off against water as the
dialysate. Dialysis is continued for 5 days until the residue consists of tiny
microsphere ensembles which are visible to the naked eyes. The residue from
the dialysis membrane is heated inside the vacuum oven for half an hour, to
evaporate the dialysate (water) from the residue. The sample is then analysed
under the transmission electron microscope.

##  II Image analysis of proteinoid ensembles

Figure 1: SEM images of the proteinoid ensembles. Equimolar mixture of
L-Glutamic acid, L-asparginine, L-phenylalanine in water, at different
magnification indication by the scale bars in the bottom right of each image.
Figure 2: Undirected graphs of the images shown in Fig. 1 generated using
Relative Neighbourhood algorithm, by employing the Delaunay Voronoi plugin in
open-source Fiji (ImageJ) software.

Once the images of proteinoid microspheres are obtained using SEM (Fig. 1),
then the next step is to analyse the granular details of the images in order
to extract some relevant parameters out of it, that could enhance our physical
intuition of the system. The image analysis protocol is carried out using
open-source Fiji (imageJ) software, and can be divided into two major steps:

  1. 1.

Particle selection:
On the Fiji panel, select Image >Adjust >Threshold. Tick Dark background and
select Default and Red from the pull-down menus. Use the top slider so as to
threshold the image until the particles are red against a dark background,
then select Apply. To convert this image to a binary image, select Process
>Binary >Make binary. The particle selection is done by selecting Process
>Binary >Watershed.

  2. 2.

Relative Neighbourhood Graph sampling: Once the granular images of particles
with enough clarity is obtained, the next step is to convert these images into
graphs, to estimate their functional and morphological complexity. To convert
raw images of particles into graphs, Delaunay triangulation (DT) need to be
constructed (Fig. 2).

The algorithm that Fiji uses to construct Delaunay triangulation is available
as a plugin, accessible by selecting Plugins>Analyze>Delaunay Voronoi. The
distinctive characteristic of this algorithm is that it ensures that in the
set $\mathbb{P}$ of discrete points in the graph, there exists no point in
$\mathbb{P}$ which is inside the circumcircle of any triangle that is formed
by the triangulation111A triangulation of a set of points $\mathbb{P}$ is a
simplical complex that encompasses the convex hull of $\mathbb{P}$ and whose
vertices belong to $\mathbb{P}$.  of $\mathbb{P}$.

After the Delaunay Voronoi option is selected, the graph that embed all the
central points of the particles is constructed in the form of ROI. This graph
can be saved in the .roi format. It is possible to store the coordinates of
the vertices and the edges of the triangulation constructed in the graph. This
information will be later used to calculate graph complexity of each image, as
discussed in the next section. The mean distance and the variance of points in
the set $\mathbb{P}$ is also calculated using Delaunay Voronoi algorithm.

##  III Results of the analysis

Figure 3: Results of the analysis: diameter of the proteinoid microspheres
(average size and mean distance) and the distance between the particles which
is recorded in the graphs.

By employing the tools discussed in the previous section, particle sizes —
average and standard deviation — is determined. Delaunay Voronoi triangulation
method is used to determine the distances between the particles. The results
are shown in Fig. 3. Interestingly, the size recorded in the experiments is
significantly lower by two orders of magnitude than the ones reported in
experiments previously [1, 12]. It means the protocol for the preparation of
the proteinoids [21] needs to be modified.

Figure 4: Proteinoid microspheres of almost 200 times larger size than the
ones shown in Fig 1. (a,b) Proteinoids made out of 1:1 mixture of aspartic
acid and L-arginine, (c) Table showing diameter of the proteinoids and
distance between them recorded in the graph.

In another batch of preparation of proteinoids (Protocol 2) heating of the
proteinoids is done at lower temperatures than before - at 70∘ C. Lower
temperature ensures that the particles remain coarse-grained and are not
destroyed (unlike at higher temperatures). Another advantage of producing
proteinoids at $70^{\circ}$C is that the larger size of particles (in the
order of $\sim 100$ µm) allows one to use standard objective microscopes in
the lab, and thus the need of using SEM vanishes. The new results for the
modified protocol: images of proteinoids obtained in microscope; undirected
graph corresponding to each image; results (diameter of particles and
distances between them) is given in Fig. 4.

To summarise the difference in two protocols: Protocol 1 used an equimolar
mixture of L-Glutamic acid, L-asparginine, L-phenylalanine by heating the
powder (upto 290∘C) in the initial stage this until it gets almost black to
get O(100) nanometres sized proteinoid particles. On the other hand, Protocol
2 used 1:1 mixture of aspartic acid and L-arginine to produce O(100) microns
sized particles by heating the powder (upto 70∘C) in the intial stage until it
turns reddish brown.

##  IV Complexity and information transmission analysis

In the previous section, Delaunay triangulation method was used to determine
the size of the particles and the distance between them. In this section, the
discussion on the functional and morphological complexity of the proteinoid
particles will be pursued.

The metrics to understand the complexity of proteinoids are manifold. The
primary choice behind the choice of these metrics stems from the fact that the
interconnection between proteinoid particles is abstracted as undirected
graphs. The primary metrics of relevance here are the total number of edges
$e$ and total number of nodes (vertices) $v$ in the graph. A graph $G$ is
usually characterised by the number of vertices and edges that it contains,
and usually expressed as $G(e,v)$. Once the graph is defined, the secondary
set of metrics comes into picture. First in the list is the average degree
given by

$\textrm{Deg}_{av}=\frac{\textrm{Total edges}}{\textrm{Total nodes}}$ (1)

defined and previously discussed here [23] for an undirected graph. The second
in the list is the maximum number of independent cycles $u$ given by

$u=\frac{e}{2}-v+p$ (2)

where $p$ is the number of subgraphs, previously discussed here [24] and edges
are counted once (hence, divided by two). Third metric in the list is the
diameter of the graph $D_{G}$ given by shortest distance between two vertices
which are at the largest distance from each other, and the distance is counted
in terms of nodes that connects the vertices.

Apart from complexity measures, information transmission is another parameter
that is important to understand the amount of information flow between nodes
in a graph. An important parameter to characterise the amount of information
transmission is given by the average connections per node $\textrm{Conn}_{av}$
given by

$\textrm{Conn}_{av}=\frac{\textrm{Total connections}}{\textrm{Total nodes}}$
(3)

where total connections is given by the cumulative number of connections each
node has in the graph.

There are three more metrics that are slightly less direct to calculate that
the ones mentioned above. These are, namely, total effective resistance
$\textrm{res}_{\textrm{eff}}$, average shortest path $p_{short}$, and average
edge length $l_{edge}$. The definitions of each of them are as follows.

Figure 5: (a) Schematic representation of the pipeline measuring the
complexity ($\textrm{res}_{\textrm{eff}}$, $p_{short}$, $l_{edge}$) of a given
set of nodes, (b) Randomly generated undirected planar graph with 64 nodes.
Figure 6: Table of nine complexity and information transmission metrics for
the images of proteinoid ensembles captured in the present study. Image
labelled ‘random’ is shown in Fig. 5(b).

The NetworkX library in Python is used to calculate the three metrics:
$\textrm{res}_{\textrm{eff}}$; $p_{short}$; $l_{edge}$ mentioned above. At the
core, all-pairs shortest paths is calculated using Floyd-Warshall algorithm.
By including intermediary nodes, the Floyd-Warshall method aims to steadily
improve the shortest route estimates between all pairs of nodes [25].
Initially, the algorithm considers the direct edge weights between nodes as
the shortest path estimates. Then, it iteratively updates the estimates by
considering all possible intermediate nodes. For each pair of nodes, the
shortest path length is extracted from the all-pairs shortest paths array.
Then the effective resistance is calculated as the inverse of shortest path
length. The total effective resistance is the sum of all effective
resistances. The total shortest paths is calculated as the sum of all the
shortest path lengths in the all-pairs shortest paths array. The average
shortest path is then calculated by dividing the total shortest paths by the
number of possible pairs of nodes in the graph. The code iterates through all
the edges in the graph and sums their weights (lengths) as total edge lengths.
The average edge length is calculated by dividing the total edge lengths by
the number of edges in the graph. The schematic of the pipeline used to
measure complexity metrics for a given set of nodes is given in Fig. 5.
Another important metric calculated in this analysis is the percolation
threshold. Percolation threshold $\textrm{perc}_{\textrm{t}}$ is essentially a
critical value of the proportion of nodes or edges in a graph, below which the
network becomes fragmented into isolated clusters, i.e., the graph transitions
from being fully connected to disconnected, and above which, the graph can be
seen a giant fully connected graph of roughly the same size as the system
itself. The algorithm to measure is as follows:

  1. 1.

A set of edges are randomly removed using a variable $p$ (defined below).

  2. 2.

For the graph, before and after, removing the edges, calculate the largest
connected component (using a built-in nx library function)

  3. 3.

Finally, the percolation threshold is defined as dividing the size of the
largest connected component after edge removal by the size of the largest
connected component before edge removal.

Figure 7: Plots of complexity metrics
$e,v,\textrm{Deg}_{av},p_{short},l_{edge},u,D_{G},\textrm{Conn}_{av}$ against
the images Fig. 2(a-d) and Fig. 4(a-b). $D_{G}$ is defined in terms of nodes.
Figure 8: Plots of complexity metrics
$\textrm{p}_{short},l_{\textrm{edge}},\textrm{res}_{\textrm{eff}},\textrm{perc}_{\textrm{t}}$
against the images Fig. 2(a-d) and Fig. 4(a-b).

The variable $p$ represents the proportion of edges that are to be randomly
removed from the graph. To compute the percolation threshold traditionally,
one needs to vary the proportion $p$ of edges that are to be removed, and then
observe how the size of the largest connected component changes. One might
term the percolation threshold measured, thus, as a relative percolation
threshold $RPT$, a term not introduced before in the literature, which is used
across all the graphs. By iteration over all the $p$ values, one observes the
breaking point.

##  V Complexity of proteinoid ensembles: results and discussion

Using the methods described above, the nine complexity metrics:
$e,v,\textrm{Deg}_{av},p_{short},l_{edge},u,D_{G},\textrm{Conn}_{av},\textrm{res}_{\textrm{eff}},\textrm{perc}_{\textrm{t}}$
are evaluated for each of the image of the proteinoid, and the results are
shown in Fig. 6. The results for each of the parameter against the images are
plotted in Fig. 7\- 8.

From the figures, it is apparent that Fig. 2(c) has the highest number of
total edges $e$ and total nodes $v$, making it the most dense graphs and
proteinoid ensembles out of all the images tested. However, quantifying
complexity using edges and nodes is not a good indicator to understand the
functional complexity of a system. The only advantage of measuring $e$ and $v$
is that a set of random undirected graphs (5(b)) can be simulated to compare
the complexity measures against, and thus, have scaled values of complexity
instead of absolute. In Fig. 6, a random undirected graoh with nodes $v=81$
and edges $e=384$ is constructed to compare against the experimental images in
Fig. 2 and 4. Highest complexity of Fig. 2(c) at 5.89 against the base value
of 5 for random undirected graph (scaled complexity of 1.18) is also apparent
through the measures of $\textrm{Deg}_{av}$; the lowest scaled complexity
(average degree) is 0.8 for 4(a). As is apparent from this comparison, the
average degree metric for the Protocol 2 is significantly lower than Protocol
1, meaning that the information flow is slower and the network is lesser
robust and resilient against random failures and targeted attacks. Another
important complexity metric is $\textrm{Conn}_{av}$, which is highest for Fig.
2(c), with a scaled complexity value of 1.08; the lowest scaled value is 0.55
for 4(a). This again affirms that the information flow and robustness-
resilience of the proteinoid hardware is highest when Protocol 1 is used as
opposed to Protocol 2. Apart from the metrics above, the diameter of the graph
(in units of nodes) is highest for Fig. 2(c) (scaled value of 2.64) which
means that the proteinoid ensemble in Fig. 2(c) is both widely and densely
distributed, in comparison to the scaled value of 0.18 for 4(a). It should be
however noted that the absolute average number of connections for proteinoid
ensembles fall somewhere between 4.5 and 5.89 (scaled falls in the range
0.9-1.2); in the case of Fig. 4(a,b) it can reduce further down to scaled 0.8.
It again confirms a building hypothesis that the protocol of preparation of
proteinoid has a direct influence on the range of $\textrm{Conn}_{av}$,
particularly Protocol 1 builds proteinoid hardware which is more dense, widely
spreaded with lower information loss and higher resilience-robustness. Apart
from the such primary morphological and complexity measures, we are also
interested in secondary metrics that might inform us better about the analog
hardware of proteinoid ensembles.

To investigate the information transmission, resistance of the graphs is shown
in Fig. 8(c) and it is apparent that the Fig. 2(c) has the highest resistance:
the scaled value is 42.81, thus Protocol 1 offers the maximum resistance to
pass any information through itself. In comparison, Fig. 4(b) has the lowest
scaled value of resistance at 0.05, again confirming that the Protocol 2 is
less preferable than Protocol 1. In the context of information transmission,
it would be safe to say that the lower resistance of proteinoid network
produced using protocol 1 would mean that the energy dissipation is higher and
the magnitude of electric current passing is signficantly lower by almost
three orders of magnitude than the proteinoid network produced by protocol 2.
This offers a new insight into the design of analog hardware using proteinoid
networks; it is not always the case that protocol 1 could be sometimes
desirable for its highly complex network design. To minimise energy loss in
information transmission, a less dense and more scattered proteinoid network
is preferable. This insight does bolsters the importance of measuring
secondary complexit metrics, instead of making conclusions only from primary
metrics. Such secondary measures of complexity metrics are useful for our
analog computing studies using proteinoid microspheres in the future, where we
would be interested in converting the information transmission occurring
through the proteinoid microspheres to a series of Boolean gates, and these
logical gates would become a hidden, underlying information-theoretic language
of the microspheres.

Since Fig. 2(c) is the most complex and dense image with lower rate of
information transmission, it has the least edge length as shown in Fig. 8(b).
It should be however noted that the average shortest path is not a reliable
measure of complexity of the graph within the images which are prepared using
the same protocol (Fig. 2(a-d)), however, it still can be used to compare
different protocols of preparation: $p_{short}$ for Fig. 4(a-b) (protocol 2)
is significantly lower than Fig. 4(a-b) (protocol 1). It is to be noted that
the average shortest path and average edge length complexity metrics do not
offer a direct insight into the information transmission inside the proteinoid
networks. However, they do help a hardware designer to make sure that the
proteinoid networks, thus designed, are consistent over multiple factory
makes. It is good to again reinstate that such metrics are useful only to
ensure the designer to produce identical proteinoid networks as final customer
products such that analog computing operation becomes device agnostic. This
feature is important to consider while designing industry production pipeline
for analog hardware and computing systems made out of proteinoid networks.

The last important complexity metric to consider is the percolation threshold
$\textrm{perc}_{t}$. The scaled value of $\textrm{perc}_{t}$ is lowest (0.05)
for Protocol 2 and highest (0.41) for Protocol 1. This means that it is
relatively easier for the proteinoid network produced using Protocol 2 to
become a single fully connected piece of hardware carrying out similar kind of
operations. Networks produced using Protocol 1, on the other hand, have more
tendency to remain as isolated clusters with as little connection as possible.
The usefulness of having isolated clusters is that the proteinoid hardware can
be useful to perform analog computing processes that require parallel
operations (read, cores, in digital context) to expedite the computing task.
It also allows to reduce the chances of failure in completion of the task if
some clusters fail to perform well - the system, on the whole, can be designed
to not effectively fail, and still complete the task.

##  VI Conclusion

This study identifies key complexity metrics that can characterise the
computational and design complexity for the functioning of proteinoid networks
as potential chemical hardware that can aid in performing analog computing
tasks. Analog computing, albeit initially popular in early 20th century, is
again coming back into popularity in early 21st century, because of: (i) rise
in ML algorithms that do not need to be necessarily accurate 100% in their
output, and (ii) Moore’s law about to come to an end, hence, a demand for non-
digital computing hardwares. Proteinoids, which are essentially gels, made
from poly(amino acids) are shown to be potential contender in this direction.

We devised two different protocols to produced proteinoids; Protocol 1 to
develop O(100) micron sized and Protocol 2 to develop O(100) nanometer sized
particles. We used primary and secondary complexity metrics to investigate the
functional complexity of such microspheres, where the function of concern is
the information transmission, had the proteinoid networks are being subjected
to performing arbitrary analog computing task. Protocol 1 performed better
when it comes down to characterising system in terms of average degrees,
average connections per node, and percolation threshold. However, Protocol 2,
despite being being less dense and less complex, is more useful to aid in easy
flow of information across the system with as little power dissipation as
possible. It would be, thus, safe to conclude that the protocol that would
meet the industry requirement would fall somewhere between Procotol 1 and
Protocol 2, such that the complexity metrics measured above, especially the
average resistance and the percolation threshold, are higher and consistent in
the final product.

##  VII Future directions and perspective

After the morphological and functional complexity of the proteinoid
microspheres is established and convenient protocol for the preparation of
consistent and efficient analog hardware is established, we are planning to
venture into experiments (using voltage-sensitive dyes) on mapping binary
sequences and QR codes, represented as spikes of electrical activity, by
ensembles of proteinoid microspheres. This would enable industry specialist to
treat such chemical systems and their inherent language of electrical signals
as no less than a potential hardware that can carry information across through
its network and build fundamental block of analog computing - in the form of
logic gates and QR codes - thus enabling them to customise soft matter fluidic
systems in performing analog computing and machine learning tasks.

The broader scope of the work is in aiming towards relinquishing a
mathematical treatment of microdroplets, because of the reasons that the
mathematical continuum model of treating arbitrary222It is to important to
note that analytical solution of Navier-Stokes equations are possible for
specific, special, and well-chosen cases. When it comes down to arbitrary
choice of problem, i.e., arbitrary initial conditions of velocity fields and
arbitrary domain, then, in such cases, completely analytic and general
$C^{\infty}$ solution do not exist. Hence, the potential departure to
information-theoretic solution, that perhaps is divisible into series of logic
gates, of Navier-Stokes equations is sought after. system of fluids or gels is
inherently flawed as argued recently [27]. This is line with a recent result
by a group of mathematicians that the Euler equations are Turing-complete, and
thus fluids should be seen as fundamentally programmable instead of
analytically tractable. The reason why analytical tractable of arbitrary
fluidic or gel physical systems is not possible is because the continuum
equations do not take into account the atomic/molecular contributions within
the system (argued here [28]) and in cases where such contributions are not
taken, physical manifestation of mathematical singularities plays out. This
analytical intractability has been the key reason why the millennium problem
of the 21st century - Navier-Stokes regularity problem [29] \- still remain an
elusive and unsolved problem among the community of functional analysts. It
can be safely conjectured that a strong combined effort by theoretical
computer scientists and logicians, apart from mathematical physicists, would
be needed to solve the Navier-Stokes millennium problem. With this theme in
mind, the current study also plans to investigate the design of logical gates
in our upcoming work that can represent information transmission in proteinoid
microspheres. Instead of using continuum mechanics to represent information in
this system, we would be mindful to use a well-designed system of Boolean
gates to do the same, as suggested almost a decade ago by Tao’s fluid program
[30], inspired from Conway’s Cellular Automaton [31].

## Code

The github code used in the article to estimate $p_{short}$, $l_{edge}$,
$\textrm{res}_{\textrm{eff}}$ is available here and is written solely by A.M.

## References

  * Harada and Fox [1958] K. Harada and S. W. Fox, The thermal condensation of glutamic acid and glycine to linear peptides1, Journal of the American Chemical Society 80, 2694 (1958).
  * Fox  _et al._ [1959] S. W. Fox, K. Harada, and J. Kendrick, Production of spherules from synthetic proteinoid and hot water, Science 129, 1221 (1959).
  * Hadad  _et al._ [2020] E. Hadad, S. Rudnick-Glick, E. Itzhaki, M. Y. Avivi, I. Grinberg, Y. Elias, and S. Margel, Engineering of doxorubicin-encapsulating and trail-conjugated poly (rgd) proteinoid nanocapsules for drug delivery applications, Polymers 12, 2996 (2020).
  * HARATAKE  _et al._ [1998] M. HARATAKE, R. ZHAO, J. LIAO, and R. M. OTTENBRITE, Self-assemblies as drug delivery systems, Frontiers in Biomedical Polymer Applications 1, 59 (1998).
  * Kolitz-Domb and Margel [2018] M. Kolitz-Domb and S. Margel, Recent advances of novel proteinoids and proteinoid nanoparticles and their applications in biomedicine and industrial uses, Israel Journal of Chemistry 58, 1277 (2018).
  * Sason  _et al._ [2017] E. Sason, M. Kolitz-Domb, S. Cohen, I. Grinberg, and S. Margel, Engineering of new proteinoids and proteinoid nanoparticles of narrow size distribution for anti-fog applications, J. Nanomed. Nanotechnol 8, 1 (2017).
  * Dhar  _et al._ [2020] P. Dhar, S. Narendren, S. S. Gaur, S. Sharma, A. Kumar, and V. Katiyar, Self-propelled cellulose nanocrystal based catalytic nanomotors for targeted hyperthermia and pollutant remediation applications, International journal of biological macromolecules 158, 1020 (2020).
  * Pitingolo  _et al._ [2018] G. Pitingolo, P. Nizard, A. Riaud, and V. Taly, Beyond the on/off chip trade-off: A reversibly sealed microfluidic platform for 3d tumor microtissue analysis, Sensors and Actuators B: Chemical 274, 393 (2018).
  * Sharma  _et al._ [2022a] S. Sharma, P. Mougoyannis, G. Tarabella, and A. Adamatzky, A review on the protocols for the synthesis of proteinoids, arXiv preprint arXiv:2212.02261 (2022a).
  * Fox and Waehneldt [1968] S. W. Fox and T. V. Waehneldt, The therml synthesis of neutral and basic proteinoids, Biochimica et Biophysica Acta (BBA)-Protein Structure 160, 246 (1968).
  * Przybylski and Fox [1984] A. T. Przybylski and S. W. Fox, Excitable artificial cells of proteinoid, Applied Biochemistry and Biotechnology 10, 301 (1984).
  * Adamatzky [2021] A. Adamatzky, Towards proteinoid computers. hypothesis paper, Biosystems 208, 104480 (2021).
  * Przybylski [1985] A. T. Przybylski, Excitable cell made of thermal proteinoids, BioSystems 17, 281 (1985).
  * Vaughan  _et al._ [1987] G. Vaughan, A. T. Przybylski, and S. W. Fox, Thermal proteinoids as excitability-inducing materials, BioSystems 20, 219 (1987).
  * Zhu  _et al._ [2020] J. Zhu, T. Zhang, Y. Yang, and R. Huang, A comprehensive review on emerging artificial neuromorphic devices, Applied Physics Reviews 7, 011312 (2020).
  * Schrauwen  _et al._ [2007] B. Schrauwen, D. Verstraeten, and J. Van Campenhout, An overview of reservoir computing: theory, applications and implementations, in  _Proceedings of the 15th european symposium on artificial neural networks. p. 471-482 2007_ (2007) pp. 471–482.
  * Lukoševičius and Jaeger [2009] M. Lukoševičius and H. Jaeger, Reservoir computing approaches to recurrent neural network training, Computer science review 3, 127 (2009).
  * Roberts and Adamatzky [2022] N. Roberts and A. Adamatzky, Mining logical circuits in fungi, Scientific Reports 12, 15930 (2022).
  * Miquel  _et al._ [1971] J. Miquel, S. Brooke, and S. W. Fox, Assembly of microspheres from acidic proteinoids and histones or histone-like proteinoids, Biosystems 3, 299 (1971).
  * Young  _et al._ [1965] R. S. Young _et al._ , Morphology and chemistry of microspheres from proteinoid., The origins of pre-biological systems and of their molecular matrices. (1965).
  * Mougkogiannis and Adamatzky [2023] P. Mougkogiannis and A. Adamatzky, Towards proteinoid computers. hypothesis paper, Sci. Rep. 13, 1992 (2023).
  * Note [1] A triangulation of a set of points $\mathbb{P}$ is a simplical complex that encompasses the convex hull of $\mathbb{P}$ and whose vertices belong to $\mathbb{P}$.
  * [23] Graphs, https://bookdown.org/omarlizardo/_main/2-7-average-degree.html.
  * [24] Maximum number of independent cycles, https://transportgeography.org/contents/methods/graph-theory-measures-indices/number-cycles-graph.
  * [25] Floyd-Warshall algorithm, Wikimedia, Wikimedia foundation, https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm.
  * Note [2] It is to important to note that analytical solution of Navier-Stokes equations are possible for specific, special, and well-chosen cases. When it comes down to arbitrary choice of problem, i.e., arbitrary initial conditions of velocity fields and arbitrary domain, then, in such cases, completely analytic and general $C^{\infty}$ solution do not exist. Hence, the potential departure to information-theoretic solution, that perhaps is divisible into series of logic gates, of Navier-Stokes equations is sought after.
  * Sharma  _et al._ [2022b] S. Sharma, G. Marcucci, and A. Mahmud, A complexity perspective on fluid mechanics, arXiv preprint arXiv:2212.00153 (2022b).
  * Sharma and Marcucci [2022] S. Sharma and G. Marcucci, From navier-stokes millennium-prize problem to soft matter computing, arXiv preprint arXiv:2212.01492 (2022).
  * Fefferman [2000] C. L. Fefferman, Existence and smoothness of the navier-stokes equation, The millennium prize problems 57, 67 (2000).
  * Tao [2016] T. Tao, Finite time blowup for an averaged three-dimensional navier-stokes equation, Journal of the American Mathematical Society 29, 601 (2016).
  * Adamatzky [2010] A. Adamatzky,  _Game of life cellular automata_ , Vol. 1 (Springer, 2010).