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¡center¿ ¡h1¿EPTCS 375¡/h1¿ ¡h1¿¡font size=5¿Proceedings 11th International
Workshop on¡/font¿ ¡br¿Theorem Proving Components for Educational Software
¡br¿¡font size=3¿Haifa, Israel, 11 August 2022¡/font¿¡/h1¿ Edited by: Pedro
Quaresma, João Marcos and Walther Neuper ¡/center¿ ¡hr class=sep¿
¡table width=100¡tr¿¡td class=fill¿¡i¿¡a href=”#EPTCS375.0”¿
Preface¡/a¿¡/i¿¡/br¿ ¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡b¿Invited Talk:¡/b¿ ¡a
href=”#EPTCS375.1”¿ Satisfiability Modulo Theories in an Undergraduate
Class¡/a¿¡/i¿¡/br¿ Yoni Zohar¡/td¿¡td align=”right”¿1¡/td¿¡/tr¿ ¡tr¿¡td
class=fill¿¡i¿¡a href=”https://doi.org/10.4204/EPTCS.375.2”¿ Computer Assisted
Proofs and Automated Methods in Mathematics Education¡/a¿¡/i¿¡/br¿ Thierry
Noah Dana-Picard¡/td¿¡td align=”right”¿2¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”https://doi.org/10.4204/EPTCS.375.3”¿ A Rule Based Theorem Prover: an
Introduction to Proofs in Secondary Schools¡/a¿¡/i¿¡/br¿ Joana Teles, Vanda
Santos and Pedro Quaresma¡/td¿¡td align=”right”¿24¡/td¿¡/tr¿ ¡tr¿¡td
class=fill¿¡i¿¡a href=”https://doi.org/10.4204/EPTCS.375.4”¿ ANITA: Analytic
Tableau Proof Assistant¡/a¿¡/i¿¡/br¿ Davi Romero Vasconcelos¡/td¿¡td
align=”right”¿38¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”https://doi.org/10.4204/EPTCS.375.5”¿ A Proof Tree Builder for Sequent
Calculus and Hoare Logic¡/a¿¡/i¿¡/br¿ Joomy Korkut¡/td¿¡td
align=”right”¿54¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”https://doi.org/10.4204/EPTCS.375.6”¿ On Exams with the Isabelle Proof
Assistant¡/a¿¡/i¿¡/br¿ Frederik Krogsdal Jacobsen and J#x00f8;rgen
Villadsen¡/td¿¡td align=”right”¿63¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”https://doi.org/10.4204/EPTCS.375.7”¿ Automated Grading of Automata with
ACL2s¡/a¿¡/i¿¡/br¿ Ankit Kumar, Andrew Walter and Panagiotis Manolios¡/td¿¡td
align=”right”¿77¡/td¿¡/tr¿ ¡tr¿¡td class=fill¿¡i¿¡a
href=”https://doi.org/10.4204/EPTCS.375.8”¿ Towards an Accessible Mathematics
Working Environment Based on Isabelle/VSCode¡/a¿¡/i¿¡/br¿ Klaus Miesenberger,
Walther Neuper, Bernhard St#x00f6;ger and Makarius Wenzel¡/td¿¡td
align=”right”¿92¡/td¿¡/tr¿ ¡/table¿ ¡hr class=sep¿¡h1 align=left¿¡a
name=’EPTCS375.0’¿Preface¡/a¿¡/h1¿

¡p¿The ThEdu series pursues the smooth transition from an intuitive way of
doing mathematics at secondary school to a more formal approach to the subject
in STEM education, while favouring software support for this transition by
exploiting the power of theorem-proving technologies. What follows is a brief
description of how the present volume contributes to this enterprise.¡/p¿

¡p¿The 11th International Workshop on Theorem Proving Components for
Educational Software (ThEdu’22), was a satellite event of the 8th Federated
Logic Conference (FLoC 2022), July 31-August 12, 2022, Haifa, Israel. ThEdu’22
was a vibrant workshop, with two invited talks, by Thierry Dana-Picard
(Jerusalem College of Technology, Jerusalem, Israel) and by Yoni Zohar (Bar
Ilan University, Tel Aviv, Israel), and four regular contributions. An open
call for papers was then issued, and it attracted seven submissions. Those
submissions have been accepted by our reviewers, who jointly produced at least
three careful reports on each of the contributions. The resulting revised
papers are collected in the present volume.¡/p¿

¡p¿ The contributions in this volume are a faithful representation of the wide
spectrum of ThEdu, ranging from those more focused on the automated deduction
research, not losing track of the possible applications in an educational
setting, to those focused on the applications, in educational settings, of
automated deduction tools and methods.¡/p¿

¡p¿ Yoni Zohar reported on his experience in teaching an automated reasoning
class focused on SMT, in his invited talk, the abstract of which is here
contained in HTML format only. Thierry Dana-Picard, in a comprehensive article
covering the area of Computer Algebra Systems (CAS) and Dynamic Geometry
Systems (DGS) and their connections with automated reasoning, presents tools
and methods for high-school mathematics education, in particular geometry,
discussing the transformation of Mathematics Education into an exploration-
discovery-conjecture-proof scheme, avoiding usage as a black box. At the same
school level, Joana Teles, Vanda Santos and Pedro Quaresma discuss the
possible use of the deductive database method for geometry in a 7th year class
(asymp;12 year-old students), with a specifically designed set of rules, as a
way to motivate students for deductive reasoning and to exhibit the beauty of
mathematical proof. The teaching for formal reasoning at the university level
is addressed by several authors presenting several systems with different
goals. Davi Vasconcelos presents the system ¡em¿ANITA¡/em¿ (Analytic Tableau
Proof Assistant), developed for teaching analytic tableaux to computer science
students. The tool is written in Python and may be used either as a desktop
application or as a Web application. Joomy Korkut presents a Web-based
pedagogical proof assistant, the ¡em¿Proof Tree Builder¡/em¿ that may be
applied to sequent calculus and Hoare logic proofs. With an intuitive
interface, the system can be used to lessen the burden of manual production of
proof trees. Frederik Krogsdal Jacobsen and Joslash;rgen Villadsen present an
approach for testing student learning outcomes in a course on automated
reasoning using the Isabelle proof assistant. The approach allows teachers to
test both general understanding of formal proofs in various logical proof
systems and understanding of proofs, in particular, in the higher-order logic
of Isabelle/HOL. Ankit Kumar, Andrew Walter and Panagiotis Manolios present
the ¡em¿Automatic Automata Checker¡/em¿ (A2C), an open-source library that
enables one to construct executable automata using definitions that mimic
those found in standard textbooks. Instructors can conveniently specify
solutions in the form of their own constructions with the system generating
actionable feedback, which helps students better understand the material.
Finally, Klaus Miesenberger, Walther Neuper, Bernhard Stouml;ger and Makarius
Wenzel describe plans for a project, made in close collaboration between
experts in accessibility of studies for visually impaired individuals with
experts in developing educational mathematics software and experts in
designing and implementing interactive proof assistants. This way they come
back to one of the initial goals of ThEdu, namely to support early mathematics
education with formal mathematics software. ¡/p¿

¡p¿ For their help in reviewing the papers, the editors would like to thank
the other members of the Programme Committee of ThEdu’22: Francisco Botana
(University of Vigo at Pontevedra, Spain); David Cerna (Czech Academy of
Sciences Institute of Computer Science, Czech Republic); Filip Maricacute;
(University of Belgrade, Serbia); Adolfo Neto (UTFPR, Brazil); Giselle Reis
(Carnegie Mellon University Qatar, Qatar); Vanda Santos (University of Aveiro,
Portugal); Anders Schlichtkrull, Aalborg University, Denmark; Wolfgang
Schreiner (Research Institute for Symbolic Computation (RISC), Austria); M.
Pilar Veacute;lez (Nebrija University, Spain) and Joslash;rgen Villadsen
(Technical University of Denmark, Denmark). ¡/p¿

¡p¿ We, the volume editors, hope that this collection of papers will further
promote the development of theorem-proving based software, and that it will
allow to improve the mutual understanding between computer scientists,
mathematicians and stakeholders in education. ¡/p¿

¡p¿ While this volume goes to press, the next edition of the ThEdu workshop is
being prepared: ThEdu’23 will be a satellite event of the 29th international
Conference on Automated Deduction (CADE 2023), July 1-4, 2023, Rome, Italy.
¡/p¿

¡p¿ PC Chairs: ¡br/¿ Joatilde;o Marcos (UFRN, Federal University of Rio Grande
do Norte, Brazil)¡br/¿ Walther Neuper (JKU, Johannes Kepler University, Linz,
Austria)¡br/¿ Pedro Quaresma (University of Coimbra, Portugal)¡br/¿ ¡/p¿ ¡hr
class=sep¿¡h1 align=left¿¡a name=’EPTCS375.1’¿Satisfiability Modulo Theories
in an Undergraduate Class¡/a¿¡/h1¿

¡b¿Yoni Zohar¡/b¿ (¡i¿Bar Ilan University, Tel Aviv, Israel¡/i¿) ¡p¿
¡p¿Satisfiability Modulo Theories (SMT) solvers are widely used in various
applications, most prominently verification of hardware and software. They are
focused on the task of checking the satisfiability of first-order formulas
over a wide range of background theories. Currently, mainstream SMT-solvers
offer easy to use APIs and textual interfaces, and are accessible even to
users with little logical background. Nevertheless, their algorithms are based
on interesting and varied theoretical subjects.¡/p¿

¡p¿For these reasons, SMT as a field is a good candidate for introducing
automated reasoning and logic to students with little or no background in
logic: its wide applicability provides a great motivation to study them, the
solvers’ accessibility offers hands-on exercises and the logical foundations
contain a wide variety of topics of different levels and depth.¡/p¿

¡p¿In this talk I will describe my experience in teaching an automated
reasoning class focused on SMT. I will describe my original hypotheses and
plans before the class, what happened during the class, and some reflections
that occurred in the process.¡/p¿

¡hr class=sep¿ ¡/body¿¡/html¿


# Epithelial Tissue Growth Dynamics: Universal or Not?

Mahmood Mazarei  Department of Mathematics, The University of Western Ontario,
1151 Richmond Street, London, Ontario, Canada N6A 5B7  The Centre for Advanced
Materials and Biomaterials Research, The University of Western Ontario, 1151
Richmond Street, London, Ontario, Canada N6A 3K7  Jan Åström  CSC Scientific
Computing Ltd, Kägelstranden 14, 02150 Esbo, Finland  Jan Westerholm  Faculty
of Science and Engineering, Åbo Akademi University, Vattenborgsvägen 3,
FI-20500, Åbo, Finland  Mikko Karttunen  The Centre for Advanced Materials and
Biomaterials Research, The University of Western Ontario, 1151 Richmond
Street, London, Ontario, Canada N6A 3K7  Department of Chemistry, The
University of Western Ontario, 1151 Richmond Street, London, Ontario, Canada
N6A 5B7  Department of Physics and Astronomy, The University of Western
Ontario, 1151 Richmond Street, London, Ontario, Canada N6A 3K7

###### Abstract

Universality of interfacial roughness in growing epithelial tissue has
remained a controversial issue. Kardar-Parisi-Zhang (KPZ) and Molecular Beam
Epitaxy (MBE) universality classes have been reported among other behaviors
including total lack of universality. Here, we utilize a kinetic division
model for deformable cells to investigate cell-colony scaling. With seemingly
minor model changes, it can reproduce both KPZ- and MBE-like scaling in
configurations that mimic the respective experiments. This result neutralizes
the apparent scaling controversy. It can be speculated that this diversity in
growth behavior is beneficial for efficient evolution and versatile growth
dynamics.

Growth of biological matter, e.g., tumor invasion, depends on complex
processes such as the mechanism(s) of proliferation, the physical properties
of the microenvironment, and cellular migration that can be dominated either
by single cell or collective motion that depend on intercellular interactions
and intracellular regulation [1].

Characterization of interfacial growth of cellular colonies is important for
understanding the factors that control growth and how they manifest themselves
in the kinetics and morphology of cell aggregation. Numerous experimental and
computational studies have investigated the effects of biochemical regulation
and mechanical factors such as cell-to-cell adhesion and friction, and cell
division [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Typically, growth is characterized
by scaling analysis which identifies the underlying mechanisms of growth
dynamics by critical exponents.

Self-affinity of the interface width, $w(l,t)$, obeys the Family-Vicsek
scaling relation [12, 13]

$w(l,t)\sim t^{\beta}F(lt^{-\frac{1}{z}}),$ (1)

where the exponent $z$ describes the scaling relation between the critical
time and length scales, and can be obtained from the scaling relation
$z=\frac{\alpha}{\beta}$, where the exponent $\alpha$ characterizes the
roughness of the interface. The exponent $\beta$ is obtained through the
scaling function $F(u)=F(lt^{-\frac{1}{z}})$ which has the following
properties: There is a crossover at $u=l_{\mathrm{*}}$. For $u\ll
l_{\mathrm{*}}$ the scaling function increases as a power law,
$F(u)=u^{\beta}$, where $\beta$ is the growth exponent and characterizes the
time-dependent dynamics surface roughening. For $u\gg l_{\mathrm{*}}$ the
width saturates, and $F(u)$ becomes a constant [14]. With these three critical
exponents, interfacial growth is often classified into different dynamic
universality classes.

The KPZ equation was the first nonlinear continuum equation used to study
surface growth. It is described by the stochastic differential equation [15]

$\partial_{t}h(x,t)=-\lambda\left[\partial_{x}h(x,t)\right]^{2}+\nu\partial_{x}^{2}h(x,t)+\xi(x,t),$
(2)

where the height ($h(x,t)$) depends on position and time, and $\lambda,\nu$
and $D$ are physical constants. The first term on the RHS reflects growth that
occurs locally normal to the interface and renders the KPZ equation nonlinear.
The second term smooths the interface by surface tension $\nu$, and the last
term, $\xi(x,t)$, is Gaussian noise given by $\langle\xi(x,t)\rangle=0$ and
$\langle\xi(s,x)\xi(t,y)\rangle\\!=\\!2D\delta(s-t)\delta(x-y)$. The KPZ
universality class is characterized by the exponents
$\alpha^{\mathrm{KPZ}}\\!=\\!\frac{1}{2}$,
$\beta^{\mathrm{KPZ}}\\!=\\!\frac{1}{3}$, and
$z^{\mathrm{KPZ}}\\!=\\!\frac{3}{2}$ [15].

Mathematically, surface tension and lateral growth determine the asymptotic
scaling of the KPZ equation. In some growth processes, however, surface
diffusion controls the scaling behavior, and the growth process is described
by the MBE model [16, 17]

$\partial_{t}h(x,t)=-K\partial_{x}^{4}h(x,t)+F+\xi(x,t),$ (3)

where $K$ is the surface diffusion coefficient, $F$ is the growth rate, and
$\xi(x,t)$ is Gaussian white noise as in the KPZ equation. The roughness
exponents for the MBE universality class for a one dimensional interface are
$\alpha^{\mathrm{MBE}}\\!=\\!\frac{3}{2}$,
$\beta^{\mathrm{MBE}}\\!=\\!\frac{3}{8}$, and $z^{\mathrm{MBE}}\\!=\\!4.0$.

Brú et al. studied cellular growth using cells from 15 different in vitro cell
lines and 16 in vivo types of tumor cells obtained from patients [7, 8]. They
determined the growth to belong to the MBE universality class in all cases
with exponents $\alpha\\!=\\!1.5\\!\pm\\!0.15$,
$\beta\\!=\\!0.38\\!\pm\\!0.07$, and $z\\!=\\!4.0\\!\pm\\!0.5$ thus suggesting
universal growth dynamics for cells. This conclusion was strongly criticized
by Buceta and Galeano [18] who dismissed universality of tumor growth dynamics
stating serious flaws in Brú et al.’s scaling analysis. In their rebuttal Brú
et al. [9] restated their conclusions and wrote ”the characteristics of MBE
dynamics discussed in Brú et al. (2003,1998) have not only been rigorously
demonstrated but have served as the basis for a successful antitumor therapy
currently under development.” A recent study of growth of different brain
tumors in vivo using fractal and scaling analysis shows similarities with some
of the results of Brú et al. [19].

In contrast to MBE-like dynamics, Huergo et al. [10, 11, 20] reported KPZ
scaling for both linearly and radially spreading interfaces of HeLa (cervix
cancer) and Vero cell colonies. Galeano et al. studied the development of
plant cell species Brassica oleracea and B. rapa under various growing
conditions and obtained $\alpha\\!=\\!0.86\pm 0.4$, and $z\\!=\\!5.0$ [21].
Santalla et al. [22] grew colonies of Bacillus subtilis and Escherichia coli
using a high agar concentration regime with various nutrients and discovered
branching interfaces with exponents $\beta\\!=\\!0.5$ and $\alpha\\!=\\!0.75$
that are inconsistent with both MBE and KPZ.

Substrate disorder can also influence growth dynamics. Vicsek et al. [23]
studied the growth of E. coli and B. subtilis colonies and found the roughness
exponent $\alpha\\!=\\!0.78\pm 0.07$ which is inconsistent with both the KPZ
and MBE models as well as with the quenched KPZ (qKPZ) model [14] that
includes disorder. It has also been demonstrated that the behavior of
bacterial colonies in the medium-to-high nutrient concentration regime can be
very rich due to the appearance of quenched disorder in growth patterns [24].
In that context, Huergo et al. also examined the 2D growth dynamics of
quasilinear Vero cell colony fronts in a methylcellulose-containing culture
medium. Their scaling analysis yielded $\alpha\\!=\\!0.63\pm 0.04$,
$\beta\\!=\\!0.75\pm 0.05$, and $z\\!=\\!0.84\pm 0.05$, suggesting qKPZ
dynamics [25].

On the computational and theoretical side, Santalla and Ferreira [26] used an
off-lattice Eden model modified to account for nutrient diffusion. Under
scarce nutrient supply, they observed initially a KPZ regime that transitioned
via a qKPZ transient to unstable growth. Block et al. studied the growth of 2D
cellular monolayers for a class of cellular automaton models. Their results
suggest KPZ dynamics over a wide range of parameters and different cell
migration dynamics [27] contradicting the MBE dynamics reported by Brú et al.
[7]. Another contradiction was reported by Azimzade et al. who developed a
tumor growth model based on the nonlinear Fisher-Kolmogorov-Petrovsky-Piskunov
equation, a reaction-diffusion equation, to investigate the impact of the
cellular environment and spatial correlations on tumor invasion [28]. They
concluded that kinetic growth models, such as KPZ, cannot characterise tumor
invasion fronts, and that the structure of the tumor interface depends
intimately on the initial conditions [28].

A large number of different models has been used to describe cellular growth
[29]. Here, we use the CellSim3D off-lattice growth model and simulator to
study epithelial tissue growth [30, 31]. In this model, cells can migrate,
deform, divide, and interact with each other and their environment
mechanically via adhesion and friction. Its 2D version has been shown to
produce, e.g., cell-cell force distributions, force dipoles, spontaneous
orientation of cells in the direction of highest stiffness and cellular
migration in agreement with experiments [32]. CellSim3D software leverages
graphics processor units, enabling simulations of systems $>\\!\\!100,000$
cells easily. The analysis below uses averages over 10 independent
simulations. Details, parameters and animations demonstrating the model are
provided as Supplemental Material. In brief, in CellSim3D, epithelial tissues
can be modeled as quasi-2D systems of 3D cells confined into a plane,
corresponding to the experimental confinement of cells between two plates; the
bottom plate models basal tissue and the top plate prevents excessive
buckling.

Figure 1: Interface evolutions of cell colonies starting from a horizontal
line and a radially expanding interface. (a) Line configuration at weak ($10$)
and (b) strong cell-cell adhesion ($2000$). (c) Radially expanding interface
with at weak ($10$) and (d) strong adhesion ($2000$). All interfaces have
overhangs. Scaling analysis was done using overhang-corrected interfaces [14].
For the units, see Table S1.

The interface width is defined as the standard deviation of the height over a
length scale $l$ at time $t$ as [14]

$w(l,t)=\bigg{\\{}\frac{1}{N}\sum_{i=1}^{N}[h_{i}(t)-\langle
h_{i}\rangle_{l}]^{2}\bigg{\\}}_{L}^{\frac{1}{2}},$ (4)

where $L$ is the contour length, which increases with time as $L=2\pi\langle
h(t)\rangle$ for radially expanding fronts and is constant for linear fronts.
For radially expanding fronts, $h_{\mathrm{i}}(t)$ is the distance from the
center of mass to the point $i$ of the interface at time $t$, $\langle
h_{\mathrm{i}}\rangle_{l}$ is the local average of the subsets of arc length
$l$, and $\\{\cdot\\}_{L}$ is the overall average. We complement the
universality class analysis by an examination of the structure factor,

$S(k,t)=\langle\hat{h}(k,t)\hat{h}(-k,t)\rangle,$ (5)

where $k$ is the wavenumber, and $\hat{h}(k,t)$ is the Fourier transform of
the interface profile $h(x,t)$ [14]. The advantage of this method over the
real space is that only long-wavelength modes contribute to its scaling.
Hence, it is less affected by finite-size effects. This method provides the
global roughness exponent $\alpha$ and the dynamic exponent $z$ via the
Family-Vicsek scaling form for $S(k,t)$,

$S(k,t)=k^{-2\alpha+1}s(kt^{\frac{1}{z}}),\mathrm{\,\,where}$ (6)
$s(u=kt^{\frac{1}{z}})=\left\\{\begin{array}[]{ll}\mathrm{const}&\mbox{for
$u\gg 1$};\\\ u^{-2\alpha+1}&\mbox{for $u\ll 1$}.\end{array}\right.$ (7)

At $u\\!=\\!1$ there is a crossover, for $u\\!\gg\\!1$ the curves measured at
different times collapse and for $u\\!\ll\\!1$ they split.

Figure 2:  (a) Velocity ($\langle v\rangle$) of the interface determined from
the mean colony radius $\langle R\rangle$ and the mean interface height
$\langle h\rangle$ vs. time for different cell-cell adhesion strengths.
Radially expanding interface: (purple circles) at weak ($10$) and (blue
circles) strong adhesion ($2,000$). Line configuration: (green triangles) at
weak and (red triangles) at strong adhesion. (b) The fractal dimension
($d_{\mathrm{f}}$) determined by plotting box counts vs. box size. For
radially expanding interface: (purple circles) at weak ($10$) and (blue
circles) strong adhesion ($2,000$). For line configuration: (green triangles)
at weak and (red triangles) strong adhesion. For units, see Table S1.

Figure 3:  Data collapse for line growth at high adhesion at three different
times. (a) Using the structure factor (Eq. 6) and MBE exponents,
$\alpha^{\mathrm{MBE}}\\!=\\!\frac{3}{2}$ and $z^{\mathrm{MBE}}\\!=\\!4$. (b)
With KPZ exponents, $\alpha^{\mathrm{KPZ}}\\!=\\!\frac{1}{2}$ and
$z^{\mathrm{KPZ}}\\!=\\!\frac{3}{2}$. The $y$-axis is scaled with the factor
$(\mathrm{k}\mathrm{t}^{\frac{1}{\mathrm{z}}})^{\frac{3}{2}}$ to have the same
range as MBE scaling. (c) Using the Family-Vicsek relation for width (Eq. 1)
with MBE exponents, $\beta^{\mathrm{MBE}}\\!=\\!\frac{3}{8}$ and
$z^{\mathrm{MBE}}\\!=\\!4$, and (d) with KPZ exponents,
$\beta^{\mathrm{KPZ}}\\!=\\!\frac{1}{3}$ and
$z^{\mathrm{KPZ}}\\!=\\!\frac{3}{2}$. For units, see Table S1.

We first consider the growth of linear fronts at two different adhesion
strengths, $10$ (weak) and $2000$ (strong), see Table S1 for parameters. The
initial configurations had a line of 240 cells, and the final populations
consisted of $\approx$200,000 cells. Snapshots are shown in Fig. 1. As this
figure shows, increasing the cell-cell adhesion changes the morphology of the
colony and the interface. Figure 2a shows that the interfaces grow at constant
velocities with the growth rate decreasing with increasing adhesion.

Next, the box counting method was utilized to determine the fractal dimensions
($d_{\mathrm{f}}$) of the interfaces for both line configurations (triangles)
and radially growing (circles) systems. Figure 2b shows that $d_{\mathrm{f}}$
is in the same range for all simulations. In both geometries, however,
$d_{\mathrm{f}}$ is slightly higher for the case of strong adhesion. This is
consistent with the experiments of Torres Hoyos et al. [19] who reported
smaller $d_{\mathrm{f}}$ for malignant and invasive cancer cells as compared
to the benign and more solid (higher adhesion) tumors.

Figures 3a-d show the scaling results for line growth both through the width
function (Eq. 1) and the structure factor (Eq. 6). As the figure shows, both
approaches suggest KPZ dynamics; using the Family-Vicsek relation for the
structure factor (Eq. 6) displays slightly better collapse for KPZ than for
MBE, Figs. 3a,b, and using width, Eq. 1, the data collapses to a single
function using the KPZ exponents (Fig. 3d).

Figure S1 shows that interface roughness, $w(t)$, follows a power-law
$t^{\beta}$ with $\beta^{\mathrm{weak}}\\!=\\!0.28\pm 0.01$ and
$\beta^{\mathrm{strong}}\\!=\\!0.25\pm 0.02$ for weak and strong adhesion,
respectively. Figures S2 and S3 show the local and global roughness exponents.
As the figures show, determining the roughness exponent is questionable
especially in the case of weak adhesion. The value
$\alpha_{\mathrm{loc}}^{\mathrm{strong}}\\!=\\!0.62\pm 0.02$, was calculated
using width, and $\alpha_{\mathrm{glob}}^{\mathrm{weak}}\\!=\\!0.75\pm 0.04$
and $\alpha_{\mathrm{glob}}^{\mathrm{strong}}\\!=\\!0.52\pm 0.02$, using the
structure factor. As the results show, $\alpha_{\mathrm{glob}}$ decreases with
increasing adhesion.

Figure 4:  Data collapse for the radially growing interface at low adhesion at
three different times. (a) Using the structure factor (Eq. 6) with MBE,
$\alpha^{\mathrm{MBE}}\\!=\\!\frac{3}{2}$ and $z^{\mathrm{MBE}}\\!=\\!4$ and
(b) KPZ exponents, $\alpha^{\mathrm{KPZ}}\\!=\\!\frac{1}{2}$ and
$z^{\mathrm{KPZ}}\\!=\\!\frac{3}{2}$. The $y$-axis is scaled with the factor
$(\mathrm{k}\mathrm{t}^{\frac{1}{\mathrm{z}}})^{\frac{3}{2}}$ to have the same
range as MBE scaling. (c) Using the Family-Vicsek relation for width (Eq. 1)
with MBE exponents, $\beta^{\mathrm{MBE}}=\frac{3}{8}$ and
$z^{\mathrm{MBE}}\\!=\\!4$, and (d) with KPZ exponents,
$\beta^{\mathrm{KPZ}}\\!=\\!\frac{1}{3}$ and
$z^{\mathrm{KPZ}}\\!=\\!\frac{3}{2}$. For units, see Table S1.

Next, we focus on radially expanding isotropic fronts. The initial
configuration was one cell at the center of the simulation box and the final
populations were $\approx$200,000 cells. We define $R_{\mathrm{i}}(t)$ to be
the distance from the center of mass of the colony to the $i$th site at the
interface. Snapshots at different times and adhesion strengths are shown in
Fig. 1; increasing the adhesion between the cells changes the colony
morphology and increases overhangs. The radii grow at constant velocities,
$\langle v\rangle\\!=\\!4.24\pm 0.01$ for weak and $\langle
v\rangle\\!=\\!2.15\pm 0.01$ for strong adhesion, Fig. 2a. As in the linear
case, increasing the adhesion causes the front velocities to decrease. Again,
the fractal dimension (Fig. 2b) shows slight increase with increasing
adhesion, $d_{\mathrm{f}}^{\mathrm{weak}}\\!=\\!1.13\pm 0.01$ and
$d_{\mathrm{f}}^{\mathrm{strong}}\\!=\\!1.21\pm 0.01$.

Figures 4a-d show the scaling results for radial growth both through the width
function (Eq. 1) and structure factor (Eq. 6). Using the Family-Vicsek
relation for width (Fig. 4c) with MBE exponents displays good collapse. In
addition, the structure factors at different times show good collapse with MBE
exponents.

Figure S4 shows that fitting the width vs time gives the growth exponents
$\beta^{\mathrm{weak}}\\!=\\!0.40\pm 0.04$ and
$\beta^{\mathrm{strong}}\\!=\\!0.42\pm 0.06$, For varying adhesion strengths,
the local roughness exponent, $\alpha_{\mathrm{loc}}$, are all within the same
range: $\alpha_{\mathrm{loc}}^{\mathrm{weak}}\\!=\\!0.66\pm 0.01$ and
$\alpha_{\mathrm{loc}}^{\mathrm{strong}}\\!=\\!0.7\pm 0.01$, Fig. S5. The
scaling exponent $\alpha_{\mathrm{glob}}$, measured from the structure factor,
shows a decrease with increasing adhesion strength,
$\alpha_{\mathrm{glob}}^{\mathrm{weak}}\\!=\\!0.95\pm 0.04$ and
$\alpha_{\mathrm{glob}}^{\mathrm{strong}}\\!=\\!0.71\pm 0.02$, Fig. S6.

Finally, to test the generality of the above observations, cell-medium
friction, intermembrane friction, cell division rules, and the number of cell
types with different stiffness were tested using the radially growing system.
Figure S7 shows that $\alpha_{\mathrm{glob}}$, is insensitive to changes in
factors such as cell-medium friction, intermembrane friction, cell division
rules, and the number of cell types with different stiffness. These results
suggest that dynamics of the radially growing colonies are well described by
MBE-like scaling when at weak cell-cell adhesion.

In conclusion, we have demonstrated that weak cell-cell adhesion and an
isotropically growing colony display MBE-like scaling for the boundary
roughness. This is consistent with the experiments of Brú et al. [7]; we
digitized their data and show its scaling in Fig. S8. In contrast, a colony
growing from a single line of cells, and with strong cell-cell adhesion
displays KPZ-like scaling. This is in agreement with the experiments of Huergo
et al. [10]. In all the studied cases the fractal dimensions of the interface
are within the range 1.13–1.26, which is also consistent with prior
experiments [7, 10, 11, 20, 21], regardless of the growth scaling behavior.

All the studied cases show linear growth with constant velocity. This is
consistent with experiments: Brú et al. reported radially spreading cell
colonies to exhibit exponential growth in the early stages followed by linear
growth [7]. Huergo et al. showed that once the number of cells exceeds
700–1,000, radially spreading colonies grow with a constant velocity [11, 20].
Constant velocity also applies for line growth [10]. Physically, these data
suggest that cells are partially contact-inhibited and that most activity
occurs within a limited band along the interface challenging the notion of
Gomperzian growth of cancer [33]. Experiments by Costa et al. indicate that in
vitro cultivated cells may exhibit sigmoidal growth [3].

Finally, it can be assumed that the first multi-cellular life-forms were
rather simple cell colonies. It is interesting to speculate that if cell-
colony growth dynamics would be rigidly confined to a single universality
class, adaptive evolution would likely be significantly harder in contrast to
more versatile growth dynamics. This is consistent with earlier results in the
sense that very rich growth behaviour and diverse tissues appear with modest
changes, and, e.g., non-trivial dependencies in initial conditions, nutrients,
apoptosis, disorder and mechanical forces from various sources cause changes
in both quantitative and qualitative behaviors, see, e.g., Refs. [2, 34, 28,
31, 32].

###### Acknowledgements.

We thank the Natural Sciences and Engineering Research Council of Canada (MK),
Canada Research Chairs Programs (MK) and Western University’s Science
International Engagement Fund (MM) for financial support. The use of the
computational resources provided by the Finnish Grid and Cloud Infrastructure
FGCI, funded by the Academy of Finland, is gratefully acknowledged.

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