ABSTRACT An abstract model of electric circuit self-assembly that is amenable to exact analysis has been proposed in previous works in the circuit tile assembly model (cTAM) to understand

self-assembled and self-controlled growth as emergent phenomena that are capable of complex behaviors, like self-replication. In the cTAMs, a voltage source represents a finite supply of

energy that drives growth by attachment of a single circuit tile type until it is unable to overcome randomizing factors in the environment, represented by a threshold of hybridization at

tip voltages. Here, the cTAM is extended to allow attachment of copies of any tile from a predetermined finite heterogeneous set of tile types, which may include resistors, inductances

results may apply to many other areas. SIMILAR CONTENT BEING VIEWED BY OTHERS SYSTEMS OF AXON-LIKE CIRCUITS FOR SELF-ASSEMBLED AND SELF-CONTROLLED GROWTH OF BIOELECTRIC NETWORKS Article Open

access 04 August 2022 PATTERN RECOGNITION IN THE NUCLEATION KINETICS OF NON-EQUILIBRIUM SELF-ASSEMBLY Article Open access 17 January 2024 ISOTHERMAL SELF-ASSEMBLY OF MULTICOMPONENT AND

EVOLUTIVE DNA NANOSTRUCTURES Article Open access 31 July 2023 INTRODUCTION Self-assembly models are inspired by natural phenomena where interactions among component parts and the environment

build complex structures, including examples such as biomolecules (DNA and proteins) and living organisms. Theoretically, self-assembly can be understood as an algorithmic process,

resulting in complex and powerful behavior that is capable of Turing universal computation1,2, and has produced new methods for building nanostructures3. Originally motivated by DNA-based

self-assembly3, the circuit tile assembly model (cTAM) was introduced to analyze and understand how self-controlled growth4 and self-replication5 (i.e., an electric potential \(\nu _0\) that

is consumed as copies of a tile type attach to a seed tile and partial assemblies to form a growing family of circuits in an circuit ladder4, as illustrated in Figs. 1, 2 and 3 in section 

2.1). In the cTAM, tiles attach if a voltage (Direct Current, or DC) at the terminus (growing tip) is greater than or equal to a threshold \(\tau\). The assembly process is assumed

monotonic, _i.e._, once a tile is attached, the attachment will never dissolve. This assembly process produces a family of circuits whose electrical properties change as the circuit grows,

which in turn, modulates electric signals that are communicated throughout. In contrast to models based on differential equations, the cTAM is abstract and discrete, with the potential to

reveal the essence of electrical effects in natural phenomena. As an abstract model, the cTAM is amenable to analysis for exact prediction of its electrical properties and thus, circuit size

(such as self-controlled growth to a maximum predictable size as a function of the voltage source \(\nu _0\) or threshold \(\tau\))4,6,7,8, and their effects on growth of the assembled

circuit (such as self-replication5). The goal of this paper is two-fold. First, to extend the model to allow for the growth of more complex patterns in mature circuits by allowing the

attachment of any of a predetermined finite heterogeneous set of tile types, the htcTAM model. Second, to provide a full analytic solution of the growing family of circuits at any point in

the assembly process and the exact properties of the grown terminal circuits for size and response, for both DC and AC power sources. In Section 2.1, the htcTAM is defined precisely. Exact

closed-form expressions for the node potentials as a function of its parameters and size of the circuit are derived in Section 2.2; they are applied to the analysis of RLC circuit

self-assembly in Section 3, where the products of htcTAM self-assembly in a system of ladder circuits and their mature size are characterized. Finally, a discussion of the significance of

the htcTAM for development and function of structures in biological and other more complex circuits systems is discussed in Section 4. THE GENERAL CIRCUIT TILE ASSEMBLY MODEL AND ITS

SOLUTION In this section, the circuit model and the self-assembly model being used are defined precisely. A full solution for the distribution of potentials as well as some properties of the

models assembled are then derived using nodal analysis for a constant power source \(\nu _0\) of direct current. The results will then be applied to do likewise for the most general kind of

heterogeneous circuits in definition 1 in section 3. THE HETEROGENEOUS HTCTAM Definitions of a circuit and the homogeneous cTAM and the axonal cTAM (acTAM) have been given in7,5,6. The

heterogeneous htcTAM is a generalization of both and the definition is given in full here to make this paper self-contained. DEFINITION 1 _(htcTAM Circuit)_ An _heterogeneous circuit_ is a

tuple $$\begin{aligned} \Psi = (N, E, C, g, \partial N) \end{aligned}$$ on a graph (_N_, _E_), where _N_ denotes the set of nodes corresponding to electrical nodes in the circuit, _E_

denotes the set of edges, _C_ is a set of circuit components (chosen from _positive valued_ resistors, capacitors, inductances and voltage sources) assigned to edges \(e_{(i,j)} \in E\)

where \(\{i, j\} \in N\), and _g_ maps some subset of nodes \(\partial N\) to some subset of glues labeled from a finite set \(\Sigma\), _i.e._ \(g: \partial N \rightarrow \Sigma\).

\(\partial N = N_{in} \cup N_{out}\) consists of two finite subsets of nodes, _input nodes_ \(N_{in}\) and _output nodes_ \(N_{out}\) of the circuit, and are the points at which glues bind

tiles together on the boundary of the circuit. The size of the circuit is the number of tiles in it. A homeogeneous htcTAM is an htcTAM with only kind of resistor tiles and no capacitors or

inductor tiles, and will be referred to as an htcTAM. DEFINITION 2 _(htcTAM)_ An _heterogeneous Tile Assembly Model_ (htcTAM) is a tuple \(\mathcal {C} = (\Gamma , S, \nu _0, \zeta , \tau

)\), where \(\Gamma\) is a finite set of tiles, \(S \subset \Gamma\) is a set of seed tiles, \(\nu _0\) is the potential at the power source (of either direct or alternating current),

\(\zeta : \Gamma (N_{in}) \times \Gamma (N_{out}) \rightarrow \{0,1\}\), is a glue indicator function that determines whether glues on input nodes of a tile match or bind to glues on output

nodes of htcTAM circuits, and \(\tau \in \mathbb {R_+}\) is the threshold voltage that determines the criterion for further attachments. The simplest homogeneous hcTAM consists of two

circuit tiles, namely a seed tile (Fig. 1) and an unlimited number of copies of one attaching ladder (i.e. resistor) tile besides the seed tile (Fig. 2.) The ladder tiles consist of an

impedance _Z_ in series with a parallel combination of a second impedance \(Z^{\prime }\). Ladder tiles attach to the seed, and subsequently to other ladder tiles if and only if the node

potential at the output nodes of last tile in the assembly is at least \(\tau\) (Fig. 3.) The vector of values of the electric potential distribution \(\mathbf {\nu }\) in the emerging

homogeneous hcTAM (and even the heterogeneous htcTAM) circuit of size _n_ tiles will be denoted \(\nu ^{(n)}(t)\) (or just \(\nu (t)\) if a certain size _n_ is being assumed.) It is a

distribution of electrical potentials \(\nu = ( \nu _k(t) )_k\) on _n_ nodes labeled \(k=0,\ldots ,n\) between \(Z_k\) impedances, including the potential \(\nu _0\)(t) at the seed. In the

general heterogeneous model, the ladder tiles include resistors, capacitors, inductors or any of them, characterized by various _impedances_ \(Z_k\) and \(Z_k^\prime\) at tile in position

_k_ whose values may (or may not) differ from one to the next tile. Ladder tiles of various types bind to form assemblies in the shape of growing ladders (Fig. 3.) In a _reaction-rate

limited regime_, ladder tiles are present in saturation and always bind whenever the tip potential is at least \(\tau\), whereas in a _diffusion-limited regime_, ladder tiles only arrive and

attach at set time intervals. The specific mechanism by which tiles attach can be left unspecified under either assumption without affecting the results below. (One specific implementation

would follow the well known aTAM model of DNA self-assembly1,2, in which a tile has a pair of oligonucleotides _a_, _b_ on the output nodes of the circuit that may bind to their

corresponding Watson-Crick complements \(a^{\prime }, b^{\prime }\) on input nodes of the attaching ladder tile. Other models can use protein-protein interactions resulting from electric

potentials forming across ion channels9.) NETWORK POTENTIALS IN HTCTAM SYSTEMS Kirchoff’s Current Law (KCL) or conservation of charge states that the sum of the currents entering and exiting

a node in a circuit must be 0. In particular, the seed (Fig. 1) has a distribution of potentials at its three nodes (source \(\nu _0\), ground \(\nu _{-1}\), and node potential \(\nu _1\)

at the tip of the tile, between \(Z_1\) and the parallel \(Z_1^\prime\) pair.) Attachment of successive ladder tiles causes a (nearly instantaneous, speed of light) propagation of the

signals to the other tiles, which reconfigures the node potentials at the previous nodes into a new steady state after a brief transient. Over time, the self-assembly process in an htcTAM

model generates a family of circuits of increasing size (number of tiles) with a dynamic electric potential distribution \(\nu ^n_k\) at nodes _k_ in a ladder of size _n_ (\(k \in \{0,\ldots

,n\}\)), as illustrated in Fig. 3. Because of the series-parallel impedances of the circuits in the family, the tip potentials \(\nu ^n_n\) are a decreasing function of time or size, so

they eventually become unable to support new attachments, as was shown in the DC case in4,7. Thus an htcTAM really defines a growing family of circuits of increasing size exhibiting emerging

characteristic behavior. Questions of interest arise as to the nature of the distribution of potentials as the circuit grows in length, as well as the length of the largest ladder obtained

in the self-assembly process when a stable mature circuit has been achieved. These questions are addressed next for the general case of a circuit with a constant DC power source \(\nu _0\)

and impedances in the tiles in the remainder of this section, while other circuit types will be addressed in the following section. The solution of the family of resistive circuits in the

htcTAM model can be characterized in general using the equations for the node voltages \(\nu _k^n\) obtained from Kirchoff’s Current Law for a given circuit of size _n_. (We will drop the

superindex _n_ when the context is not ambiguous.) Given the circuit in Fig. 3, the circuit tiles and nodes are indexed by 0 (power source) through _n_ (tip voltage at the terminus), with

corresponding potential distribution \(\mathbf {\nu } = (\nu _0, \nu _1, \cdots , \nu _k, \cdots , \nu _n).\) For the first node, a typical intermediate node _k_ and the last node, where the

impedances on the tiles are \(Z_k, Z_k^{\prime }\), the node equations are, respectively $$\begin{aligned} \frac{\nu _1 - \nu _{0}}{Z_1} + \frac{\nu _1 - 0}{Z_1^{\prime }} + \frac{\nu _1 -

\nu _{2}}{Z_{2}} = 0, \nonumber \\ \frac{\nu _k - \nu _{k-1}}{Z_k} + \frac{\nu _k - 0}{Z_k^{\prime }} + \frac{\nu _k - \nu _{k+1}}{Z_{k+1}} = 0, \nonumber \\ \frac{\nu _n - \nu _{n-1}}{Z_n}

+ \frac{\nu _n - 0}{Z_n^{\prime }} = 0 \,. \end{aligned}$$ (1) Regrouping for the \(\nu _k\)’s, ordering the equations by node and multiplying each equation by the corresponding \(Z_k\)

yield $$\begin{aligned} (a_1 + r_1) \nu _1 - r_1 \nu _2 = \nu _0, \nonumber \\ - \nu _{k-1} + (a_k + r_k) \nu _k - r_k \nu _{k+1} = 0, \nonumber \\ - \nu _{n-1} + a_n \nu _{n} = 0,

\end{aligned}$$ (2) where \(a_k\) and \(r_k\) are given by \(a_k:= 1 + \frac{Z_k}{Z_k^{\prime }} > 1\) and \(r_k:= \frac{Z_k}{Z_{k+1}} > 0\) for \(k = 1, \cdots , n\) respectively. The

distribution of potentials \(\mathbf {\nu }^T = (\nu _1, \cdots , \nu _n)^T\), including the tip potential \(\nu _n\), is determined as the solution of the system of equations

$$\begin{aligned} \textbf{M}_n \mathbf {\nu } = \textbf{b}_n , \end{aligned}$$ (3) where the matrix of coefficients is $$\begin{aligned} \textbf{M}_n = \begin{bmatrix} a_1+r_1 & -r_1

& 0 & \cdots & 0 \\ -1 & a_2 +r_2 & -r_2 & 0 & 0 \\ 0 & -1 & a_3+r_3 & -r_3 & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0

& \cdots & -1 & a_{n-1} + r_{n-1} & -r_{n-1} \\ 0 & \cdots & \cdots & -1 & a_n \end{bmatrix}, \end{aligned}$$ (4) the _n_-D source/independent term vector is

\(\textbf{b}_n = \begin{bmatrix} \nu _0 \;\; 0 \;\; \cdots \;\; 0 \end{bmatrix}^T\), _T_ it the transpose operator and \(\nu _0\) is the source potential (DC or AC.) The matrix

\(\textbf{M}_n\) is a nonsingular tridiagonal matrix that is similar to a _symmetric_ matrix $$\begin{aligned} \textbf{J}_n = \textbf{D}_n^{-1} \textbf{M}_n \textbf{D}_n , \end{aligned}$$

where the change of basis \(\textbf{D}_n\) is given by diagonal matrix $$\begin{aligned} \textbf{D}_n = \text {diag}( \sqrt{r_1 r_2 \cdots r_{n-1}},\; \sqrt{r_2 \cdots r_{n-1}},\; \cdots ,\;

\sqrt{r_{n-1}},\; 1) \end{aligned}$$ and the symmetric matrix is obtained from \(\textbf{M}_n\) by replacing the \(-1\) and \(-r_k\)’s with the corresponding \(-\sqrt{r_k}\)’s in the two

nonzero minor diagonals, as can be easily verified (or, see https://en.wikipedia.org/wiki/Tridiagonal_matrix, under subheading “Similarity to symmetric tridiagonal matrix”.) Hence,

$$\begin{aligned} \textbf{M}_n = \textbf{P}_n \mathbf {\Lambda }_n \textbf{P}_n^{-1}, \end{aligned}$$ (5) where \(\mathbf {\Lambda }_n\) is a diagonal matrix of eigenvalues of

\(\textbf{M}_n\) and \(\textbf{P}_n\) is an orthogonal matrix whose columns are the corresponding orthogonal eigenvectors of \(\textbf{M}_n\). Therefore, the potentials \(\nu _k\) could be

calculated as linear combinations of these eigenvalues, i.e. $$\begin{aligned} \mathbf {\nu } = \textbf{P}_n \mathbf {\Lambda }^{-1}_n \textbf{P}_n^{-1} \textbf{b}_n . \end{aligned}$$ These

formulas provide a general closed-from solution of the htcTAM family of circuits circuit in the case of a constant DC power source for a resistive circuit. Nevertheless, the distribution of

potentials can be calculated somewhat more easily and elegantly in the space of \(2 \times 2\) matrices as follows. One can solve for \(\nu _k\)’s \((1< k < n)\) in eq. (2) (with the

index _k_ rather than \({k-1}\)) as $$\begin{aligned} \nu _1&= (a_2 + r_2) \nu _2 - r_2 \nu _3, \nonumber \\ \nu _{k}&= (a_{k+1} + r_{k+1})\nu _{k+1} - r_{k+1}\nu _{k+2}, \nonumber

\\ \nu _{n-1}&= a_n \nu _{n} \end{aligned}$$ (6) so that $$\begin{aligned} \textbf{N}_k:= \begin{bmatrix} \nu _k \\ \nu _{k+1} \end{bmatrix} = \begin{bmatrix} a_{k+1} + r_{k+1} &

-r_{k+1} \\ 1 & 0 \end{bmatrix} \; \begin{bmatrix} \nu _{k+1} \\ \nu _{k+2} \end{bmatrix} = \begin{bmatrix} a_{k+1} + r_{k+1} & -r_{k+1} \\ 1 & 0 \end{bmatrix} \;

\textbf{N}_{k+1} , \end{aligned}$$ (7) _i.e._, the pairs satisfy the recursive matrix identities \(\textbf{N}_k = \mathbf {\Psi }_{k+1} \textbf{N}_{k+1}\) for \(0 \le k \le n-2\), where

\(\mathbf {\Psi }_{k+1}\) is the \(2 \times 2\) matrix in eq. (7). This formula shows _how a pair of consecutive potential values in_ \(\textbf{N}_k\) _at the corners of a tile is affected

by the following pair of potentials down the ladder_ in terms of the parameters of the tile, or conversely, by multiplying by the inverse of \(\mathbf {\Psi }_{k+1}\), _by the following pair

of potentials up the ladder_ closer to the seed tile. Iterated substitution thus shows how to solve for the potential as a function of the last pair \(\textbf{N}_{n-1}\), or even of the

last potential since \(\nu _{n-1} = a_n \nu _n\) from eq. (6), _i.e._, for \(0 \le k \le n-1\), $$\begin{aligned} \textbf{N}_k = \left( \prod _{i=k}^{n-2} \mathbf {\Psi }_{i+1} \right)

\textbf{N}_{n-1} = \left( \prod _{i={k+1}}^{n-1} \mathbf {\Psi }_i \right) \begin{bmatrix} \nu _{n-1}\\ \nu _{n} \\ \end{bmatrix} = \; \left( \prod _{i={k+1}}^{n-1} \mathbf {\Psi }_i \right)

\begin{bmatrix} a_n \\ 1 \\ \end{bmatrix} \, \nu _n . \end{aligned}$$ (8) Note that \(\nu _n\) can be easily obtained using Cramer’s rule from the system of eqs. (3) by evaluating the

determinants \(|\mathbf {M_n}|\) and \(|\mathbf {M_n^{\prime }}|\), where the latter is obtained by replacing the last column of \(\mathbf {M_n}\) by the source/independent column vector,

\(\mathbf {b_n}\,\) and computing the determinant by the last column, _i.e._, $$\begin{aligned} \nu _n = (-1)^{n-1} \nu _0 / |\mathbf {M_n}| . \end{aligned}$$ (9) since, after deleting the

first row and last column, the minor determinant is \((-1)^{n-1}\) for the resulting upper triangular minor, an \((n-1) \times (n-1)\)-matrix with \(n-1 \; \; -1\)s in the main diagonal.

Now, multiplying on the left in eq. (8) by the inverses of the \(\Psi _i\)’s, $$\begin{aligned} \mathbf {\Psi }_i^{-1} = \frac{1}{r_i} \begin{bmatrix} 0 & r_i \\ -1 & a_i + r_i

\end{bmatrix}, \end{aligned}$$ (10) in the appropriate sequence (consecutive matrices \(\Psi _i, \Psi _{i+1}\) do _not_ commute unless \(a_{i}=a_{i+1}\) and \(r_{i}=r_{i+1}\)), yields the

opposite relations: $$\begin{aligned} \textbf{N}_{n-1} = \left( \prod _{i={n-1}}^{k+1} \mathbf {\Psi }_i^{-1} \right) \; \textbf{N}_{k} = \left( \prod _{i={n-1}}^{k+1} \mathbf {\Psi }_i^{-1}

\right) \; \begin{bmatrix} \nu _{k} \\ \nu _{k+1} \end{bmatrix}, \; 0 \le k \le n-2, \end{aligned}$$ (11) and so (with \(k=0\) and \(|\Psi _i|=r_i\)), $$\begin{aligned} \textbf{N}_{n-1} =

\left( \prod _{i=n-1}^{1} \mathbf {\Psi }_i^{-1} \right) \; \textbf{N}_{0} = \frac{1}{r_1 r_2 \cdots r_n} \left( \prod _{i={n-1}}^{1} \begin{bmatrix} 0 & r_i \\ -1 & a_i + r_i

\end{bmatrix} \right) \; \begin{bmatrix} \nu _0 \\ \nu _1 \\ \end{bmatrix}. \end{aligned}$$ (12) This is a second closed-form general solution of the family of heterogeneous htcTAM circuits.

Multiplying out the \(2 \times 2\) matrices produces multivariable polynomials in the \(a_k\)’s and \(r_k\)’s of degree _n_ to obtain the tip voltage \(\nu _n\), for example, from the

source voltages \(\nu _0\) and \(\nu _1\). Furthermore, a third closed-form solution can be obtained in terms of the pairs of real, positive and distinct eigenvalues \(\lambda _1^{(i)},

\lambda _2^{(i)}\) of the \(\mathbf {\Psi }_i^{-1}\)’s since their characteristic polynomials are like \(\lambda ^2 - (a+r) \lambda + r =0\), and so their eigenvalues are like

$$\begin{aligned} \lambda = (a + r) \pm \sqrt{(a + r)^2 - 4r} , (a+r) - 2\sqrt{r} = (a-1) + 1- 2\sqrt{r} +r = (a-1) + (\sqrt{r}-1)^2 > 0 \end{aligned}$$ for all \(r_k\)’s since \(a_k >

1\). Their eigenvectors are like \([1 \;\; \frac{\lambda -1}{\lambda -a}]^T\). Therefore, these matrices are diagonalizable and of the form \(\mathbf {\Psi }_i^{-1} = \textbf{P}_i \mathbf

{\Lambda }^{(i)} \textbf{P}_i^{-1}\), where \(\Lambda ^{(i)}\) is a \(2 \times 2\) diagonal matrix of eigenvalues of \(\mathbf {\Psi }_i\) and the columns of \(\textbf{P}_i\)’s are their

corresponding eigenvectors. Thus, $$\begin{aligned} \begin{bmatrix} \nu _{n-1} \\ \nu _n \\ \end{bmatrix} = \textbf{N}_{n-1} = \frac{1}{r_1 r_2 \cdots r_n} \; \textbf{P}_1 \mathbf {\Lambda

}^{(1)} \textbf{P}_1^{-1} \cdots \textbf{P}_{n-1} \mathbf {\Lambda }^{(n-1)} \textbf{P}_{n-1}^{-1} \; \begin{bmatrix} \nu _0 \\ \nu _1 \\ \end{bmatrix}. \end{aligned}$$ This solution affords

a geometric interpretation of the effect of each tile on the transfer function of the growing circuit ladders in the htcTAM. The effect of each tile on the 2-D vector of consecutive

potential drops \(\textbf{N}_k\) amount to a geometric transformation, either a _reflection_ or a _rotation_ followed by a _dilation/contraction_ by a factor equal to the corresponding

eigenvalue along the eigendirections of the \(2\ \times 2\) matrix representing the tile. These last two approaches require, in addition to the calculation of the eigenvalues and

eigenvectors of the \(2 \times 2\) matrices, the computation of the two potentials \(\nu _n\) and \(\nu _1\), which can be computed using Cramer’s rule, as mentioned above, and so require

calculating \(\Delta _1:= |\mathbf {M_n}|\) as well. For a fixed _n_, the lower major subdeterminants \(\Delta _{k}\) obtained by dropping the first \(k-1\) rows and columns in \(\mathbf

{M_n}\) (e.g., \(\Delta _{n}=a_n\) and \(\Delta _{n-1}=a_n(a_{n-1}+r_{n-1} -r_{n-1})\)) can also be calculated by a method similar to that for the \(\nu _k\)’s above since they satisfy

analogous recurrence relations given by $$\begin{aligned} \Delta _{k}&= (a_{k} + r_{k})\Delta _{k+1} - r_{k}\Delta _{k+2}, \;\;\; \text{ for } \;\;\; 1 \le k \le n - 2 \,,

\end{aligned}$$ (13) whence $$\begin{aligned} \begin{bmatrix} \Delta _{1} \\ \Delta _{2} \\ \end{bmatrix} = \left( \prod _{i=1}^{n-2} \; \begin{bmatrix} a_{i} + r_{i} & -r_{i} \\ 1 &

0 \end{bmatrix} \right) \; \begin{bmatrix} \Delta _{n-1} \\ \Delta _n\\ \end{bmatrix}. \end{aligned}$$ (14) Again, multiplying out these matrices will produce a closed-form solution for the

remaining \(\Delta _k\)’s for \(k < n-1,\) including \(\Delta _1=|\textbf{M}_n|\), and hence (using Cramer’s rule with the full matrix \(\textbf{M}_n\)) for $$\begin{aligned} \nu _1 =

\Delta _2 \nu _0 / \Delta _1 \;\; {\textrm{and}} \;\; \nu _n = (-1)^{n-1} \nu _0 / \Delta _1 . \end{aligned}$$ The remaining \(\nu _k\)’s can then be obtained from similar products for the

initial pair \([\nu _0 \;\; \nu _1]^T\) from eq. (12). SOLUTION FOR HETEROGENEOUS RESISTIVE LADDER CIRCUITS To test the correctness of the solution, the method described above was applied to

a resistive ladder with period 2. (Similar solutions can be found for an arbitrary period _m_.) In this case, only two matrices \(\mathbf {\Psi }_1\) and \(\mathbf {\Psi }_2\) are needed in

eq. (8), multiplied in alternating fashion so they can be simplified to $$\begin{aligned} \mathbf {\Psi }_1:= \begin{bmatrix} a_1 + r_1 & -r_1 \\ 1 & 0 \end{bmatrix} ,

\end{aligned}$$ (15) and $$\begin{aligned} \mathbf {\Psi }_2:= \begin{bmatrix} a_2 + r_2 & -r_2 \\ 1 & 0 \end{bmatrix} , \end{aligned}$$ (16) where \(a_1 = 1+\frac{Z_1}{Z_1^{\prime

}}\), \(r_1 = \frac{Z_1}{Z_2}\), \(a_2 = 1+\frac{Z_2}{Z_1^{\prime }}\), and \(r_2 = \frac{Z_2}{Z_1}\) since \(Z_3 = Z_1\) and \(Z_3^{\prime } = Z_1^{\prime }\). For \(Z_1^{\prime } = 10\

m\Omega\), \(Z_1 = 10\ \mu \Omega\), \(Z_2^{\prime } = 1\ \Omega\), and \(Z_2 = 2\ \Omega\), the electric potential at the tip of the ladder \(\nu _n\) is plotted for different lengths _n_

in Fig. 4, and the electric potentials at the intermediate nodes \(\nu _k\) for different lengths _n_ in Fig. 5. As expected from the results in the previous section, the potential at the

terminal node \(\nu _n^n\) goes monotonically down and growth cannot continue after a circuit size _N_, which is consistent with the bound in eq. (21) below. SOLUTIONS FOR RLC LADDER

CIRCUITS To demonstrate the solution to the family of circuits in the case of a homogeneous alternating power source \(\nu _0\) of frequency \(\omega\), some homogeneous RLC

(Resistor-Inductor-Capacitor) and periodic circuits will be solved and characterized. BOUNDS FOR HOMOGENEOUS RLC CIRCUIT SELF-ASSEMBLY GROWTH In Fig. 3, for a homogeneous RLC circuit, the

circuit parameters are \(Z^{\prime } = \frac{1}{s L}\) and \(Z = R + \frac{1}{s C}\), where \(s = \sigma + j \omega\) is the complex frequency. Thus, \(a_k = 2 + \frac{R+\frac{1}{sC}}{s^2 L

C}\) and \(r_k = 1\). Using eqs. (7) and (8), the solution is given by the powers of the \(2 \times 2\) matrix \(\mathbf {\Psi }\) for arbitrary _n_, namely, $$\begin{aligned} \mathbf {\Psi

}:= \begin{bmatrix} 2 + \frac{R + \frac{1}{sC}}{s^2 L C} & -1 \\ 1 & 0 \end{bmatrix} , \end{aligned}$$ (17) The natural frequency for a single tile is \(\omega _0 = 1/\sqrt{L C}\),

which is \(1 \times 10^5\) (rad/s) for \(L=10\ \mu H\) and \(C=10\ \mu F\). In a RLC circuit, the value of the resistance _R_, representing energy dissipation, affects the oscillatory

response of the system, and can be overdamped, underdamped, or critically damped10. In an overdamped circuit, oscillations at \(\omega _0\) are suppressed. In an underdamped circuit,

oscillations at \(\omega _0\) are present when a driving force is applied, but slowly decrease. A critically damped circuit is the boundary between overdamped and underdampled, and the

steady-state response is reached as quickly as possible. Even though steady-state behavior is the focus here, the RLC circuits are analyzed for each of these cases. For the homogeneous RLC,

\(|V_0 |=10\ \textrm{V}\), \(L=10\ \mu \textrm{H}\), \(C=10\ \mu \textrm{F}\), and \(R= 6\ \Omega\) for overdamped, \(R= 10\ \textrm{m}\Omega\) for underdamped, and \(R= 2\ \Omega\) for

critically damped. The absolute value of \(\nu _n\) is plotted versus frequency in Fig. 6. The poles of the response are evident for the underdamped case. Next, \(\nu _n\) is plotted versus

_n_ (the size of the ladder) at \(\omega _0\) (Fig. 7). All cases show that \(\nu _n\) decreases with length, though in the underdamped case, the decrease is small. The underdamped case is

plotted versus time at steady state showing that the peak \(v_n\) does decrease with time (Fig. 8.) Next, an alternating RLC circuit of period two is analyzed. In Fig. 3 for the alternating

RLC circuit, \(Z^{\prime }_1 =Z^{\prime }_2 = \frac{1}{s L}\), \(Z_1 = R_1 + \frac{1}{s C_1}\), and \(Z_2 = R_2 + \frac{1}{s C_2}\), and thus, \(a_1 = 1 + \frac{R_1+\frac{1}{sC_1}}{s L}+

r_1\) with \(r_1 = \frac{R_1+\frac{1}{s C_1}}{R_2+\frac{1}{s C_2}}\), and \(a_2 = 1 + \frac{R_2+\frac{1}{sC_2}}{s L} + r_2\) with \(r_2 = \frac{R_2+\frac{1}{s C_2}}{R_1+\frac{1}{s C_1}}\).

There are two \(2 \times 2\) matrices from which to build the solution for arbitrary _n_ from eqs. (7) and (8), namely, $$\begin{aligned} \varvec{\Psi }:= \begin{bmatrix} 1 +

\frac{R_1+\frac{1}{sC_1}}{s L}+ r_1 & -1 \\ 1 & 0 \end{bmatrix} , \end{aligned}$$ (18) and $$\begin{aligned} \varvec{\Psi }:= \begin{bmatrix} 1 + \frac{R_2+\frac{1}{sC_2}}{s L}+ r_2

& -1 \\ 1 & 0 \end{bmatrix} , \end{aligned}$$ (19) The natural frequency for a single case of the alternating tiles is \(\omega _0 = 1/\sqrt{L (C_1 + C_2)}\), which is approximately

\(1 \times 10^6\) (rad/s) for the cases studied. As for the homogeneous case, the absolute value of \(\nu _n\) is plotted versus frequency in Fig. 9 for parameters that make the response

overdamped, underdamped, and critically damped for each of the two tiles, respectively. The poles of the response are evident for the underdamped case. Next, \(\nu _n\) is plotted versus _n_

(the size of the ladder) at \(\omega _0\) (Fig. 10.) All cases show that \(\nu _n\) decreases with length, though in the underdamped case, the decrease is small. The underdamped case is

plotted versus time at steady state showing that the peak \(v_n\) does decrease with time (Fig. 11.) BOUNDS FOR HETEROGENEOUS RLC CIRCUIT SELF-ASSEMBLY GROWTH Using the expressions derived

in previous sections, bounds on the length of the largest self-assembled ladder can be derived. In the htcTAM model, \(\nu _0\) represents the source of energy for the growth or the strength

of the signal from a sensor, and the threshold \(\tau\) represents those randomizing forces in the environment that oppose either growth or signal propagation. Thus, the length of the

ladder is a measure of both. To determine bounds on the length _N_ of the longest ladder obtained by the self-assembly processs, the terminal voltage at the tip in eq.  (9) has to be less

than the threshold, _i.e._, $$\begin{aligned} \nu _n = \frac{(-1)^{n-1} \nu _0}{|\textbf{M}_n|} < \tau . \end{aligned}$$ (20) Since \(\textbf{M}_n\) is diagonalizable and symmetric with

_n_ distinct real eigenvalues \(\lambda _k\), all of which are positive and distinct, its determinant is given by the product \(\Delta _n = \prod _{k=1}^n \lambda _k > \lambda _{\min

}^n\), where \(\lambda _{\min }\) is the smallest eigenvalue. Replacing and solving for the inequality in eq.  (20) shows that when _n_ exceeds \(\nu _0 /(\tau \, \lambda _{\min }^n)\), no

further tiles will attach and, therefore, the maximum size _N_ of a given ladder circuit under the self-assembly htcTAM model is bounded by $$\begin{aligned} N \le \log (\nu _0/\tau ) / \log

(\lambda _{\min }) . \end{aligned}$$ (21) DISCUSSION AND CONCLUSIONS Ladder circuits are important both as an abstract model for various phenomena, _e.g._ axons, biological circuit models,

transimission lines, filters, topological insulators, and in a number of other applications. For example, there is theoretical interest and results in resistive ladders11,12,13,14. Various

configurations of RLC ladders are models for passive filters15 and transmission lines16. Recently, LC ladders have been demonstarted as models for topological insulators17 and even, quantum

computation18,19. Most of these results either analyze a single circuit or assume infinite length circuits. None of them consider a dynamic familiy of growing circuits, let alone as products

of a self-assembly process. While there is some similarity to how two-port networks are used to build solutions to transmission lines with matrix multiplication, the matrices\(\mathbf {\Psi

}\) directly implement the recurrence relation for the determinant of the matrices in Cramer’s rule, specifically \(|\mathbf {M_n}|\). Instead of current and voltages at the terminals of

the two-port network, the \(2\times 2\) matrices here directly produce a subsequent node potential from a previous one. In this paper, new methods to analyze growing families of finite

heterogeneous ladder circuits in the htcTAM self-assembly model5 have been presented, including a geometric interpretation (as rotations and dilations in the 2-D Euclidean plane) of the

effect of the impedances on successive tiles, as well as a quantitatively precise interpretation of the influence of each successive tile in the transfer function of the ladder from the

source and each tile to the next. We have also established an upper bound on the growth of such circuits even in the case of RLC circuits. Moreover, the results apply to the family of

circuits indexed by length and are capable of producing results for circuit composed of arbitrary components in different topologies, _i.e_ series and parallel square configurations obtained

by using circuit tiles that “turn corners” without essentially remaining equivalent (in terms of the electrical potential distribution ) to a ladder circuit with appropriate changes in the

impedances of the tiles. This might be an important tool to apply to ladders manufactured at the micro- or nano-scale through self-assembly methods. For example, self-assembly would likely

produce ladders of different lengths with different electrical properties. In addition, these models could be extended to mechanical or fluid properties given the standard mappings of

electric circuits to those systems20,21. Thus, the results of this paper provide a technique that might be useful to characterize properties of a wide range of important material systems.

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  Download references AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Computer Science, The University of Memphis, Memphis, TN, USA Max Garzon * Department of Electrical and

Computer Engineering, The University of Memphis, Memphis, TN, USA Russell Deaton Authors * Max Garzon View author publications You can also search for this author inPubMed Google Scholar *

Russell Deaton View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS MG conceived the idea and developed a new theoretical methodology to resolve

the family of circuits. RD confirmed the correctness of the solutions experimentally on RLC circuits as reported herein. Both authors jointly evaluated the results, prepared the manuscript

and agreed to this submission. CORRESPONDING AUTHOR Correspondence to Max Garzon. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION

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