#  Continued-fraction characterization of
Stieltjes moment sequences with support in $[\xi,\infty)$

Alan D. Sokal1,2 and James Walrad1
1Department of Mathematics, University College London, London WC1E 6BT, UK
2Department of Physics, New York University, New York, NY 10003, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS>

(22 March 2024)

###### Abstract

We give a continued-fraction characterization of Stieltjes moment sequences
for which there exists a representing measure with support in $[\xi,\infty)$.
The proof is elementary.

Key Words: Classical moment problem, moment sequence, Stieltjes moment
sequence, binomial transform, continued fraction.

Mathematics Subject Classification (MSC 2020) codes: 44A60 (Primary); 05A15,
30B70, 30E05 (Secondary).

Let us recall that a sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ of real numbers is
called a Hamburger (resp. Stieltjes, resp. Hausdorff) moment sequence [18, 1,
4, 19, 17] if there exists a positive measure $\mu$ on ${\mathbb{R}}$ (resp.
on $[0,\infty)$, resp. on $[0,1]$) such that $a_{n}=\int\\!x^{n}\,d\mu(x)$ for
all $n\geq 0$; we call $\mu$ a representing measure for ${\bm{a}}$. A
Hamburger (resp. Stieltjes) moment sequence is called H-determinate (resp.
S-determinate) if there is a unique such measure $\mu$; otherwise it is called
H-indeterminate (resp. S-indeterminate). Please note that a Stieltjes moment
sequence can be S-determinate but H-indeterminate [1, p. 240] [19, p. 96].

One fundamental characterization of Stieltjes moment sequences was found by
Stieltjes [22] in 1894 (see also [26, pp. 327–329]): A sequence
${\bm{a}}=(a_{n})_{n\geq 0}$ of real numbers is a Stieltjes moment sequence if
and only if there exist real numbers $c\geq 0$ and
$\alpha_{1},\alpha_{2},\ldots\geq 0$ such that

$\sum_{n=0}^{\infty}a_{n}t^{n}\;=\;\cfrac{c}{1-\cfrac{\alpha_{1}t}{1-\cfrac{\alpha_{2}t}{1-\cdots}}}$
(1)

in the sense of formal power series. (That is, the ordinary generating
function $f(t)=\sum\limits_{n=0}^{\infty}a_{n}t^{n}$ can be represented as a
Stieltjes-type continued fraction with nonnegative coefficients.) Moreover,
when $c\neq 0$ the coefficients ${\bm{\alpha}}=(\alpha_{i})_{i\geq 1}$ are
unique if we make the convention that $\alpha_{i}=0$ implies $\alpha_{j}=0$
for all $j>i$; we shall call such a sequence ${\bm{\alpha}}$ standard.

If $K$ is a closed subset of $[0,\infty)$, we denote by ${\mathcal{S}}_{K}$
the set of Stieltjes moment sequences for which there exists a representing
measure $\mu$ with support in $K$.111 We stress the words “there exists”: it
is not necessarily the case that all representing measures have support in
$K$, when $K$ is unbounded. Indeed, a stunning contrary situation occurs
whenever the Stieltjes moment sequence is S-indeterminate: see Remark 3 at the
end of this paper.  Can we characterize, in terms of the coefficients
${\bm{\alpha}}$, the sequences in ${\mathcal{S}}_{K}$?

For the case $K=[0,1]$ — or by a rescaling, $K=[0,\xi]$ for any $\xi>0$ — this
question was answered by Wall [24, Theorems 4.1 and 6.1] in 1940:

######  Theorem 1 (Wall 1940).

Fix $\xi\geq 0$, and let ${\bm{a}}=(a_{n})_{n\geq 0}$ be a sequence of real
numbers. Then the following are equivalent:

  * (a)

${\bm{a}}$ is a Stieltjes moment sequence for which there exists a
representing measure with support in $[0,\xi]$. (In this situation the
representing measure is unique.)

  * (b)

There exist real numbers $c\geq 0$, $g_{0}=0$ and $g_{1},g_{2},\ldots\in[0,1]$
such that (1) holds in the sense of formal power series, with

$\alpha_{n}\;=\;\xi\,(1-g_{n-1})g_{n}\;.$ (2)

  * (c)

Same as (b), but with $g_{0}\in[0,1]$.

Wall [24, 25] gave two analytic proofs of this result; and recently one of us
[20] gave an elementary proof of an algebraic/combinatorial flavor.222 All of
these proofs considered only (a)$\iff$(b). But (b)$\implies$(c) is trivially
true; and (c)$\implies$(a) holds because the sequence generated by
$g_{0}\in(0,1]$ is bounded above by the corresponding sequence with $g_{0}=0$,
hence satisfies $a_{n}\leq\xi^{n}$ according to (b)$\implies$(a).

Here we would like to answer the analogous question for the case
$K=[\xi,\infty)$. We refer to these moment sequences as $\bm{\xi}$-Stieltjes.
Our result is the following:

######  Theorem 2.

Fix $\xi\geq 0$, and let ${\bm{a}}=(a_{n})_{n\geq 0}$ be a sequence of real
numbers. Then the following are equivalent:

  * (a)

${\bm{a}}$ is a $\xi$-Stieltjes moment sequence.

  * (b)

There exist real numbers $c\geq 0$, $g_{0}=0$ and $g_{1},g_{2},\ldots\geq 0$
such that (1) holds in the sense of formal power series, with

$\alpha_{2k-1}&=\xi\,(1+g_{2k-2})\,+\,g_{2k-1}\\\
\alpha_{2k}={g_{2k-1}g_{2k}\over 1+g_{2k-2}}$ (3)

  * (c)

Same as (b), but with $g_{0}\geq 0$.

Note in particular that in order for ${\bm{a}}$ to be a $\xi$-Stieltjes moment
sequence, it is necessary that $\alpha_{i}\geq\xi$ for all odd $i$; but it is
not sufficient: see Remark 1 below.

Our proof of Theorem 2 will use arguments similar to those used in [20] to
prove Theorem 1, but the logic is in many ways simpler (though the algebra is
a bit messier).

Besides Stieltjes-type continued fractions (1) [henceforth called S-fractions
for short], we shall also make use of Jacobi-type continued fractions
(J-fractions)

$f(t)\;=\;\cfrac{1}{1-\gamma_{0}t-\cfrac{\beta_{1}t^{2}}{1-\gamma_{1}t-\cfrac{\beta_{2}t^{2}}{1-\cdots}}}$
(4)

(always considered as formal power series in the indeterminate $t$).333 Our
use of the terms “S-fraction” and “J-fraction” follows the general practice in
the combinatorial literature, starting with Flajolet [11]. The classical
literature on continued fractions [16, 26, 12, 13, 7] generally uses a
different terminology. For instance, Jones and Thron [12, pp. 128–129,
386–389] use the term “regular C-fraction” for (a minor variant of) what we
have called an S-fraction, and the term “associated continued fraction” for (a
minor variant of) what we have called a J-fraction.  We shall need two
elementary facts about these continued fractions:

1) The contraction formula: We have

$\cfrac{1}{1-\cfrac{\alpha_{1}t}{1-\cfrac{\alpha_{2}t}{1-\cfrac{\alpha_{3}t}{1-\cdots}}}}\;\;=\;\;\cfrac{1}{1-\alpha_{1}t-\cfrac{\alpha_{1}\alpha_{2}t^{2}}{1-(\alpha_{2}+\alpha_{3})t-\cfrac{\alpha_{3}\alpha_{4}t^{2}}{1-(\alpha_{4}+\alpha_{5})t-\cfrac{\alpha_{5}\alpha_{6}t^{2}}{1-\cdots}}}}$
(5)

as an identity between formal power series. In other words, an S-fraction with
coefficients ${\bm{\alpha}}$ is equal to a J-fraction with coefficients
${\bm{\gamma}}$ and ${\bm{\beta}}$, where

$\gamma_{0}&=\alpha_{1}{}\\\
\gamma_{n}=\alpha_{2n}+\alpha_{2n+1}\quad\hbox{for $n\geq 1$}{}\\\
\beta_{n}=\alpha_{2n-1}\,\alpha_{2n}{}$ (6)

See [26, pp. 20–22] for the classic algebraic proof of the contraction formula
(5); see [10, Lemmas 1 and 2] [9, proof of Lemma 1] [8, Lemma 4.5] for a very
simple variant algebraic proof; and see [23, pp. V-31–V-32] for an
enlightening combinatorial proof, based on Flajolet’s [11] combinatorial
interpretation of S-fractions (resp. J-fractions) as generating functions for
Dyck (resp. Motzkin) paths with height-dependent weights.

2) Binomial transform: Fix a real number $\xi$, and let
${\bm{a}}=(a_{n})_{n\geq 0}$ be a sequence of real numbers. Then the
$\xi$-binomial transform of ${\bm{a}}$ is defined to be the sequence
${\bm{b}}=(b_{n})_{n\geq 0}\stackrel{{\scriptstyle\rm
def}}{{=}}B_{\xi}{\bm{a}}$ given by

$b_{n}\;=\;\sum_{k=0}^{n}\binom{n}{k}\,a_{k}\,\xi^{n-k}\;.$ (7)

Note that if $a_{n}=\int\\!x^{n}\,d\mu(x)$, then
$b_{n}=\int(x+\xi)^{n}\,d\mu(x)$. In other words, if ${\bm{a}}$ is a Hamburger
moment sequence with representing measure $\mu$, then
${\bm{b}}=B_{\xi}{\bm{a}}$ is a Hamburger moment sequence with representing
measure $T_{\xi}\mu$ (the $\xi$-translate of $\mu$).

Now suppose that the ordinary generating function of ${\bm{a}}$ is given by a
J-fraction:

$\sum_{n=0}^{\infty}a_{n}t^{n}\;=\;\cfrac{1}{1-\gamma_{0}t-\cfrac{\beta_{1}t^{2}}{1-\gamma_{1}t-\cfrac{\beta_{2}t^{2}}{1-\cdots}}}\;.$
(8)

Then the $\xi$-binomial transform ${\bm{b}}$ of ${\bm{a}}$ is given by a
J-fraction in which we make the replacement $\gamma_{i}\to\gamma_{i}+\xi$:

$\sum_{n=0}^{\infty}b_{n}t^{n}\;=\;\cfrac{1}{1-(\gamma_{0}+\xi)t-\cfrac{\beta_{1}t^{2}}{1-(\gamma_{1}+\xi)t-\cfrac{\beta_{2}t^{2}}{1-\cdots}}}\;.$
(9)

See [2, Proposition 4] for an algebraic proof of (9); or see [21] for a simple
combinatorial proof based on Flajolet’s [11] theory.

Proof of Theorem 2.  When $\xi=0$ the theorem is trivial, because any standard
nonnegative sequence ${\bm{\alpha}}$ can be obtained in the form (3) with a
suitable nonnegative ${\bm{g}}$. (We say “standard” because, under (3),
$\alpha_{2k-1}=0$ forces $\alpha_{2k}=0$.) So we assume henceforth that
$\xi>0$.

Note now that a sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ is a $\xi$-Stieltjes
moment sequence if and only if it is the $\xi$-binomial transform of some
Stieltjes moment sequence ${\bm{b}}$; and of course this means that
${\bm{b}}=B_{-\xi}{\bm{a}}$.

(a)$\implies$(b): Let ${\bm{b}}=B_{-\xi}{\bm{a}}$ be a Stieltjes moment
sequence, and let ${\bm{\alpha}}=(\alpha_{i})_{i\geq 1}$ be the standard
nonnegative coefficients in the S-fraction (1) of its ordinary generating
function. Then by the contraction formula (5), the ordinary generating
function of ${\bm{b}}$ is also given by a J-fraction with the coefficients
(6). Therefore, the ordinary generating function of ${\bm{a}}=B_{\xi}{\bm{b}}$
is given by a J-fraction with the coefficients (6) replaced by
$\gamma_{i}\to\gamma_{i}+\xi$: that is,

$\gamma_{0}&=\alpha_{1}+\xi\\\
\gamma_{n}=\alpha_{2n}+\alpha_{2n+1}+\xi\quad\hbox{for $n\geq 1$}\\\
\beta_{n}=\alpha_{2n-1}\,\alpha_{2n}$ (10)

Now set $g_{0}=0$, and define $g_{1},g_{2},g_{3},\ldots\geq 0$ inductively by
the rules

$g_{2k-1}&=\alpha_{2k-1}\\\
g_{2k}=\displaystyle{1+g_{2k-2}\over\xi\,(1+g_{2k-2})\,+\,g_{2k-1}}\>\alpha_{2k}{}$
(11)

Since $\xi>0$, the prefactor in (LABEL:def.g.b) is strictly positive; so the
map ${\bm{\alpha}}\mapsto{\bm{g}}$ is a bijection from nonnegative sequences
${\bm{\alpha}}$ to nonnegative sequences ${\bm{g}}$. (Also $g_{i}>0$ if and
only if $\alpha_{i}>0$, though we will not need this.)

Now define ${\bm{\alpha}}^{\prime}=(\alpha^{\prime}_{i})_{i\geq 1}$ by (3),
i.e.

$\alpha^{\prime}_{2k-1}&=\xi\,(1+g_{2k-2})\,+\,g_{2k-1}\\\
\alpha^{\prime}_{2k}={g_{2k-1}g_{2k}\over 1+g_{2k-2}}$ (12)

Substituting (11) into (12), we find after some algebra that

$\alpha^{\prime}_{1}&=\alpha_{1}+\xi\\\
\alpha^{\prime}_{2k}+\alpha^{\prime}_{2k+1}=\alpha_{2k}+\alpha_{2k+1}+\xi\quad\hbox{for
$k\geq 1$}\\\
\alpha^{\prime}_{2k-1}\,\alpha^{\prime}_{2k}=\alpha_{2k-1}\,\alpha_{2k}$ (13)

In other words, ${\bm{\alpha}}^{\prime}$ are precisely the S-fraction
coefficients that are related by contraction with the J-fraction coefficients
(10). This shows that the ordinary generating function of ${\bm{a}}$ is given
by an S-fraction with the coefficients ${\bm{\alpha}}^{\prime}$.

(b)$\implies$(a): Given a sequence ${\bm{\alpha}}^{\prime}$ satisfying (12)
for some nonnegative sequence ${\bm{g}}$, we can find a nonnegative sequence
${\bm{\alpha}}$ satisfying (11) because the map ${\bm{\alpha}}\mapsto{\bm{g}}$
is a bijection. The foregoing argument, read backwards, shows that the
ordinary generating function of ${\bm{b}}=B_{-\xi}{\bm{a}}$ has an S-fraction
representation (1) with nonnegative coefficients ${\bm{\alpha}}$, and hence is
a Stieltjes moment sequence.

(b)$\implies$(c) holds trivially.

(c)$\implies$(b): Let $g_{0}>0$ and $g_{1},g_{2},\ldots\geq 0$, and define
${\bm{\alpha}}$ by (3). We will show that for any
$g^{\prime}_{0}\in[0,g_{0}]$, there exist
$g^{\prime}_{1},g^{\prime}_{2},\ldots\geq 0$ that give rise via (3) to the
same coefficients ${\bm{\alpha}}$; in particular, we can choose
$g^{\prime}_{0}=0$. Indeed, we will prove inductively that

$0\;\leq\;g^{\prime}_{2k}&\leq g_{2k}{}\\\ g^{\prime}_{2k+1}\geq
g_{2k+1}\;\geq\;0{}$ (14)

for all $k\geq 0$. To see this, observe first that the equations asserting the
equality ${\bm{\alpha}}={\bm{\alpha}}^{\prime}$ are

$\xi g_{2k-2}+g_{2k-1}&=\xi g^{\prime}_{2k-2}+g^{\prime}_{2k-1}{}\\\
{g_{2k-1}g_{2k}\over 1+g_{2k-2}}={g^{\prime}_{2k-1}g^{\prime}_{2k}\over
1+g^{\prime}_{2k-2}}{}$ (15)

for $k\geq 1$.

Base case $k=0$ of (14). The $k=0$ case of (LABEL:eq.induct.2k) is the
hypothesis $g^{\prime}_{0}\in[0,g_{0}]$. And (LABEL:eq.ctob.2k-1) with $k=1$
gives

$g^{\prime}_{1}\;=\;g_{1}+\xi(g_{0}-g^{\prime}_{0})\;\geq\;g_{1}\;,$ (16)

as claimed in the $k=0$ case of (LABEL:eq.induct.2k+1). This proves the base
case of the induction.

Inductive step. Now assume that (14a,b) hold for a given $k$. Then using
(LABEL:eq.ctob.2k) with $k\to k+1$, we have

$g^{\prime}_{2k+2}\;=\;g_{2k+2}\>{g_{2k+1}\over
g^{\prime}_{2k+1}}\>{1+g^{\prime}_{2k}\over 1+g_{2k}}\;\leq\;g_{2k+2}$ (17)

by the inductive hypotheses (14a,b), provided that $g^{\prime}_{2k+1}\neq 0$.
But if $g^{\prime}_{2k+1}=0$, then the inductive hypothesis
(LABEL:eq.induct.2k+1) says that $g_{2k+1}=0$, in which case we can satisfy
both (LABEL:eq.ctob.2k) and (LABEL:eq.induct.2k) by setting
$g^{\prime}_{2k+2}=0$. And using (LABEL:eq.ctob.2k-1) with $k\to k+2$, we have

$g^{\prime}_{2k+3}\;=\;g_{2k+3}\,+\,\xi(g_{2k+2}-g^{\prime}_{2k+2})\;\geq\;g_{2k+3}$
(18)

by (17). This completes the inductive step. $\square$

Remarks. 1\. It is easy to see that, although $\alpha_{i}\geq\xi$ for all odd
$i$ is necessary for ${\bm{a}}$ to be a $\xi$-Stieltjes moment sequence, it is
not sufficient. For example, the Catalan numbers
$C_{n}=\displaystyle\frac{1}{n+1}\binom{2n}{n}$ are a Stieltjes moment
sequence with ordinary generating function

$\sum_{n=0}^{\infty}C_{n}\,t^{n}\;=\;\cfrac{1}{1-\cfrac{t}{1-\cfrac{t}{1-\cdots}}}\;,$
(19)

i.e. $\alpha_{i}=1$ for all $i$. And the Catalan numbers have a unique
representing measure $(2\pi)^{-1}\,x^{-1/2}\,(4-x)^{1/2}\,dx$ on the interval
$[0,4]$, so they are not a $\xi$-Stieltjes moment sequence for any $\xi>0$.
This can also be seen from the criterion of Theorem 2: if $\xi>0$ and
$g_{0},g_{1},g_{2},\ldots\geq 0$ satisfy

$1&=\xi\,(1+g_{2k-2})\,+\,g_{2k-1}{}\\\ 1={g_{2k-1}g_{2k}\over 1+g_{2k-2}}$
(20)

then $g_{2k}\geq[1-\xi\,(1+g_{2k-2})]\,g_{2k}=g_{2k-1}g_{2k}=1+g_{2k-2}$ for
all $k$, so $g_{2k}\geq k$. But for large $k$ this is incompatible with
(LABEL:eq.catalan.1) whenever $\xi>0$.

2\. Fix $\xi>0$, let ${\bm{a}}$ be a $\xi$-Stieltjes moment sequence, and let
${\bm{\alpha}}$ be the standard nonnegative coefficients in its S-fraction
(1). Now let $I({\bm{a}})$ be the set of $g_{0}\geq 0$ for which there exist
$g_{1},g_{2},\ldots\geq 0$ that reproduce the given coefficients
${\bm{\alpha}}$ via (3). Theorem 2(a)$\implies$(b) states that $0\in
I({\bm{a}})$. Define $g_{0}^{\rm max}=\sup I({\bm{a}})$. From eq. (3a) we see
that $g_{0}^{\rm max}\leq\alpha_{1}/\xi-1<\infty$; and the foregoing proof of
(c)$\implies$(b) shows that $I({\bm{a}})$ is an interval, either
$[0,g_{0}^{\rm max})$ or $[0,g_{0}^{\rm max}]$. We can show that the latter is
the case: take any sequence $g_{0}\uparrow g_{0}^{\rm max}$; then the
$g_{i}^{\prime}$ are rational functions of $g_{0}$, which are bounded on both
sides by (14); therefore they must converge to finite nonnegative limits as
$g_{0}\uparrow g_{0}^{\rm max}$.

There are obviously cases where $g_{0}^{\rm max}>0$; indeed, we can choose
arbitrary numbers $g_{0},g_{1},g_{2},\ldots\geq 0$ and define ${\bm{\alpha}}$
by (3) and then ${\bm{a}}$ by (1); then clearly $g_{0}^{\rm max}\geq g_{0}$.
However, there are also $\xi$-Stieltjes moment sequences ${\bm{a}}$ for which
$g_{0}^{\rm max}=0$, as we now show by explicit example.

Consider the sequence $a_{n}={{\textstyle{1\over 2}}}(1^{n}+2^{n})$, which is
a $1$-Stieltjes moment sequence. It has the ordinary generating function

$\sum_{n=0}^{\infty}a_{n}\,t^{n}\;=\;{1\over 2}\,\biggl{(}{1\over
1-t}\,+\,{1\over 1-2t}\biggr{)}\;=\;\cfrac{1}{1-\cfrac{{\textstyle{3\over
2}}t}{1-\cfrac{{\textstyle{1\over 6}}t}{1-{\textstyle{4\over 3}}t}}}\;,$ (21)

so that $\alpha_{1}={\textstyle{3\over 2}}$, $\alpha_{2}={\textstyle{1\over
6}}$, $\alpha_{3}={\textstyle{4\over 3}}$ and $\alpha_{i}=0$ for $i\geq 4$.
Solving (3) with $\xi=1$ and some fixed $g_{0}\geq 0$, we find

$g_{1}&=\frac{1}{2}-g_{0}\\\ g_{2}={1+g_{0}\over 3(1-2g_{0})}\\\
g_{3}=-\,{g_{0}\over 3(1-2g_{0})}$ (22)

So $g_{1}\geq 0$ requires that $0\leq g_{0}\leq{1\over 2}$, but this is
compatible with $g_{3}\geq 0$ only if $g_{0}=0$. This example can of course be
rescaled to allow any $\xi>0$.

3\. Let ${\bm{a}}$ be any Stieltjes moment sequence that is S-indeterminate,
i.e. for which there exists more than one representing measure supported on
$[0,\infty)$. Then it is a remarkable and surprising (at least to us) fact
that ${\bm{a}}$ is always a $\xi$-Stieltjes moment sequence for some $\xi>0$.
This follows from a result of Berg and Valent [5, Lemma 2.2.1 and Remark
2.2.2], as follows:

Let ${\bm{a}}$ be any Stieltjes moment sequence that is H-indeterminate, and
let $A,B,C,D$ be the corresponding Nevanlinna functions [5, eqns.
(2.10)/(2.12)]. Then Berg and Valent [5, Lemma 2.2.1] show that

$\alpha\,\stackrel{{\scriptstyle\rm def}}{{=}}\,\lim_{x\to-\infty}{D(x)\over
B(x)}\;\hbox{ exists and is }\leq 0\;.$ (23)

They furthermore observe [5, Remark 2.2.2] that the N-extremal measure
$\nu_{t}$ is supported on $[0,\infty)$ if and only if $t\in[\alpha,0]$, and
that $\alpha<0$ if and only if ${\bm{a}}$ is S-indeterminate. (As always, all
the N-extremal measures $(\nu_{t})_{t\in{\mathbb{R}}\cup\\{\infty\\}}$ have
disjoint supports.) Furthermore, when $\alpha<0$ it follows from [5, Lemma
2.2.1] that $\xi(t)\stackrel{{\scriptstyle\rm def}}{{=}}\inf\mathop{\rm
supp}\nolimits\nu_{t}$ is nonnegative, continuous, real-analytic and strictly
decreasing on the interval $[\alpha,0]$, with $\xi(0)=0$; in particular, we
can achieve a value $\xi^{*}\stackrel{{\scriptstyle\rm
def}}{{=}}\xi(\alpha)>0$. Further reasoning [3] based on [14, 15] shows that
$\nu_{\alpha}$ is the unique representing measure $\mu$ having
$\inf\mathop{\rm supp}\nolimits\mu=\xi^{*}$, and that there does not exist any
representing measure having $\inf\mathop{\rm supp}\nolimits\mu>\xi^{*}$.

It is an interesting problem, for concrete examples of S-indeterminate moment
sequences ${\bm{a}}$, to determine explicitly the maximum “spectral gap”
$\xi^{*}$. For Stieltjes’ [22, section 56] famous example
$a_{n}=q^{-n(n-1)/2}$ with $q\in(0,1)$, this computation was in essence
performed by Christiansen [6]. $\blacksquare$

## Acknowledgments

We are extremely grateful to Christian Berg for discussions concerning Remark
3.

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# Motor crosslinking augments elasticity in active nematics

Steven A. Redford1,2, Jonathan Colen3, Jordan L. Shivers4,5, Sasha Zemsky6,
Mehdi Molaei7, Carlos Floyd4, Paul V. Ruijgrok6, Vincenzo Vitelli3,4, Zev
Bryant6,8, Aaron R. Dinner2,4,5,*, Margaret L. Gardel2,3,4,7,*
1The Graduate Program in Biophysical Sciences University of Chicago Chicago IL
60637 U.S.A.
2Institute for Biophysical Dynamics University of Chicago Chicago IL 60637
U.S.A.
3Department of Physics University of Chicago Chicago IL 60637 U.S.A.
4James Franck Institute University of Chicago Chicago IL 60637 U.S.A.
5Department of Chemistry University of Chicago Chicago IL 60637 U.S.A.
6Department of Bioengineering Stanford University Stanford CA 94305 U.S.A.
7Pritzker School of Molecular Engineering The University of Chicago Chicago IL
60637 U.S.A.
8Department of Structural Biology Stanford University School of Medicine
Stanford CA 94305 U.S.A.
*Correspondence<EMAIL_ADDRESS><EMAIL_ADDRESS>

## abstract

In active materials, uncoordinated internal stresses lead to emergent long-
range flows. An understanding of how the behavior of active materials depends
on mesoscopic (hydrodynamic) parameters is developing, but there remains a gap
in knowledge concerning how hydrodynamic parameters depend on the properties
of microscopic elements. In this work, we combine experiments and multiscale
modeling to relate the structure and dynamics of active nematics composed of
biopolymer filaments and molecular motors to their microscopic properties, in
particular motor processivity, speed, and valency. We show that crosslinking
of filaments by both motors and passive crosslinkers not only augments the
contributions to nematic elasticity from excluded volume effects but dominates
them. By altering motor kinetics we show that a competition between motor
speed and crosslinking results in a nonmonotonic dependence of nematic flow on
motor speed. By modulating passive filament crosslinking we show that energy
transfer into nematic flow is in large part dictated by crosslinking. Thus
motor proteins both generate activity and contribute to nematic elasticity.
Our results provide new insights for rationally engineering active materials.

## Introduction

Systems composed of active agents that locally break detailed balance can
exhibit striking collective behaviors that are inaccessible to assemblies that
couple to energy sources in a nondirected fashion (e.g., thermally) [1, 2].
These behaviors include directed collective motion, enhanced information
storage, giant number fluctuations, self-sorting, and motility-induced phase
separation [3, 4, 5, 6]. A better understanding of active systems can suggest
mechanisms in natural systems [7, 8, 9], enable control of nonequilibrium
pattern formation, and guide the design of new materials [10, 11, 12, 13]. One
of the most well-studied classes of active materials is active nematics (also
known as active liquid crystals) [14, 13]. In nematics, elongated components
(mesogens) interact locally through excluded volume yielding a material that
exhibits long-ranged orientational order while maintaining translational
fluidity [15, 16]. The tendency of the mesogens to align gives a nematic an
effective elasticity that resists distortions and acts to align the field as a
whole [2]. This tendency to align is opposed by activity (i.e., mechanical
work done on the individual elements of a system). Activity induces structural
distortions and flow in the nematic field leading to a dynamical steady state.

There are various ways to characterize structure in an active nematic. These
include the spacing of topological defects, the correlation length of the
orientation of the mesogens (director field), and the correlation length of
the velocity or vorticity [17, 18, 19]. However, theory [17], simulation [18],
and experiments [20, 21] suggest that these quantities all scale identically
with activity—i.e., for a given set of conditions, active nematic dynamics are
governed by a single length scale, $\ell$. This length scale arises from the
balance of the elastic stress, $K/\ell^{2}$, where $K$ is the elastic
constant, with the active stress scale, $\alpha$, so that
$\ell=\sqrt{K/\alpha}$ [18]. While $\ell$ quantifies how much energy imparted
by activity is stored in distortions to the nematic field, the average flow
speed of the nematic captures how much energy is dissipated viscously. As
such, by force balance, the average flow speed in a nematic is expected to
scale as $v\sim\alpha\ell/\eta\sim\sqrt{K\alpha}/\eta$, where $\eta$ is the
solvent viscosity [18]. Thus exerting control over $K$ and $\alpha$ affords
control over the steady-state dynamics and structure of an active nematic.

How exactly $K$ and $\alpha$ relate to microscopic properties of the elements
that make up active nematics is not well understood. In active nematics
composed of cytoskeletal elements—semiflexible filaments, molecular motors,
and crosslinkers—activity is generated when the molecular motors hydrolyze
adenosine triphosphate (ATP) and slide pairs of filaments, giving rise to
interfilament strain and extensile force dipoles [3]. Biochemical regulation
affords control of microscale mesogen properties and active stresses allowing
for explicit tuning of hydrodynamic properties on a microscopic scale. For
example, in active nematics composed of actin filaments and myosin II motors,
the elastic constant was shown to depend on filament length [20, 14]. In
nematics composed of microtubules and kinesin, active stresses have been
modulated by changing the concentration of ATP ([ATP]) available to motors. In
this case, the impact of altering [ATP] was to affect the activity through
motor stepping speed and not the elasticity [21]. The motor employed in this
and other studies of cytoskeletal active nematics (kinesin and myosin II
filaments) have high processivities. That is, they almost never detach from
filaments before reaching their ends [22, 23]. Because a motor must link a
pair of filaments to generate extensile stress, one would expect that
differences in filament binding propensities lead to differences in force
transmission capabilities. Indeed, filament crosslinking was observed to
impact local rigidity and force transmission in other cytoskeletal contexts
[24]. However, the roles of motor processivity and, more generally,
crosslinking in active nematics have not been explored to the best of our
knowledge.

To address this gap, here we utilize synthetic myosin motors that range in
their propensities for binding filaments [25]. We tune processivity through
both [ATP] and motor oligomerization state (valency). We find that nematic
speed depends nonmonotonically on [ATP], reflecting opposite trends in
filament strain and crosslinking with [ATP]. We find that crosslinking
modulates the elasticity, and we introduce a simple model that accounts for
the observed trends. Consistent with the model, we show that the addition of
the passive crosslinker filamin also modulates elasticity and in so doing
alters the energetic balance in active flows. Our results reveal a previously
unappreciated connection between activity and elasticity through motor
proteins and show how these quantities can be tuned independently through
molecular composition.

## Results

To probe how the microscopic interactions between a motor and filament control
nematic structure and dynamics, we pair in vitro experiments with multiscale
modeling. Experimentally, we can alter processivity by changing the
availability of ATP or motor valency. Specifically we employ synthetic myosin
motors that consist of the enzymatic head from Chara myosin XI, which is
linked via a flexible linker to an engineered multimerization domain [25]. By
utilizing different multimerization domains, either engineered GCN4 coiled-
coils [26] or de novo two-helix hairpins [27], which form clusters of well-
defined sizes, we are able to query clusters with identical enzymology but
with three, four, or eight heads (Fig. 1A). In the high ATP limit the Chara
myosin XI head has a low duty ratio, meaning it spends less than half of its
catalytic cycle bound to an actin filament [28, 29, 30]. Because this duty
ratio depends on [ATP] motor velocity and the distance a motor travels before
dissociating (run length) on single filaments also depend strongly on [ATP]:
at $\text{[ATP]}=10\ \mu$M, tetrameric clusters have single-filament
velocities of $0.5\ \mu\text{m\ s}^{-1}$ with run lengths of $4\ \mu$m, while
at $\text{[ATP]}=500\ \mu$M, the velocity is $10\ \mu\text{m\ s}^{-1}$, and
the run length is $0.5\ \mu$m (Fig. S1) [25].

### A microscopic model relates motor properties to hydrodynamic parameters.

To understand how the activity depends on [ATP] in our system and in turn to
make predictions for the nematic speed and correlation length through the
relations $v\sim\sqrt{K\alpha}$ and $\ell\sim\sqrt{K/\alpha}$, we developed a
microscopic model of motors with variable valencies. Because activity is
generated via filament pair strain rate and not merely motor speed, this model
focuses on the calculation of filament strain rate, $\varepsilon$. We then use
this quantity in the scaling relation $\alpha\sim\varepsilon^{\beta}$, which
was previously observed to hold for active nematics composed of microtubules
and kinesin motors [21], given the known dependence on [ATP] of the speed of
single kinesin motors walking on single filaments [22].

Building upon a previous approach [31], we coarsely approximate the catalytic
cycle of each head using three states: (1) unbound from the filament with ATP,
(2) bound to the filament in the post-powerstroke state with ADP, and (3)
bound to the filament without a nucleotide (Fig. 1B). Transitions between
these states are irreversible. An essential idea is that a head with ATP has
low affinity for the filament. As a result, the transition from state 1 to
state 2 requires ATP hydrolysis. Similarly, the head quickly releases the
filament once it exchanges ADP for ATP, and the rate of the transition from
state 3 to state 1 is linearly dependent on [ATP]. We simulate the cycle for
each head independently. That is, if there are $n$ heads in a simulation,
there are $3n$ states to keep track of. Because the heads are independent and
rates are irreversible, there are only $n$ allowed transitions at any time. To
evolve the system forward, we perform the Gillespie algorithm over all
possible transitions at a given time [32]. This scheme allows us to simulate
clusters of independent heads with any valency.

We assume that the joint between the lever arm and the multimerization domain
is flexible and that the motor prefers to bind in its least strained position.
Thus, when a head undergoes a transition from state 1 to state 2 and binds to
a filament, we draw its position from the normal distribution
$N(x(t)+s/2,s/2)$. Here, $x(t)$ is the position of the multimerization domain
that couples independent heads together and $s$ is the average step length of
a motor. On each filament, we take $x(t)$ to be a distance $s/2$ ahead of the
rearmost bound head. Assuming fast diffusion relative to binding rates, when a
motor can bind multiple filaments we choose between them randomly with equal
probability. When a transition occurs, $x(t)$ is reevaluated. We calculate the
average velocity of a motor on a filament as the total distance a motor
travels divided by the final time in the simulation. For pairs of filaments,
strain is only recorded if motion occurs while the motor crosslinks the two
filaments [2, 33]. We compute the filament strain rate, $\varepsilon$, by
dividing the total strain by the final time in the simulation. We also compute
the probability of crosslinking, $P_{\text{cl}}$, as the fraction of time that
both filaments are bound simultaneously.

We scan the three rate constants ($k_{12}$, $k_{23}$, $k_{31}$) to identify
values that yield average single-filament speeds and run lengths (i.e., the
length traveled between the first time a head is bound to the last time) that
reproduce measured trends and approximately correspond to measured values from
experiments with tetrameric clusters (Fig. S1) [25]. Two filament results,
$\varepsilon$ and $P_{\text{cl}}$, for a tetrameric motor cluster are shown in
Figs. 1C,D. These simulations show that $P_{\text{cl}}$, decreases while
$\varepsilon$ increases with [ATP].

As described above, we use the computed strain rate to estimate the activity
by $\alpha\sim\varepsilon^{\beta}$. We use $\beta=0.1$ to account for the
flexibility of the synthetic myosin XI motor [25] (for comparison, values
ranging from 0.31 to 1.54 are considered for kinesin in [21]). Substituting
the resulting $\alpha$ into $v\sim\sqrt{K\alpha}$ and
$\ell\sim\sqrt{K/\alpha}$, we obtain an increase in $v$ and a decrease in
$\ell$ with [ATP], for fixed $K$ (Fig. 1E).

Figure 1: [ATP] and activity can be related through a microscopic model. (A)
Schematic of the experiments. We study synthetic motors with controlled
numbers of myosin XI enzymatic heads that bind and slide actin filaments of
length 2 $\mu$m at an oil-water interface. Due to the polarized binding of a
dye to actin filaments, regions with filaments oriented vertically in the
laboratory frame appear brighter than those oriented horizontally [14, 20].
The experimental images are analyzed by optical flow [34] to estimate the
horizontal and vertical components of the velocity at each pixel. From the
velocity field, we calculate the average flow speed, $v_{\text{rms}}$, and
average vortex radius $\ell_{\text{vort}}$ as in [35]. (B) We simulate the
catalytic cycle of myosin XI with three states: (1) unbound with ATP (top),
(2) bound with ADP (right), and (3) bound while nucleotide free (left). (i)
Rate constants are tuned based on prior measurements of speed and processivity
on single filaments (Fig. S1). (ii) We extend the simulation to two filaments
as described in the text and compute the filament extension rate,
$\varepsilon$, and the probability of crosslinking, $P_{\text{cl}}$, as
described in the text. These quantities are used to compute the nematic speed
and correlation length as $v=\sqrt{K\alpha}$ and $\ell=\sqrt{K/\alpha}$,
respectively. (C) $P_{\text{cl}}$ and (D) $\varepsilon$ from two-filament
simulations for a cluster with four heads. (E) Normalized $v$ (magenta) and
$\ell$ (black) for activity derived from (D) assuming constant elasticity,
$K=0.001$.

### Nematic elasticity depends on the probability of crosslinking.

To test our predictions, we use nematics composed of short (2 $\mu$m) actin
filaments labelled with tetramethylrhodamine (TMR) and synthetic motors with
Chara myosin XI enzymatic heads [25]. We form nematics by crowding the actin
filaments to a surfactant stabilized oil-water interface through depletion
forces imposed by methyl-cellulose (Fig. 1A). Once the nematic is formed, we
add $120$ pM tetrameric motors to the sample to introduce activity. We image
the sample with time-lapse fluorescence microscopy at a rate of 0.5 frames/s
for 100 s. Because of the polarization of TMR dye along filaments and the
polarization of our excitation laser, brighter (darker) patches represent
filaments oriented vertically (horizontally) in the imaging plane [14, 20].
Given the video microscopy data, we estimate the nematic velocity at each
pixel using optical flow [34], as described in Materials and Methods.

The results for one series of [ATP] are shown in Fig. 2. As we expected, the
length scale $\ell_{\text{vort}}$, calculated using correlated displacement
velocimetry, decreases as [ATP] increases (Fig. 2A,C) [36]. We use
$\ell_{\text{vort}}$ to quantify length scale because it agrees well with the
velocity correlation length but requires fewer assumptions to measure [35, 18]
(Fig. S2). While $\ell_{\text{vort}}$ decreases with [ATP], the root mean
square flow velocity, $v_{\text{rms}}$, exhibits a nonmonotonic dependence on
[ATP], with a peak at $50\ \mu$M (Fig. 2A,B). This behavior contrasts with the
model prediction (Fig. 1E), suggesting that something is missing from the
model.

Given previous work in which material elasticity depends on the concentration
of crosslinkers [37, 38], we reasoned that the elastic constant $K$ should
depend (linearly) on the effective concentration of crosslinkers, $c_{e}$:

$K\sim K_{0}+\kappa c_{e},$ (1)

where $K_{0}$ is the baseline nematic elastic modulus that arises from
excluded volume interactions between filaments [39, 37], and $\kappa$
represents the energetic penalty for filament deformation at a given
concentration of crosslinker. Here, because the only crosslinkers are motors,
we expect $c_{e}=c_{m}P_{\text{cl}}$, where $c_{m}$ is the concentration of
motors which is taken to be 1 throughout this work. Using (1) for $K$ with
$P_{\text{cl}}$ from the simulation in the scaling relations
$v\sim\sqrt{K\alpha}$ and $\ell\sim\sqrt{K/\alpha}$, we obtain nonmonotonic
$v$ and decreasing $\ell$ with increasing [ATP] (Fig. 2D,E). Physically, there
is a competition between the tendency for increased [ATP] to increase motor
speed, resulting in a higher strain rate, and to reduce motor binding,
resulting in lower $P_{\text{cl}}$. In the case of kinesin, the latter
tendency is negligible due to biochemical coupling and thus was not necessary
to consider in previous studies [21, 22].

The peak in $v$ becomes more pronounced as the second term in (1) becomes
large compared with the first (Fig. 2D). To understand how a peak in $v$ could
arise from these scaling relationships, we differentiate $v=\sqrt{K\alpha}$
with respect to [ATP] and solve for the maximum by setting the resulting
expression equal to zero. This yields

$\alpha_{\text{peak}}=-\frac{K\alpha^{\prime}}{K^{\prime}},$ (2)

where $\alpha_{\text{peak}}$ is the activity that corresponds to the maximum
velocity, and $K^{\prime}$ denotes a derivative with respect to [ATP]. Note
that because $P_{\text{cl}}$ always decreases with [ATP], $K^{\prime}\leq 0$.
For a fixed dependence of the strain rate and thus the activity on [ATP],
larger $\kappa$ results in larger $K^{\prime}$ relative to $K$ and thus
smaller $\alpha_{\text{peak}}$ (i.e., $\alpha_{\text{peak}}$ at lower [ATP]).
Consistent with this reasoning, the peak in Fig. 2D moves to the left as
$\kappa$ increases. It is also worth noting here that changes in $\beta$
affect the balance in this equation as well. If we increase $\beta$, the
nematic speed increases monotonically with [ATP], similar to a decrease in
$\kappa$ (Fig. S3). As such we set $\kappa=10K_{0}$ and $\beta=0.1$ for the
rest of this work.

Note that the variations in $v$ and $\ell$ are smaller than in experiment.
This is a reflection of our simplifying assumptions in this model. On a
hydrodynamic scale, we assume that turbulent scaling relations hold at all
concentrations, even though we expect them to only hold above a critical
[ATP]. Furthermore, our assumption that $K$ is linear in $P_{\text{cl}}$ is
likely an oversimplification. Microscopically we neglect complex coupling [40]
and correlated binding [31] in our motor stepping model, both of which would
reduce $P_{\text{cl}}$ at high [ATP]. The model can readily be tuned to adjust
for these assumptions, but we do not pursue that here for simplicity.

Figure 2: Motor crosslinking modulates nematic elasticity. (A, top row)
Polarized fluorescence micrographs of nematics (gray scale) driven by
tetrameric motor clusters from [25] with [ATP] of 6, 40 or 100 $\mu\text{M}$
(concentration of motors is 120 pM). (A, bottom row) Velocity fields estimated
from optical flow. Scale arrows are $3\ \mu\text{m/s}$. (B) Average flow
speed, $v_{\text{rms}}$, for the experiments in (A) and similar ones with
[ATP] of 16 $\mu$M. Error bars are standard deviations of speed over 100 s of
steady-state activity. (C) Critical vorticity length scale,
$\ell_{\text{vort}}$, measured as in [35], for the same experiments as in (B).
Error bars are standard deviations on 5 sets of 5 non-overlapping frames. (D
and E) Normalized $v$ and $\ell$ for tetrameric motors calculated from the
model scaling with various ratios of $\kappa$ to $K_{0}$. All calculations
presented subsequently use $\kappa=10K_{0}$ and $\beta=0.1$.

### Motor valency tunes nematic dynamics.

We now consider how the motor valency (i.e., the number of heads in a cluster)
affects the structure and dynamics of the active nematics. Simulations of
motors on single filaments show that increasing the motor valency reduces the
speed and increases the processivity, consistent with experimental
measurements [25] (Fig. S4). These trends shift the dependence of
$\varepsilon$ on [ATP] in simulations of motors on two filaments such that
higher [ATP] is required to reach the same relative extension rate (Fig. 3A).
Higher valency also leads to a greater probability of crosslinking across all
ATP concentrations and a smaller relative decrease in crosslinking across the
range of [ATP] that we consider (Fig. 3B). These microscopic trends lead to a
valency-dependent shift of the peak in $v$ to higher [ATP] (Fig. 3C, dotted
line) and a decrease in the relative change in $\ell$ between low and high
[ATP] (Fig. 3D).

Experimentally, we utilize the control afforded by the motor’s multimerization
domain to consider clusters with $n=3,4$, or $8$ heads. We take into account
the contributions of cluster valency and total number of motor heads by
considering trimeric and tetrameric motor clusters at $120$ pM and octameric
motor clusters at $60$ pM (Fig. 3E). This allows us to separate the
contributions from cluster valency and the total head number in the system. We
find that the peak in $v_{\text{rms}}$ is indeed dependent on cluster valency
and shifts to higher [ATP] as valency increases (Fig. 3E). This trend holds
across multiple independent series (Fig. S5). In fact, the shift that we find
in experiment closely matches that predicted by our simulations (Fig. 3C).
Furthermore, as valency increases, $\ell_{\text{vort}}$ at a given [ATP]
increases (Fig. 3F). Thus we can access different ATP response regimes in
these nematics by tuning motor valency. However, separating the contributions
of $P_{\text{cl}}$ and $\varepsilon$ in these experiments is not possible as
these quantities vary simultaneously as valency changes (Fig. 3A,B).

Figure 3: Motor valency tunes nematic dynamics. (A and B) Normalized
$\varepsilon$ and $P_{\text{cl}}$ calculated for clusters of variable valency.
(C and D) Normalized $v$ and $\ell$ from model scaling. The black dotted line
in (C) traces the location of the peak in nematic speed; symbols show the
positions of peak speeds in (E). Brighter colors are higher values. (E and F)
$v_{\text{rms}}$ and $\ell_{\text{vort}}$ for a range of ATP concentrations
and cluster valencies. Error bars are standard deviations on 5 sets of 5 non-
overlapping frames from a single experiment. Independent replicates are found
in Fig. S5.

### Crosslinking modulates the efficiency of nematic energy transfer.

To separate the effects of crosslinking and strain rate, we consider the
effects of adding the passive crosslinker filamin (FLN). Here, we use active
nematics driven by trimeric motors because they have the lowest baseline level
of crosslinking. To incorporate the contribution from passive crosslinkers in
the model, we simply add a contribution to the effective concentration of
crosslinkers: $c_{e}=c_{m}P_{cl}+c_{p}$, where $c_{p}$ is the concentration of
passive crosslinkers. Otherwise the model is the same (Fig. 4A). This model
predicts that the addition of passive crosslinkers leads to a shift in the
peak in $v$ to higher [ATP] (Fig. 4B). We note that this shift is different
from that in response to changing the valency in that it occurs for constant
$\varepsilon$ and $P_{cl}$. Experimentally, we find that adding crosslinker to
these samples yields a dramatically longer length scale as is expected from
increased $K$ (Fig. 4C,E). Furthermore, we find that increased concentrations
of passive crosslinker do indeed lead to a shift in the peak in
$v_{\text{rms}}$ to higher [ATP] (Fig. 4D,E). These observations support our
model, in which crosslinking linearly increases the elastic modulus. In turn,
the shift in $v_{\text{rms}}$ can be understood in terms of (2). Previously we
discussed the case of increasing $\kappa$, which increases $K^{\prime}$,
shifting $\alpha_{\text{peak}}$ to lower [ATP]. By contrast, adding passive
crosslinkers leaves $K^{\prime}$ unchanged while increasing overall $K$,
shifting $\alpha_{\text{peak}}$ to higher [ATP].

As noted before this shift is accompanied by an increase in
$\ell_{\text{vort}}$ and $v_{\text{rms}}$ (Fig. 4C,D). Thus for a given [ATP]
the nematic features fewer defects but moves faster (Fig. 4E). These changes
occur without a substantial change in $\varepsilon$, suggesting that shifts in
$K$ affect how the activity supplied by motors manifests in nematic dynamics.
Indeed, lattice Boltzmann simulations show that in the high activity regime
total energy in the nematic actually increases with $K$ (Fig. S6). This
indicates a crucial role for filament crosslinking in determining the
efficiency of energy transfer from motor stress into active nematic motion.

Figure 4: Microscopic crosslinking alters nematic energy distribution. (A)
Normalized $\varepsilon$ and $P_{\text{cl}}$ (inset) calculated for trimeric
motors. (B) Normalized $v$ from model scaling. (C and D) $v_{\text{rms}}$ and
$\ell_{\text{vort}}$ measured for trimeric driven nematics with filamin (FLN)
added as indicated. (E) Polarized fluorescence micrographs (gray, top row)
with corresponding flow fields (red arrows, bottom row) for trimeric motors at
100 $\mu\text{M}$ ATP with FLN added as indicated. Scale arrow is
3$\mu\text{m/s}$.

## Conclusions

In this work we showed that crosslinking has a profound effect on active
nematic dynamics through elasticity. Previous work with high processivity
motors focused on the motors’ role in activity despite clues to their role in
elasticity from machine learning [41] and experiments in the low [ATP] limit
[21]. Our investigation here of active nematics with low processivity motors
revealed that reduced filament crosslinking at high [ATP] leads to reduced
nematic elasticity and a nonmonotonic dependence of nematic speed on [ATP].
Indeed, we find that the contribution to elasticity from crosslinking
dominates that from excluded volume interactions. We expect this to be the
case even in cytoskeletal active nematics in which crosslinking is constant
across [ATP], as in active nematics composed of microtubules and kinesin
motors [22, 21].

Our results suggest that exquisite control over active nematics dynamics can
be achieved through the choice of molecular composition. Increasing motor
valency affects both the activity and the elasticity due to the effects on
both the strain rate and filament crosslinking. Adding passive crosslinkers in
principle allows one to tune just the elasticity. That both motors and
crosslinkers affect elasticity has long been appreciated for actin gels [42,
43]. Transient crosslinkers have also been shown to tune viscoelastic
properties in fluid actin droplets [44, 45]. Our results suggest that the
degree that motor proteins dictate elasticity can be tuned by their physical
and biochemical properties. It is thus interesting to speculate that the
fantastic diversity of naturally occurring motors and crosslinkers reflects in
part evolutionary pressures to achieve different materials properties.

Our study is a step toward quantitatively linking hydrodynamic parameters of
active materials to microscopic properties. How transferable such relations
may be is an open question. For example, even though active nematics composed
of bacteria can be described in the hydrodynamic limit with similar scaling
laws, activity is generated by microscopic mechanisms that are distinct from
the active nematics considered here [9]. As a result, the characters of their
force dipoles may also be distinct, despite both being extensile. While this
suggests that it will be necessary to go beyond scaling relations to
characterize active materials fully, it is also an opportunity for tailoring
active materials with unique properties.

## Acknowledgements

We thank Chunfu Xu for sharing the sequence of the octameric helical bundle
construct before publication. SAR is grateful to Cristian Suarez and Rachel
Kadzik for help purifying proteins and valuable discussions. This work was
partially supported by the University of Chicago Materials Research Science
and Engineering Center, which is funded by National Science Foundation under
award number DMR-2011854. MLG and ZB acknowledge support from NSF award
DMR-2215605 and NIH R01GM143792. ARD acknowledges support from NSF Award
MCB-2201235. ZB acknowledges support from NIH R01GM114627. SAR was supported
by the NIH under award T32 EB009412. Simulations were performed on
computational resources provided by the University of Chicago Research
Computing Center.

## Author Contributions

SAR, MM, ZB, ARD, and MLG designed the research. SZ and PVR designed motor
constructs and expressed motor proteins. SAR, ZB, and ARD designed the kinetic
simulation. SAR performed the experiments and kinetic simulations. JC and JLS
developed the hydrodynamic connection between simulations and scaling laws.
CSF performed lattice Boltzmann simulations. All authors contributed to and
approved the manuscript.

## Conflicts of Interest

The authors declare no conflicts of interest.

## Materials and Methods

### Experimental Procedures

#### Protein Purification

Monomeric actin was purified from rabbit skeletal muscle acetone powder
(Sigma-Aldrich, St. Louis, MO) as described previously [46] and stored in
G-buffer [2mM Tris pH 8, 0.2mM ATP, 0.5mM DTT, 0.1mM CaCl2, 1mM NaN3, pH to
8]. Actin was labelled with Tetramethylrhodamine-6-maleamide (TMR; Life
Technologies, Carlsbad, CA). F-Actin Capping Protein was purified as described
previously [47] and stored in CP buffer [10mM Tris pH 7.5, 40mM KCl, 0.5mM
DTT, 0.01% NaN3, 50% Glycerol].

#### Cloning and purification of motor constructs

The tetrameric motor construct CM11CD7462R$\sim$1R$\sim$TET is described in
[25]. Motor constructs were assembled from gene fragments encoding the _Chara
corallina_ myosin XI motor domain (residues 1–746), _Dictyostelium_
$\alpha$-actinin (residues 266–502 for the lever arm and residues 266–388 for
the flexible linker), a multimerization domain, and a C-terminal HaloTag and
Flag Tag (DYKDDDDK). The tetrameric motor construct contains the GCN4 leucine
zipper variant p-LI as the multimerization domain, which forms a parallel
tetrameric coiled-coil [26]. In the trimeric construct, the multimerization
domain was replaced with the GCN4 variant p-II, which forms a coiled-coil
trimer rather than a tetramer [26], as previously described for similar
constructs [25]. To create the octameric construct, the tetramerization domain
was replaced with a _de novo_ two-helix hairpin that was designed to assemble
into a water-soluble octameric pore (WSHC8 from [27], PDB 6O35) and the
Halotag is N-terminal to the motor. Constructs were cloned into the insect
expression vector pBiEx-1.

For protein expression, plasmids were directly transfected into Sf9 cells as
described previously [48]. Purification was performed as described in [48] and
[49]. Briefly, proteins were purified using anti-Flag resin and labeled with
Alexa Fluor 660 HaloTag Ligand (Promega). Proteins were eluted into storage
buffer containing glycerol and then immediately flash-frozen in small aliquots
and stored at –80∘C until use.

#### Assay Conditions

Actin filaments were polymerized at a 1:10 labelling ratio and a concentration
of 2 $\mu$M in a 50 $\mu$L polymerization mix. This mix contained 1X F-buffer
[10 mM imidazole, 1 mM MgCl2, 50 mM KCl, 0.2 mM egtazic acid (EGTA), pH 7.5]
with each of the concentrations of ATP studied. No additional MgCl2 was added
with ATP. To minimize photobleaching, an oxygen scavenging system (4.5 mg/mL
glucose, 2.7 mg/mL glucose oxidase (catalog no. 345486, Calbiochem, Billerica,
MA), 17000 units/mL catalase (catalog no. 02071, Sigma, St. Louis, MO) and 0.5
vol. $\%$ $\beta$-mercaptaethanol was added. Actin filaments were crowded to
the surface by including 0.3% w% 400 cP methylcellulose in the polymerization
mix. Capping protein was first thawed on ice, then diluted to 500 nM in 1X
F-buffer, and added at a final concentration of 30 nM in the mix. This
polymerization reaction was allowed to proceed for one hour on ice before it
was added to the imaging chamber.

The imaging chamber was created by first rinsing a small glass cloning
cylinder (catalog no. 09-552-20, Corning Inc.) with ethanol and then attaching
it to a silanated glass coverslip with two-part epoxy. To prevent the actin
from sticking and maintain fluidity, the coverslip was coated with a thin
layer of Novec 7500 Engineered Fluid (3M, St. Paul, MN) that included PFPE-
PEG-PFPE surfactant (catalog no. 008, RAN Biotechnologies, Beverly, MA) at 2%
w/v before the polymerization mix is added. The mixture was allowed to sit in
the sample chamber for about 30 min before imaging to allow for the formation
of the nematic.

The sample was imaged on an Eclipse-Ti inverted microscope (Nikon, Melville,
NY) in confocal mode utilizing a spinning disk (CSU-X, Yokagawa Electric,
Musashino, Tokyo, Japan) and a CMOS camera (Zyla-4.2 USB 3; Andor, Belfast,
UK). Experiments were imaged at one frame every 2 s.

### Data analysis

Flow fields were calculated between every two frames from time lapse images
with optical flow using the Classic+NL-fast method [34, 50]. This method is
based on the classic Horn–Schunck method which minimizes an objective function
penalizing intensity differences between subsequent frames (the data term) as
well as enforcing smoothness in the estimated field. Flow is estimated at
various spatial scales iteratively to capture first global and then local
motion. The optical flow code was obtained from
https://cs.brown.edu/people/mjblack/code.html.

Average flow speed $v$ was calculated from the $N$ vectors, $u_{i}$, as
$v=\sum|u_{i}|/N$. The velocity correlation length quoted in Figure S2 was
calculated as the distance $r$ at which the velocity autocorrelation function
$C_{uu}(r)=\langle u_{i}(0)\cdot u_{j}(r)/|u_{i}||u_{j}|\rangle$ reaches
$1/e$, where the average is over all pairs $(i,j)$ and $e$ is Euler’s number.

$\ell_{\text{vort}}$ was calculated with the method of correlated displacement
fields, as described in [35]. Briefly, the normalized cross correlation is
measured in two dimensions between the vorticity field $\nu$ and the velocity
field $u$. This procedure effectively measures the response of the nematic to
a unit vortical perturbation at the origin. To extract a length scale from
this response, the azimuthal average of the correlation field is taken. This
average results in a one dimensional function with a single maximal extreme.
$\ell_{\text{vort}}$ is the distance $r$ at which this maximum occurs. This
length scale has been shown in active nematics to be equal to the average
radius of a vortex in the flow field [35]. Error for this method was
calculated by measuring $\ell_{\text{vort}}$ over 5 separate non-overlapping
sets of frames from the 100 s of steady-state data considered in
$v_{\text{rms}}$. The code is available at https://github.com/Gardel-
lab/ResponseFunction.

### Motor Stepping Model

The code to run an analyze the myosin stepping model described in Results is
available at
https://github.com/Gardel-lab/myosin_stepping_model.

### Lattice Boltzmann Simulations

Simulations of active nematic hydrodynamics were performed using a custom
Julia implementation of the hybrid lattice Boltzmann algorithm [51, 52]. The
simulated equations of motion are the same as those detailed in [13, 41]. The
simulation domain consists of 400$\times$400 lattice points in two dimensions
with periodic boundary conditions. The turbulent state was generated by
initially perturbing the system and evolving for $15,000$ steps, and then data
was collected every $50$ steps for another $15,000$ steps. For each condition
we ran 5 independent trials using different random seeds for the initial
perturbation. We used the following parameters (in lattice units): a collision
time $\tau=1.5$ (corresponding to viscosity $\eta=1/3$), a flow-alignment
parameter $\xi=0.7$, a rotational diffusion constant $\Gamma=0.13$, and
polarization free energy coefficients of $A_{0}=0.1$, $U=3.5$, leading to an
equilibrium nematic polarization magnitude of $q=0.62$. The elastic constant
$K\in[0,0.1]$ and activity coefficient $\alpha\in[0,0.01]$ (where positive
$\alpha$ corresponds to extensile activity) were varied to generate the
results shown here.

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## Supplemental Figures

Figure S1: Simulations reproduce single filament velocity and run length
trends. Single filament motor velocity (blue) and single filament run length
(black) from experiments (dashed lines) and simulaitons (solid lines) over a
range of ATP concentrations. The final rates we compute after 10,000 tuning
steps are 1821 s-1, 932 s-1, 6 s${}^{-1}\mu$M ATP. Figure S2: $\ell_{vort}$
robustly captures nematic length scale. Comparison of $\ell_{\text{vort}}$ as
calculated in [35] and the traditional velocity correlation length —
$C_{vv}=1/e$ — for nematics driven by 120 pM tetramers (A) and 50 pM octamers
(B). Errorbars are averages over five separate frames for $C_{vv}$ and five
10s snippets for $\ell_{vort}$. Figure S3: Strong activity coupling decreases
nonmonotonicity in tetrameric driven nematics. $v$ (A) and $\ell$ (B) from
scaling predictions for different coupling exponents $\beta$. Figure S4: Motor
velocity and run length are cluster valency dependent phenomena. Single
filament motor velocity (A) and single filament run length (B) measured from
simulation for clusters with 3,4, or 8 heads per cluster. Figure S5: Peak
shift is robust across days and motor concentrations. $v_{\text{rms}}$ for
independent replicates of oligomerization data at $60$ pM (A) or $120$ pM
motor clusters. All data are different from those in the text except for the
octamers in (A) which are included as a reference. Error bars are standard
deviations of $v_{\text{rms}}$ over 100 s of steady-state activity. Figure S6:
Nematic elasticity increases energy in the nematic. Elastic (black) and
kinetic (red) energy for nematics in lattice Boltzmann simulations with
constant $\alpha$ = 0.01 across a range of $K$.


# Enhancing Bangla Language Next Word Prediction and Sentence Completion
through Extended RNN with Bi-LSTM Model On N-gram Language

Md Robiul Islam1, Al Amin2, Aniqua Nusrat Zereen3 3 Department of Computer
Science and Engineering, BRAC University, Dhaka, Bangladesh
{robiul.cse.uu, alaminbhuyan321<EMAIL_ADDRESS>1,2 Department of
Computer Science and Engineering, Uttara University, Dhaka, Bangladesh

###### Abstract

Texting stands out as the most prominent form of communication worldwide.
Individual spend significant amount of time writing whole texts to send emails
or write something on social media, which is time consuming in this modern
era. Word prediction and sentence completion will be suitable and appropriate
in the Bangla language to make textual information easier and more convenient.
This paper expands the scope of Bangla language processing by introducing a
Bi-LSTM model that effectively handles Bangla next-word prediction and Bangla
sentence generation, demonstrating its versatility and potential impact. We
proposed a new Bi-LSTM model to predict a following word and complete a
sentence. We constructed a corpus dataset from various news portals, including
bdnews24, BBC News Bangla, and Prothom Alo. The proposed approach achieved
superior results in word prediction, reaching 99% accuracy for both 4-gram and
5-gram word predictions. Moreover, it demonstrated significant improvement
over existing methods, achieving 35%, 75%, and 95% accuracy for uni-gram, bi-
gram, and tri-gram word prediction, respectively.

###### Index Terms:

Bangla word prediction, Bangla sentence completion, LSTM, Bangla language.

##  I Introduction

In the modern era, the ubiquity of communication devices is indisputable.
However, their intrinsic dependence on laborious and time-intensive text input
procedures constitutes a noteworthy drawback. In response to these
limitations, integrating textual word prediction systems has emerged as a
promising remedy. Such systems mitigate the challenges associated with
repetitive typing, by effectively streamlining the overall text input process.
By utilizing the predictive abilities embedded within text prediction systems,
the need for manual input of text is eliminated. As an alternative, the most
popular word suggestions are shown to the user, maximizing both user enjoyment
and efficiency. This advanced forecasting procedure creates a smooth and user-
friendly text entry interface, significantly improving the speed and fluidity
of digital communication [1]. This clever application significantly lowers the
number of keystrokes needed for word entry, which helps to mitigate writing
obstacles. The effectiveness of the result is directly correlated with the
number of keystrokes that the user saves, since this lowers the amount of time
invested as well as the effort involved in producing text [2]. Once a word or
phrase is entered, the program automatically creates a curated list of
possible lexical possibilities. When the user finds the word they want in this
list, selecting it is a simple process that only requires one click to insert
the selected word into the page [3]. Scholars have conducted numerous
investigations to recognize the benefits that come with word prediction
systems, using a range of approaches to improve word prediction algorithms and
enhance user experience in general. Specifically, within the parameters of
their research project, they have diligently investigated and applied various
methods to enhance the effectiveness and user-friendliness of word prediction
systems. Partha et. al. utilized a combination of Recurrent Neural Network
(RNN) and Long Short-Term Memory (LSTM) to predict the next word in the
Assamese language. For untranslated Assamese text, they obtained an accuracy
of 88.20%, whereas for phonetically transcribed Assamese language, they
acquired a 72.10% accuracy [4]. The model presented by the Steffen Bickel et.
al. accurately predicts the most likely word that will be typed next [5]. They
used weather forecasts, food recipes, call center communications, and personal
emails to assess their model. They modified N-gram language models to predict
the following words to evaluate the model’s performance, placing more emphasis
on precise prediction than conventional performance indicators. With a focus
on the English language, our study has investigated various word prediction
methods for other languages. Nevertheless, few thorough investigations have
been conducted with the goal of creating word prediction models for the Bangla
language or sentence prediction [6]. The model’s efficacy is intricately tied
to the quality and diversity of the dataset utilized during the training
phase. To this end, a comprehensive dataset has been meticulously curated from
diverse sources, including bdnews24 [7], Prothom Alo [8], and BBC Bangla news
[9]. Despite the considerable research on enhancing word prediction
capabilities within the Bengali language using machine learning methodologies,
persistent advancements are imperative to augment system efficiency and
precision. The ongoing evolution of these systems hinges upon their capacity
to adeptly handle expansive datasets, thereby facilitating more nuanced and
accurate prognostications.

The comprehensive allowance of this research work is -

  * •

No previous research has utilized this technique for Bangla language word
prediction.

  * •

Utilizing a large dataset for Bangla word prediction, significantly improves
the accuracy of our suggested approach.

  * •

The effectiveness of our suggested approach exceeds that of other approaches
employed for Bangla word prediction and sentence completion.

The remaining sections of this research are structured like this: part II
discusses the previous researcher’s work, part III outlines the methodology
encompassing dataset creation and implementation with pre-processing, part IV
presents the experimental result and part V discussed the constraint and
upcoming directions.

##  II Related Works

Predicting the next word is a big focus in Natural Language Processing (NLP)
research. In this context, discerning the following word presents a
multifaceted challenge. A notable obstacle arises from the inherent nature of
machines, which predominantly interpret information through ASCII values,
essentially binary code represented as 0s and 1s. The need to represent
linguistic elements in numerical form becomes imperative. Researchers have
diligently addressed these challenges associated with the next-word
prediction. Their efforts have been directed towards finding effective
solutions to the intricacies posed by machine comprehension limited to binary
representations. They aim to transform words into numerical entities, paving
the way for more accurate and sophisticated next-word predictions in natural
language scenarios. The majority of the work has been LSTM based. To tackle
predicting the following word in sequential information, Mikolov et al.
suggested a language model using RNN [10], tasks that were previously
challenging for RNN to solve were solved by Felix et al. using LSTM [11].
Sutskever et al. used an RNN model to predict the next word in a sequence of
data. Their recently presented model was a development of RNNSearch [12], and
it was trained from a question-answering model to a language model via
backpropagation. Zhou et al. proposed the C-LSTM model a while back, which
combines RNN with LSTM for classification and prediction on phrase
representations [13]. A method for creating Bengali text using a sequence
paradigm was presented by Banik et al. [14]. Another approach was presented by
Abujar et al. [15], who employed bi-directional RNN to generate Bengali text
In their study, Barman et al. employed an RNN-based methodology to develop a
next-word prediction model [4]. In essence, they created a model of the LSTM,
a unique kind of RNN. They used their work to suggest the subsequent word with
the highest likelihood reliably next. The N-gram has been used to reduce the
prediction time while typing in the Kurdish language, and the model was
successful in prediction. This model has been developed in the R programming
language. The maximum accuracy recorded in this model is 96.3% [16].

##  III Methodology

This section describes the proposed methodology as well as the operating
principles of the machine learning algorithms. The Bi-LSTM model predicted the
next word and sentence in Bangla. Fig. 1. illustrates the full process of the
methodology.

Figure 1: Proposed methodology

We began by introducing the dataset in subsection A, followed by data
preprocessing techniques in subsection B, and culminated in the model
implementation details in subsection C.

###  III-A Dataset Outline

To evaluate the proposed approach, we assembled a large corpus of Bangla text,
totaling 1.7 GB, gathered from diverse sources. Table I provides a statistical
overview of the collected data.

TABLE I: Data Summary Data Source | Distinct words | Total
---|---|---
Bdnews24 [7] | 45789 | 3892651
Prothom Alo [8] | 97000 | 2087916
BBC News Bangla [9] | 67004 | 4045683

###  III-B Data Preprocessing

After collecting the data, we used several functions to remove unwanted
commas, numbers, etc. from our dataset and five distinct datasets were
generated from the cleaned dataset: Unigram, Bigram, Trigram, 4-gram, and
5-gram. Fig. 2. effectively illustrates the data-cleaning workflow and the
creation of different n-gram datasets. Language models, known as N-grams, are
statistical models that calculate the likelihood of a specific word sequence
occurring in a language. In this study, we have used n-gram language models to
capture the sequential relationships found in Bangla text [17]. When
attempting to predict the next word or words, the quantity of input words can
typically vary from one attempt to the next. One word, a 1-gram or uni-gram,
is produced whenever there is only one input word [18]. Bi-gram is used when
the input consists of two words, and the output is one word [18]. Furthermore,
a Three-gram, sometimes known as a Tri-gram, is created when three words are
input and one word is output [18], and analogously for Four-gram and Five-
gram. Usually, the last four or five words are enough to grasp the sequence’s
dependency. To better understand n-gram language models, consider an example
of a Bangla sentence. Analogous to uni-grams and bi-grams, tri-grams, and
further n-grams maintain the pattern of taking multiple input values and
generating a single output value, demonstrating a consistent approach to
sequence processing.

Figure 2: Data Preprocessing Workflow TABLE II: Bi-gram Model Data Sample Input_1 | Input_2 | Output
---|---|---
baNNGlaedsh | EkiT | smrRd/dh
EkiT | smrRd/dh | sNNGs/krRitr
smrRd/dh | sNNGs/krRitr | edsh
TABLE III: Tri-gram Model Data Sample Input_1 | Input_2 | Input_3 | Output
---|---|---|---
baNNGlaedsh | EkiT | smrRd/dh | sNNGs/krRitr
EkiT | smrRd/dh | sNNGs/krRitr | edsh

Table II and Table III showing the Bi-gram and Tri-gram models. It shows how
it predicts the next word based on the inputted word. For Bi-gram, only two
words are used to predict the next word; in the case of Tri-gram, there are
three words used to predict the next word.

###  III-C Implementation

The n-gram language model determines the likelihood of each potential next
word and chooses the one with the greatest likelihood to be the next word or
words. The utilization of a probabilistic viewpoint enables us to determine
the probability of the next value, empowering us to predict the most probable
next value based on the input value ekRmilen eDRan , equation (1) has been
used to get the highest likelihood of the next word.

$\begin{split}\text{Word after that}&=\textbf{Max}\bigg{(}P(\text{{\bngx
Hamlar}}|\text{{\bngx ekRmilen eDRan}}),\\\ &\phantom{=}P(\text{{\bngx
ghTnaiT}}|\text{{\bngx ekRmilen eDRan}}),\\\ &\phantom{=}P(\text{{\bngx
Aasel}}|\text{{\bngx ekRmilen eDRan}})\bigg{)}\end{split}$

Due to the frequent emergence of zero probability, the model cannot recommend
the next word with the highest probability, leading to the failure of the
entire method in predicting the most suitable value. The effectiveness of
n-gram models diminishes when confronted with large datasets featuring lengthy
input sequences or a considerable number of n-grams. To alleviate the impact
of data sparsity, Back-off, and Katz Back-off techniques were employed to
refine the probability distribution, enabling the effective management of
n-grams with low counts [19]. RNNs keep updating their memory with past
information to maintain a consistent understanding.

Long sequences cause RNNs to lose the impact of previous layers, which gives
rise to the vanishing gradient problem. Two well-known gating mechanisms, LSTM
(Long Short Term Memory) and GRU (Gated Recurrent Unit) were used in response
to the vanishing gradient problem in RNN. These mechanisms allow the network
to efficiently capture long-term dependencies in sequential data. The use of
three gating mechanisms—input, forget, and output—by Bi-LSTM in this study
justifies its adoption since they improve the model’s capacity to learn from
and retain information from distant past inputs, hence resolving the drawbacks
associated with vanishing gradients in RNN. GRU employs update and reset gates
to handle sequential data, whereas LSTM’s ability to maintain internal memory
over time makes it more effective for processing complex datasets compared to
GRU, which doesn’t have a specialized forget gate.

Figure 3: Structure of Bi-LSTM recurrent neural networks

Two separate recurrent neural networks (RNNs) are integrated to create
bidirectional RNNs. This setup lets the network look at information from both
directions (like before and after) for each point in the sequence. When
employing bidirectional processing, the input data is traversed in two
directions: from past to future and from future to past. The key distinction
from unidirectional RNNs lies in using a bidirectional long short-term memory
(Bi-LSTM) structure, where the backward LSTM retains information from the
future. By combining the hidden states from the forward and backward LSTMs,
the model can capture contextual information from the sequence’s preceding and
succeeding elements. This bidirectional approach, exemplified by Bi-LSTMs,
yields notable performance improvements as it enhances the network’s
understanding of contextual nuances. Fig. 3 showing the Bi-LSTM layer
architecture.

Figure 4: The proposed Bi-LSTM model architecture

Fig. 4. shows the architecture of the trained models with five hidden layers
named Embedding, Bi-LSTM Layer1, Bi-LSTM Layer2, Fully Connected Layer1, Fully
Connected Layer2. For Bi-LSTM layers we used 100 LSTM units with tanh
activation function and for Fully Connected Layer1 we used ReLU and for Fully
Connected Layer2 we used softmax activation function. In the Embedding layer v
denotes input dimension or vocabulary length, o represent the output dimension
and i represent the input length.

###  III-D Word Prediciton

Upon effectively completing training on all five datasets (U-gram to 5-gram),
we have obtained five unique trained models optimized for processing input of
varying lengths.

Figure 5: Next word prediction from the model

These models take in sequences of words of different lengths and guess what
word is likely to come next after the input sequence. When the input sequence
contains only one word, the Uni-gram model is utilized for training. With a
single-word input as its input, the model will most likely anticipate the next
word. Systematically, In the case of two input words, the inputted sequence is
directed to the trained Bi-gram model, designed to handle two-word inputs and
predict an output word. Similarly, for the other trained models. Fig. 5.
demonstrates the word prediction methodology for input sequences of different
lengths using five distinct trained models. In cases where the input word
sequence exceeds five words, the 5-gram model relies solely on the last five
words to predict the next word. The last five words typically provide enough
context to determine the next word.

###  III-E Sentence Prediction

Our innovative model combines N-gram training with Bi-LSTM-based RNNs. It
iteratively predicts and appends the next word to the input sequence,
capturing evolving context for accurate predictions. The process ends upon
encountering sentence-ending punctuation. In Bangla, sentence termination is
signified by the symbols "|" for declarative statements and "?" for
interrogative statements. Once the model detects an end punctuation mark, it
ceases prediction and outputs the complete sentence, representing a cohesive
and meaningful continuation of the given word sequence. Fig. 6. shows the word
and sentence prediction by the proposed model.

Figure 6: Next word prediction and sentence completion

##  IV Result Evaluation

We have evaluated our proposed approach using a corpus dataset, training five
different models with identical configurations for up to 300 epochs. Fig. 7.
depicts the accuracy across epochs, while Fig. 8. showcases the loss over the
same epochs. Our 4-gram and 5-gram models exhibit an average accuracy of 99%
and 99.74%, respectively, accompanied by average losses of 2.04% and 1.11%.

TABLE IV: Comparative Analysis Reference | Accuracy
---|---
Barman et. al. [4] | 88.20%
Hamarashid et. al. [16] | 96.30%
Proposed Model | 99.00%

Table IV shows the comparison between Barman et. al. [4] and Hamarashid et.
al. [16] and our proposed system. In Author [16], they used Tri and a Hybrid
Approach of Sequential LSTM and N-gram; their accuracy was 88.20%. On the
other hand, we have a maximum efficiency of 99% for high-order sequences.

Figure 7: 4-Gram and 5-Gram model accuracy per epochs Figure 8: 4-Gram and
5-Gram model loss per epochs

##  V Conclusion

The word prediction and sentence completion model shows promising accuracy
based on the provided dataset. Using multiple pattern-discovery techniques is
essential for effectively removing noisy data in NLP. Next-word prediction is
an example of an NLP application because it involves text mining. The Bi-LSTM
model was used for forecasting over multiple epochs, resulting in a
significant decrease in loss. Specific preprocessing steps and model
modifications could be implemented to improve the model’s prediction
performance. We will make our data set public so that researchers can conduct
research in the near future. A more diverse dataset will improve our work.

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