Keywords: Number theory, elliptic integrals, Ramanujan's class invariants, approximations to π

come

ON THE ELEGANCE OF RAMANUJAN'S SERIES FOR

, as well as several other examples of Ramanujan's innite series. As a matter of fact, the derivation of such formulae has involved specialized knowledge of identities of classical functions and modular functions.

The Archimedes' constant π appears in many formulae [START_REF] Berggren | π : a source book[END_REF] in various areas of mathematics and physics, such as :

James Gregory (1671

) +∞ n=0 (-1) n 2n + 1 = π 4 , ( 0.1)
Leonhard Euler (1734)

+∞ n=0 1 n 2 = π 2 6 , (0.2)
Carl Friedrich Gauss (1809)

+∞ -∞ e -x 2 dx = √ π , (0.3)
Stephen Hawking (1974)

T = 1 8πk B c 3 GM . ( 0.4)
The irrationality of π was rst proven by Jean-Henri Lambert in 1761. Finally in 1882, Ferdinand von Lindemann established its transcendence, thus laying to rest the problem of squaring the circle .

Aesthetics in mathematics ?

In 2014, researchers in neurobiology [START_REF] Zeki | The experience of mathematical beauty and its neural correlates[END_REF] from the University College London (in United Kingdom) used functional MRI to image the brain activity of 15 mathematicians (aged from 22 to 32 years, postgraduate or postdoctoral level, all recruited from colleges in London) when they viewed mathematical formulae. Each subject was given 60 mathematical formulae -including (0.1), (0.2) or (0.3) that correspond successively to arctan(1), ζ(2) and Γ 1 2

-to study at leisure and rate as ugly [-1], neutral [0] or beautiful [+1]. Note the absence of the nonsimple continued fraction :

William Brouncker (1655)

4 π = 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + . . . (1.1)
in their list. Results of the study showed that the one most consistently rated as ugly was Equation ( 14) :

1 π = √ 8 9801 +∞ n=0 (4n)! (n!) 4 1103 + 26390n 396 4n , ( 1.2)
an innite series due to Ramanujan -with an average rating of -0, 7333 ! Truly, beauty is in the eye of the beholder.

Since the starting point of (1.2) lays upon the new foundations of elliptic integrals instilled by the works of both Niels Henrik Abel and Carl Gustav J. Jacobi [START_REF] Gustav | Fundamenta nova theoriae functionum ellipticarum[END_REF] in the 19 th century, we might remember the premonitory words of Felix Klein :

When I was a student, Abelian functions were, as an eect of the Jacobian tradition, considered the uncontested summit of mathematics, and each of us was ambitious to make progress in this eld. And now ? The younger generation hardly knows Abelian functions.

Historically, the identity (1.2) appeared in [START_REF] Ramanujan | Modular equations and approximations to π[END_REF]. Afterwards, it fell into near oblivion, until the end of 1985 when it was revived in a modern computational context. Seven decades after its publication, Bill Gosper Jr. used it for computing 17, 5.10 6 decimal digits of π -and briey held the world record. But a signicant issue remained : no mathematical proof existed back then that the series (1.2) actually converges to 1 π

. It was somehow a leap of faith, yet an educated one. In fact, he veried beforehand that the sum was correct to 10 million places by comparing this same number of digits of his own calculation to a previous calculation done by Yasumasa Kanada and al.

2. Preliminaries 2.1. Jacobi's elliptic integrals.

Let k ∈]0, 1[ denote the elliptic modulus, then the quantity k = √ 1 -k 2 is called the complementary modulus. Complete elliptic integrals of the rst and second kinds are respectively dened as :

K(k) = π 2 0 dθ 1 -k 2 sin 2 θ = π 2 2 F 1   1 2 , 1 2 1 k 2   (2.1) and E(k) = π 2 0 1 -k 2 sin 2 θ dθ = π 2 2 F 1   - 1 2 , 1 2 1 k 2   , (2.2)
while their derivatives are given by :

dK dk = E -k 2 K kk 2
and

dE dk = E -K k . (2.
3)

It is also customary to dene the complementary integrals K and E as :

K (k) = K(k ) and E (k) = E(k ) .
Finally, these 4 quantities K, K , E and E are linked by the remarkable Legendre relation :

K(k)E (k) + E(k)K (k) -K(k)K (k) = π 2 . ( 2.4)
2.2. Jacobi's theta functions.

The theta functions [START_REF] Gustav | Fundamenta nova theoriae functionum ellipticarum[END_REF], [START_REF] Lawden | Elliptic functions and applications[END_REF] are classically dened as :

θ 2 (q) = +∞ n=-∞ q (n+ 1 2 ) 2 , θ 3 (q) = +∞ n=-∞ q n 2 and θ 4 (q) = +∞ n=-∞ (-1) n q n 2 = θ 3 (-q)
(2.5)

for |q| < 1. After rewriting the nome q in terms of the elliptic modulus k :

q = exp -π K (k) K(k)
, it is valuable to regard k as a function of q. Thus, we have inversely :

k = θ 2 2 (q) θ 2 3 (q) , k = θ 2 4 (q) θ 2 3 (q) and K(k) = π 2 θ 2 3 (q) . (2.6)
2.3. Ramanujan-Weber's class invariants.

Let us introduce Ramanujan's class invariants :

G = 1 2kk 1 /12 and g = k 2 2k 1 /12 , ( 2.7)
as well as the Klein's absolute invariant :

J = (4G 24 -1) 3 27G 24 = (4g 24 + 1) 3 27g 24 = 4 27 1 -(kk ) 2 3 (kk ) 4 . (2.8)
In terms of Ramanujan's class invariants, we can explicitly write the elliptic moduli as :

k = 1 2 1 + 1 G 12 -1 - 1 G 12 , k = 1 2 1 + 1 G 12 + 1 - 1 G 12 , or k = g 6 g 12 + 1 g 12 -g 12 , k = √ 2k g 6 .
2.4. Singular value functions λ * and α. Denition 2.1. Let λ * (r) = k(e -π √ r ) be as in (2.6), then the singular value function of the second kind is dened by :

α(r) = E (k) K(k) - π 4 K(k) 2 (2.9)
for positive r.

Since lim r→+∞ λ * (r) = 0, then α(r) converges to

1 π
with exponential rate :

0 < α(r) - 1 π √ r λ * (r) 2 16 √ r e -π √ r .
Using the functional equation (2.4) and the fact that

K λ * (r) K λ * (r) = √
r, we get :

α(r) = π 4 K(k) 2 - √ r E(k) K(k) -1 .
On substituting E with the dierential equation ( 2.3), we may establish that :

α(r) = 1 π π 2K(k) 2 - √ r kk 2 1 K(k) dK dk -k 2 ,
so that :

1 π = √ rkk 2 2 π 2 K(k) dK dk + α(r) - √ rk 2 2 π K(k) 2 (2.10)
where k = λ * (r). Also, observe that α(r) is algebraic for r ∈ Q + (as seen in Tables 1 and2 in the next section, or in the computation of g 2 58 and k 58 in Subsection 3.2.3). Actually, it is well-known that the quantities λ * (r), G r , g r and α(r) are algebraic numbers expressible by surds when r is a positive rational number.

Quadratic and cubic transformations of the hypergeometric function

2 F 1 .
Let us recall the denition of the hypergeometric series :

2 F 1 a, b c z = +∞ n=0 (a) n (b) n (c) n z n n! , (2.11)
where parameters a, b and c are arbitrary complex numbers, and (a

) n = Γ(a + n) Γ(a)
denotes the Pochhammer symbol. However, if and only if the numbers :

± (1 -c) , ± (a -b) , ± (a + b -c) (2.12)
have the property that one of them equals

1 2
or that two of them are equal, then there exists a so-called quadratic transformation.

Proposition 2.2. For k ∈ 0,

1 √ 2
, we have :

2 π K(k) = 2 F 1   1 4 , 1 4 1 (2kk ) 2   (2.13) and 2 π K(k) 2 = 3 F 2   1 2 , 1 2 , 1 2 1, 1 (2kk ) 2   . (2.14)
Proof. The rst identity (2.13) derives from Kummer's identity :

2 F 1 2a, 2b a + b + 1 2 z = 2 F 1 a, b a + b + 1 2 4z(1 -z) (2.15)
and can be veried by showing that both sides satisfy the appropriate hypergeometric dierential equation, are analytic and agree at 0. The second identity (2.14) is a special case of Clausen's product identity :

2 F 1   1 4 + a, 1 4 + b 1 + a + b z   2 F 1   1 4 -a, 1 4 -b 1 -a -b z   = 3 F 2   1 2 , 1 2 + a -b, 1 2 -a + b 1 + a + b, 1 -a -b z   (2.16)
for hypergeometric functions.

In like fashion, a cubic transformation exists if and only if either two of the numbers in (2.12) are equal to

1 3
or if :

1 -c = ±(a -b) = ±(a + b -c
) . Thus, quadratic and cubic transformations of 2 F 1 lead to a variety of alternate hypergeometric expressions for K and K 2 . Proposition 2.3. We also have :

2 π K(k) = 1 k 2 F 1   1 4 , 1 4 1 - 2k k 2 2   for k ∈ [0, √ 2 -1] , 2 π K(k) = 1 √ k 2 F 1   1 4 , 1 4 1 - k 2 2k 2   for k 2 ∈ [0, 2( √ 2 -1)] , 2 π K(k) = 1 √ 1 + k 2 2 F 1   1 8 , 3 8 1 2 g 12 + g -12 2   for k ∈ [0, √ 2 -1] , 2 π K(k) = 1 √ k 2 -k 2 2 F 1   1 8 , 3 8 1 - 2 G 12 -G -12 2   for k ∈ 0, 1 - √ 2 -1 2 3 /4
, and

2 π K(k) = 1 1 -(kk ) 2 1 /4 2 F 1   1 12 , 5 12 1 1 J   for k ∈ 0, 1 √ 2 .
Proof. See e.g. [START_REF] Goursat | Sur l'équation diérentielle linéaire qui admet pour intégrale la série hypergéométrique[END_REF] or [START_REF] Bateman | Higher transcendental functions[END_REF].

Proposition 2.4. For k restricted as in Proposition 2.3 :

2 π K(k) 2 = 1 k 2 3 F 2   1 2 , 1 2 , 1 2 1, 1 - 2k k 2 2   , 2 π K(k) 2 = 1 k 3 F 2   1 2 , 1 2 , 1 2 1, 1 - k 2 2k 2   , 2 π K(k) 2 = 1 1 + k 2 3 F 2   1 4 , 3 4 , 1 2 1, 1 2 g 12 + g -12 2   , 2 π K(k) 2 = 1 k 2 -k 2 3 F 2   1 4 , 3 4 , 1 2 1, 1 - 2 G 12 -G -12 2   ,
and

2 π K(k) 2 = 1 1 -(kk ) 2 3 F 2   1 6 , 5 6 , 1 2 1, 1 1 J   .
Proof. Apply the Clausen's identity (2.16) to Proposition 2.3.

In each case, we have provided series for

2 π K and 2 π K 2
in terms of the Ramanujan's invariants. Indeed, we have :

2 π K(k) 2 = m(k)F ϕ(k)
for algebraic m and ϕ, while F (ϕ) has a hypergeometric-type power series expansion +∞ n=0 a n ϕ n . Then :

2 π 2 K dK dk = 1 2 dm dk F + m dϕ dk dF dϕ
and substitution in (2.10) lead to :

1 π = +∞ n=0 a n 1 2 √ rkk 2 dm dk + α(r) - √ rk 2 m + 1 2 n √ rkk 2 m ϕ dϕ dk ϕ n .
(2.17)

Thus for rational r, the braced term in (2.17) is of the form A + nB with A and B algebraic. .

By combining Propositions 2.2, 2.3 and 2.4 with the formula (2.17), it is now straightforward to build the next 6 series :

(series in G N ) 1 π = +∞ n=0 1 n! 1 2 n 3 α(N ) - √ N k 2 N + n √ N k 2 N -k 2 N 1 G 12 N 2n (3.1) (series in g N ) 1 π = +∞ n=0 (-1) n 1 n! 1 2 n 3 α(N ) k 2 N + n √ N 1 + k 2 N k 2 N 1 g 12 N 2n (3.2) (series in g 4N = 2 1 /4 g N G N ) 1 π = +∞ n=0 (-1) n 1 n! 1 2 n 3 α(N ) - √ N k 2 N 2 1 k N + n √ N k N + 1 k N 1 g 12 4N 2n (3.3) On setting x N = 2 g 12 N + g -12 N = 4k N k 2 N (1 + k 2 N ) 2 and y N = 2 G 12 N -G -12 N = 4k N k N 1 -(2k N k N ) 2 : (series in x N ) 1 π = +∞ n=0 1 4 n 1 2 n 3 4 n (n!) 3 α(N ) x N (1 + k 2 N ) - √ N 4g 12 N + n √ N g 12 N -g -12 N 2 x 2n+1 N (3.4) (series in y N ) 1 π = +∞ n=0 (-1) n 1 4 n 1 2 n 3 4 n (n!) 3 α(N ) y N (k 2 N -k 2 N ) + √ N k 2 N G 12 N 2 + n √ N G 12 N + G -12 N 2 y 2n+1 N (3.5)
And eventually the series in J N :

1 π = 1 3 √ 3 +∞ n=0 1 6 n 1 2 n 5 6 n (n!) 3 2 α(N ) - √ N k 2 N 4G 24 N -1 + √ N 1 - 1 G 24 N + 2n √ N 8G 24 N + 1 1 - 1 G 24 N 1 J 1 /2 N 2n+1 (3.6) that is valid for N > 1.

Applications.

Let us rst evaluate the Pochhammer symbols. It is well-known that : then the fundamental unit u d > 1 is uniquely characterized as the minimal real number :

u d = a + b √ ∆ k 2 (3.7)
where (a, b) is the smallest solution to m 2 -∆ k n 2 = ±4 in positive integers. This equation is essentially Pell-Fermat's equation.

Of course, the most challenging part in the formula (2.17) lies in the evaluation of the singular value function α. For positive rational r, many values of α(r) are obtainable. But details would be slightly beyond the scope of this paper, with deep roots in number-theoretic objects and techniques such as modular equations, multipliers, modular forms, the Dedekind's η function, and so on. Alternatively, we shall rely on Weber [START_REF] Weber | Lehrbuch der Algebra[END_REF] and Ramanujan [START_REF] Ramanujan | Modular equations and approximations to π[END_REF]. Some of the nicest singular values are collected in the following tables.

N k N 1 G 12 N α(N ) u N 3 √ 3 -1 2 √ 2 1 2 √ 3 -1 2 2 + √ 3 5 √ 5 -1 -3 - √ 5 2 √ 5 -1 2 3 √ 5 -2 √ 5 -2 2 1 + √ 5 2 7 3 - √ 7 4 √ 2 1 8 √ 7 -2 2 8 + 3 √ 7 9 ( √ 2 -3 1 /4 )( √ 3 -1) 2 (2 - √ 3) 2 3 -3 3 /4 √ 2( √ 3 -1) 2 - 13 10 √ 13 -34 -5 + √ 13 2 √ 2 √ 13 -3 2 3 √ 13 -74 √ 13 -258 2 3 + √ 13 2 15 (2 - √ 3)(3 - √ 5)( √ 5 - √ 3) 8 √ 2 1 8 √ 5 -1 2 4 √ 15 - √ 5 -1 2 4 + √ 15 25 ( √ 5 -2)(3 -2 × 5 1 /4 ) √ 2 √ 5 -1 2 12 5 1 -2 × 5 1 /4 (7 -3 √ 5) 2 - 37 290 √ 37 -1762 + 29 -5 √ 37 2 √ 2 ( √ 37 -6) 3 √ 37 -(171 -25 √ 37) √ 37 -6 2 6 + √ 37
Table 1. Selected singular values, class invariants G N and fundamental units u N for N odd.

In Table 1, observe that G 4 N = u N for N = 5, 13 and 37.

N k N 1 g 12 N α(N ) uN /2 u N 2 √ 2 -1 1 √ 2 -1 - 1 + √ 2 6 (2 - √ 3)(5 -2 √ 6) 1 /2 ( √ 2 -1) 2 ( √ 2 + 1)(2 - √ 3)(5 -2 √ 6) 1 /2 (3 - √ 2) 2 + √ 3 5 + 2 √ 6 10 ( √ 2 -1) 2 ( √ 10 -3) √ 5 -1 2 6 √ 5 + 1 2 3 ( √ 2 -1) 2 ( √ 10 -3)(3 √ 5 -4) 1 + √ 5 2 3 + √ 10 18 (7 -4 √ 3)(5 √ 2 -7) ( √ 3 - √ 2) 4 3( √ 3 + √ 2) 2 (7 -4 √ 3)(5 √ 2 -7)(7 -2 √ 6) - 1 + √ 2 22 (10 -3 √ 11)(197 -42 √ 22) 1 /2 ( √ 2 -1) 6 ( √ 2 + 1) 3 (10 -3 √ 11)(197 -42 √ 22) 1 /2 (
(N ) = g 6 N k N f N , where f N is an element of some quadratic eld Q( √ d) with d | N .
Many more singular moduli are given in [START_REF] Borwein | π and the AGM : a study in analytic number theory and computational complexity[END_REF] or [START_REF] Petrovi¢ | Comprehensive table of singular moduli[END_REF].

The case

N = 7.
Table 1 provides :

G 12 7 = 8 and α(7) = √ 7 2 -1 , so that k 2 7 = 8 -3 √ 7 16
. By putting these values in the series (3.1) which is valid for N > 1, we obtain :

1 π = 1 16 +∞ n=0 (2n)! 3 (n!) 6 5 + 7 × 6n 64 2n . ( 3.8)
This is equivalent to Equation (29) in Ramanujan's original paper [START_REF] Ramanujan | Modular equations and approximations to π[END_REF]. Being composed of fractions whose numerators grow like ∼ 2 6n and whose denominators are exactly 16 × 2 12n , the above series can be employed to calculate the second block of n binary digits of π without calculating the rst n binary digits.

Note that the series (3.5) is valid for N 4. On using the invariant y 7 = 16 63 in (3.5), we get : .

1 π = 1 9 √ 7 +∞ n=0 ( -
Putting these numerical values into (3.5) yields : .

1 π = +∞ n=0 ( -
From the relation : This concludes the proof of Equation (44) in [START_REF] Ramanujan | Modular equations and approximations to π[END_REF]. As observed by Ramanujan himself, the series (1.2) is extremely rapidly convergent by adding 8 decimal digits a term ! As an exercise, the reader is encouraged to determine the other series of [START_REF] Ramanujan | Modular equations and approximations to π[END_REF] with the singular values in Tables 1 and2. A solution is provided in the companion le https://clwmypage.files.wordpress.com/2021/01/ramanujan-reciprocal-pi.pdf.

+∞ m,n=-∞ (-1) m m 2 + 2rn 2 -4 +∞ m,n=-∞ (-1) m m 2 + 8rn 2 = - π √ 2r log k r 4 , ( 3

Conclusion

Srinivasa Ramanujan recorded the bulk of his mathematical results in several notebooks of looseleaf paper and mostly written up without proofs. Hence, his works were often shrouded in a veil of divine magic and mystery. As being a deeply religious Hindu, he credited his substantial capacities to divinity, and stated that formulae were revealed to him by his family goddess, Namagiri Thayar. During the 20 th century, the many results in Ramanujan's Notebooks inspired numerous papers by later mathematicians trying to prove what he had previously found.

As demonstrated, the general formula (2.17) produces multiple reciprocal series for π in terms of the function α(r) and related modular quantities. Thus, we showed that the amazing sum (1.2) is a specialization (when N = 58) of (2.17) coupled with the

invariant ϕ(k) = 4k(1 -k 2 ) (1 + k 2 ) 2 2 .
For the sake of simplicity, we have intentionally skipped here some technical aspects, namely about modular equations of order p (with p prime), modular forms, Eisenstein series, the Dedekind's η function, etc. References [START_REF] Borwein | Ramanujan, modular equations, and approximations to π, or how to compute one billion digits of π[END_REF] and [START_REF] Nayandeep | Ramanujan's series for 1 π : a survey[END_REF] (as well as multiple references therein) are accessible expository papers in connection with Ramanujan's series for

1 π
. For a deeper insight, material based on the context of elliptic and modular curves can be found e.g. in [START_REF] Borwein | π and the AGM : a study in analytic number theory and computational complexity[END_REF], [START_REF] Huat | Rational analogues of Ramanujan's series for 1 π[END_REF] or [START_REF] Chen | On Chudnovsky-Ramanujan type formulae[END_REF].

  presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of π :

3 .π 3 . 1 .

 331 Examples of hypergeometric-like series representations for 1 Deriving Ramanujan's series for 1 π

Table 2 .

 2 Selected singular values, class invariants g N and fundamental units uN /2 and u N for N even.For N = 6, 10, 18, 22 and 58, observe that the values of the function α in Table2are all expressed in the form α

	33 -17	√	2) 10 + 3	√	11 197 + 42	√	22

  Since this zeta sum over a 2-dimensional lattice (with the exception of the origin) can be decomposed into a sum of products of L-series, we have :

	3.2.3. The case N = 58.
	Let r ∈ Q * + , it turns out that :
											+∞ m,n=-∞	(-1) m m 2 + rn 2 = -	π √ r	log(2g 4 r ) .	(3.12)
			+∞ m,n=-∞	(-1) m+1 m 2 + 58n 2 =	π √ 58	log 2 +	d|29	1 -	2 d	L -232 d (1)L d (1) =	π √ 58	log 2 + 2L -8 (1)L 29 (1)
	where	2 d	denotes the Kronecker symbol of 2 and d > 0. Hence :
					π √ 58	log(2g 4 58 ) =	π √ 58	(log 2 + 2 log u 29 ) =⇒ g 2 58 = u 29 =	5 +	√ 2	29
							1) n 4 4n	(4n)! (n!) 4	α(37) y 37 (k 2 37 -k 2 37 )	+	√	37	k 2 37 G 12 37 2	+ n	√	37	G 12 37 + G -12 37 2	y 2n+1 37
				=	1 3528	+∞ n=0 (-1) n (4n)! (n!) 4	1123 + 37 × 580n 14112 2n	(3.11)
	which can be identied with Equation (39) of [12].

Acknowledgements. The author would like to thank the anonymous referee for his / her constructive comments and suggestions which signicantly improved the present manuscript.

in terms of the central binomial coecient. For the remaining symbols, we may require the following lemma : Lemma 3.1. For any n ∈ N, we have :

, as well as

Proof. Let p, q ∈ N * , observe that :

p + (m -1)q .

Subsequently :

, whereas

This leads naturally to an other famous instance of Ramanujan-Sato series, to wit :

when N = 163. On the quest for digits of π, the series (4.1) was used by Alexander J. Yee and Shigeru Kondo to calculate more than 12, 1.10 12 decimal places for a new record-breaking computation in 2013.

It was only recently that Heng Huat Chan and Shaun Cooper [START_REF] Huat | Rational analogues of Ramanujan's series for 1 π[END_REF] discovered a general approach that used the underlying modular congruence subgroup Γ 0 (N ) to generate a set of all-new Ramanujan-Sato series, such as :

Chan & Cooper (2012) which can be considered as a counterpart of (1.2).


Improved Treatment for Alzheimer's by Enhancing Tyrosine Phosphorylation of the DAB1 Protein through Lauric Acid

Introduction An Overview on Alzheimer's

Just 10 years ago, the National Institute of Health stated that Alzheimer's was the sixth leading cause of death [START_REF] Liu | Apolipoprotein E and Alzheimer disease: risk, mechanisms and therapy[END_REF]. The NIH has just recently revised this fact to say that Alzheimer's is the third leading cause of death, ranked just behind malignant neoplasms and myocardial infarctions. Furthermore, Alzheimer's is the most common form of dementia that happens in the elderly. Dementia is linked with the loss of thinking, remembering, and reasoning as it disrupts a person's daily life. In 1906, Dr. Alois Alzheimer noticed the brain tissue of a woman who had these symptoms [START_REF] Domalaon | Development of dilipid polymyxins: Investigation on the effect of hydrophobicity through its fatty acyl component[END_REF]. After she died, he examined her brain and found many abnormal clumps (now called amyloid plaques) and tangled bundles of fibers (now called neurofibrillary, or tau, tangles).

These plaques and tangles in the brain are still considered some of the main features of Alzheimer's disease. Another feature is the loss of connections between nerve cells (neurons) in the brain [START_REF] Liu | Apolipoprotein E and Alzheimer disease: risk, mechanisms and therapy[END_REF]. Scientists have concluded that damage to the brain starts a decade or more before memory and other cognitive problems appear. During this preclinical stage of Alzheimer's disease, people seem to be symptom-free, but toxic changes are taking place in the brain. Abnormal deposits of proteins form amyloid plaques and tau tangles throughout the brain, and once-healthy neurons stop functioning, lose connections with other neurons, and die. The damage initially appears to take place in the hippocampus, the part of the brain essential in forming memories (Mayo Clinic, 2018). As more neurons die, additional parts of the brain are affected, and they begin to shrink (see figure 1). By the final stage of Alzheimer's, damage is widespread, and brain tissues have shrunk significantly (Mayo Clinic, 2018). Alzheimer's disease has three major stages that have symptoms relating to the loss of memory [START_REF] Howell | Dab1 tyrosine phosphorylation sites relay positional signals during mouse brain development[END_REF]. Mild Alzheimer's disease is the first stage. Problems often include wandering and getting lost, trouble handling money and paying bills, repeating questions, taking longer to complete normal daily tasks, and personality and behavior changes. People are often diagnosed in this stage. In moderate Alzheimer's, damage occurs in areas of the brain that control language, reasoning, sensory processing, and conscious thought [START_REF] Franco | Reelin regulates cadherin function via Dab1/Rap1 to control neuronal migration and lamination in the neocortex[END_REF]. As this progresses, memory loss and confusion grow worse and people begin to have problems recognizing family and friends. They may be unable to learn new things, carry out multistep tasks such as getting dressed, or cope with new situations. In addition, people at this stage may have hallucinations, delusions, and paranoia. People with severe Alzheimer's cannot communicate and are completely dependent on others for their care [START_REF] Dong | FRET analysis of protein tyrosine kinase c-Src activation mediated via aryl hydrocarbon receptor[END_REF]. Near the end, the person may be in bed most or all of the time as the body shuts down (see figure 2).

Causes of Alzheimer's Disease

Even though Alzheimer's is such a life threatening disease, the definite cause is still not found. The disease is divided into 2 subtypes based on the age of onset: early-onset AD (EOAD) and late-onset AD (LOAD) [START_REF] Niu | Reelin promotes hippocampal dendrite development through the VLDLR/APOER2-Dab1 pathway[END_REF]. A combination of genetics, lifestyle, and environmental factors have shown to cause the early/ late onset of Alzheimer's. Most people with Alzheimer's have the late-onset form of the disease, in which symptoms become apparent with an age later than 60 or 65 years [START_REF] Dibattista | Very low density lipoprotein receptor regulates dendritic spine formation in a RasGRF1/CaMKII dependent manner[END_REF]. The apolipoprotein E (APOE) gene is involved in late-onset Alzheimer's. This gene has several forms. One of them, APOE ε4, increases a person's risk of developing the disease and is also associated with an earlier age of disease onset [START_REF] Domalaon | Development of dilipid polymyxins: Investigation on the effect of hydrophobicity through its fatty acyl component[END_REF]. Early-onset AD accounts for approximately 1% to 6% of all cases and ranges roughly from 30 years to 60 or 65 years.

APOE has major functions including the mediation of lipid transport from one tissue to the other, regulates lipid homeostasis, and modulates Aβ metabolism, aggregation, and deposition [START_REF] Franco | Reelin regulates cadherin function via Dab1/Rap1 to control neuronal migration and lamination in the neocortex[END_REF]. APOE4 is linked to many different diseases including atherosclerosis but it's major link is to Alzheimer's Disease (AD). APOE is produced by the liver and macrophages which mediates cholesterol metabolism as well (Mayo Clinic, 2018). In the liver, specific astrocytes transport cholesterol to neurons by APOE receptors who are member of the LDLR family. From studies, APOE seems to regulate Aβ aggregation and deposition. Studies show that APOE4 causes the knockdown of the APOE gene [START_REF] Niu | Reelin promotes hippocampal dendrite development through the VLDLR/APOER2-Dab1 pathway[END_REF]. This causes Aβ aggregation and plaques to be formed. Not only this, but relationships of the DAB1 protein and APOE gene account for most case of Alzheimer's along with the APOE4 allele (Mayo Clinic, 2018).

APOE is composed of 299 amino acids and has a molecular mass of ~34 kDa. Differences between the three APOE isoforms are limited to amino acids 112 and 158, where either cysteine or arginine is present: APOE2 (Cys112, Cys158), APOE3 (Cys112, Arg158), and APOE4 (Arg112, Arg158) (NeuroImage, 2015). The single amino acid differences at these two positions affect the structure of APOE isoforms and influence their ability to bind lipids, receptors and Aβ. Human and animal studies clearly indicate that APOE isoforms and DAB1 downregulation differentially affect Aβ aggregation and clearance [START_REF] Howell | Dab1 tyrosine phosphorylation sites relay positional signals during mouse brain development[END_REF]. Several Aβ-independent functions are also associated with APOE isoforms. APOE4 and DAB1 downregulation on the APOE gene is what causes differences in interaction and causes and increase in aggregation [START_REF] Franco | Reelin regulates cadherin function via Dab1/Rap1 to control neuronal migration and lamination in the neocortex[END_REF]. Genome-wide association studies have confirmed that the downregulation of APOE is the strongest genetic risk factor for AD. The presence of this is associated with increased risk for both early-onset AD and LOAD [START_REF] Domalaon | Development of dilipid polymyxins: Investigation on the effect of hydrophobicity through its fatty acyl component[END_REF].

AD is associated with both functional abnormalities of the hippocampus and cortical atrophy in the memory network (see figure 3). Patients with AD who have a downregulation of APOE exhibit greater medial temporal lobe atrophy, particularly in the hippocampal area. Structural MRI studies found that, compared with normal gene function, those that have a downregulation of the APOE gene have accelerated age-related loss in cortical thickness and hippocampal volume that are tightly coupled to decline in cognitive performance [START_REF] Franco | Reelin regulates cadherin function via Dab1/Rap1 to control neuronal migration and lamination in the neocortex[END_REF]. Functional MRI (fMRI) studies reported that APOE downregulation disrupts resting state fMRI connectivity and the balance between brain networks, in the absence of amyloid pathology. Such changes have been hypothesized to represent that downregulation of APOE is a great risk factor for AD [START_REF] Domalaon | Development of dilipid polymyxins: Investigation on the effect of hydrophobicity through its fatty acyl component[END_REF].

Figure 3: Downregulation of the APOE and its effects on the system. Adapted from MedIndia

The Role of VLDLR Gene

Studies suggest that VLDLR and APOER2 (LRP8) genes further lead to the downregulation of the APOE gene [START_REF] Dibattista | Very low density lipoprotein receptor regulates dendritic spine formation in a RasGRF1/CaMKII dependent manner[END_REF]. These genes are downregulated because of messages from DAB1 protein [START_REF] Dong | FRET analysis of protein tyrosine kinase c-Src activation mediated via aryl hydrocarbon receptor[END_REF]. The DAB1 protein interacts with cadherin proteins on the receptor for VLDLR and APOER2, which causes tyrosine phosphorylation. When tyrosine phosphorylation occurs, more signals are being sent to maintain neuron signals and stability, preventing amyloid plaques to be formed in Alzheimer's.

Connection between VLDLR, APOER2 (LRP8), and DAB1

Bioinformatics was used to examine if there is a connection between the VLDLR, DAB1, and LRP8 (APOER2). NCBI GEO2R BLAST, a bio-informatics tool, was used to find a dataset of VLDLR and its effect to Alzheimer's. GSE8425 was found and had samples of VLDLR deficiency in the hippocampus. It was suggested that the samples lead to Alzheimer in patients. Six samples were given which were examined. Three of the samples were of a regular hippocampus and the other three were samples from VLDLR knockout. The three samples of the regular hippocampus were named control and the others were called VLDLR (NCBI Bank). Lower p values link to the increase of significance (Geo Accession Viewer). Further, VLDLR gene is highly downregulated in the VLDLR downregulated group rather than the control group (see figure 4).

Research Question and Hypothesis

The purpose of this project is to examine if Lauric Acid could be proven to be a novel way of minimizing effects for the downregulation of a string with genes DAB1, VLDLR, and APOE; a gene which is the main link to Alzheimer's. Unsaturated fatty acids and Lauric Acid exhibit similar patterns in structure. The research also suggests that the DAB1 protein bound to cadherin proteins on VLDLR which issued tyrosine phosphorylation of the DAB1 protein. This process later sent signals to the APOE protein in order to maintain neuron function. Thus based on the extant literature, the following research question and hypothesis is proposed:

Research Question: Does Lauric Acid when added to DAB1 protein lead to increase in tyrosine phosphorylation thus reducing the risk of Alzheimer?

Hypothesis: Adding Lauric Acid to the DAB1 protein will increase the intensity of tyrosine phosphorylation of cadherin proteins and DAB1 protein thus decrease the risk of Alzheimer's because of the structural changes in the DAB1 protein.

Materials Required

Step 1: Schrodinger -List of 100 different fatty acids, Schrodinger biologics suite

Step 2: Culture NIH-3T3 Cells -Modified eagle's medium, bovine calf serum, 0.25% (w/v) Trypsin, 0.53 mM EDTA solution, micropipettes, incubator, 3T3 cell line

Step 3: Preparation of Plasmid -3T3 cells, buffer P1 (50mM Tris-Cl, pH 8.0, 10mM EDTA, 100ug/mL RNase A), buffer P2 (200mM NaOH, 1% SDS), buffer N3 -neutralization buffer for spin columns (4.2 M Gu-HCl, 0.9 M potassium acetate, pH = 4.8), centrifuge, QIA prep, buffer PB -binding buffer (5 M Gu-HCl, 30% isopropanol), buffer PEwash buffer (10 mM Tris-HCl pH = 7.5, 80% ethanol), buffer EB -elution buffer (10mM Tris-Cl, pH = 8.5)

Step 4: Transfection using PEI -3T3 cells, DMEM/10% FBS medium, plasmid, PEI

Step 5: Crosslinking of Beads -Elution buffer (0.1 M Glycine-HCL, pH = 2.5), binding buffer (0.1 M Sodium Phosphate buffer, pH = 8.0), cross linking buffer (0.2 M Triethanolamine, pH = 8.2), cross linker -Dimethyl Pimelimidate Dihydrochloride, blocking buffer (0.1 M Ethanolamine, pH = 8.2), immunoglobulin (in binding buffer), storage buffer (Phosphate-Buffered Saline), Amicon ultra .5 centrifugal filter units for volumes up to 500 microliters

Step 6: Immunoprecipitation using Protein A Beads -Immunoprecipitation buffer (150 mM NaCl, 10 mM Tris-HCl (pH 7.4), 1 mM EDTA, 1 mM EGTA (pH 8.0), 0.2 mM sodium orthovanadate, 0.2 mM PMSF, 1% Triton X-100, 0.5% NP-40) , 3T3 cells, centrifuge, protein A beads, vortex, magnet

Step 7: Protein Quantification -20μL of Copper Reagent,1000μL of BCA Reagent, 8 tubes, 40μL of BSA Standard, 500μL of immunoprecipitation buffer, pipette, DAB1 protein, ToxCast Assay Network (TCAN)

Step 8: Combining Fatty Acid and DAB1 Protein by BSA -DAB1 protein extracted 3T3 cell line, BSA, Lauric Acid, Sodium Palmitate

Step 9: Combine DAB1 protein/ fatty acid mixture and Cadherin Protein and FRET -DAB1 protein/ fatty acid mixture, Cadherin Protein, Nikon TE300 Eclipse Wide-Field microscope with computer, saline solution (pH = 7.4), FRET microscope

Methods

Step 1: Computational Using Schrodinger's construction, 100 fatty acid structures were made and the DAB1 protein and cadherin proteins were inputted. Using Schrodinger options like MM-GBSA and Protein Preparation, the ligand (fatty acid) and DAB1 protein were bound together. Further, this complex was bound to the cadherin proteins. Results were analyzed with the below equation and can be shown in Graph 1 in the "Results" section: G = H -T S ΔG= Gibbs free energy of each fatty acid ΔH= Enthalpy (Total Heat) ΔS= Entropy (Randomness) T= Temperature

Figure 6: Lauric Acid and DAB1 complex bound to the cadherin proteins displays the most spontaneity as it has the least Gibbs free energy in comparison to the other fatty acid complexes.

Step 2: Culture NIH-3T3 Cells Frozen 3T3 cells were thawed in a 37°C water bath. 70% ethanol was used to decontaminate the cells. They were later transferred to a centrifuge tube containing 9.0 mL complete growth medium, and were spun down using a centrifuge at 125 rpm for 5-7 minutes. The cell pellet was resuspended and dispensed into a 75 cm 2 culture flask. The culture was incubated at 37°C with 5% CO 2 in atmosphere. Cells were later sub-cultured, trypsinized with 0.25% Trypsin and 0.53 mM EDTA, and re-suspended in fresh media in new culture flasks. The culture incubated at 37°C and the sub-culture was repeated every two days.

Step 3: Preparation of Plasmid

Plasmid was inoculated by adding it into 2-5 mL of LB medium. The mixture was incubated for 24 hours at 37°C, 300 rpm. The plasmid was harvested by centrifuging at 6000 rpm for 15 mins at 4°C. The pellet was resuspended in 10 ml of resuspension buffer (50 mM Tris•Cl, pH 8.0; 10 mM EDTA; 100 µg/ml RNase A), 10 ml of lysis buffer (200 mM NaOH, 1% SDS (w/v)) and neutralization buffer (3.0 M potassium acetate, 15-25°C; pH 5.5), inverted 4-6 times, and incubated on ice for 20 mins. The plasmid was centrifuged at 20,000 rpm for 30 min at 4°C. The supernatant was removed and kept in a tube. Micro-centrifuge tubes were equilibrated by applying 10 mL of equilibration buffer (750 mM NaCl; 50 mM MOPS, pH 7.0; 15% isopropanol (v/v); 0.15% Triton X-100 (v/v)). The supernatant was added and washed two times with 30 mL wash buffer (1.0 M NaCl; 50 mM MOPS, pH 7.0; 15% isopropanol (v/v)). 15 mL of elution buffer (1.25 M NaCl; 50 mM Tris•Cl, pH 8.5; 15% isopropanol (v/v)) was added along with 10.5 mL of isopropanol and centrifuged for 15,000 rpm at 4°C for 20 mins. The supernatant was discarded and the above step was repeated with ethanol. DNA was later redissolved in 20 mL of elution buffer.

Step 4: Transfection using PEI Further, cells were split in 5 ml DMEM/10% FBS medium (24 hrs before transfection). 3 ug total plasmid DNA was diluted in 200 µL of serum-free DMEM w/o phenol red. PEI (1 ug/µL) was added and mixed by vortexing.. The serum was incubated for 15 mins and 9ul DNA/PEI mix was added. The transfected cells were harvested at 48 hrs post-transfection.

Step 5: Crosslinking of Beads

Step 5a: Buffer Exchange for Antibody into Binding Buffer. An antibody (0.2 mg/mL concentration) was thawed under warm water. Contents of one vile were centrifuged for 15 mins at 14,000 rpm. 480 µL of Sodium Phosphate Buffer (0.1 M Sodium Phosphate buffer, pH = 8.0) was added to the resulting 20 µL and centrifuged. The concentrate was disposed into a new centrifuge tube and was centrifuged at 1000g for 2 mins.

Step 5b: IgG Binding to Protein A Magnetic Beads. 500 µL of 0.1 M Sodium Phosphate Buffer (binding buffer) was added to beads and were vortexed to re-suspend. The magnet was applied for 30 secs on the side and aspirated the supernatant. The beads were washed by adding 80 µL of Sodium Phosphate Buffer and 20 µg of DAB1 antibody to beads. The beads were mixed and incubated at 4°C with agitation for 30 mins. The magnet was applied and supernatant aspirated. The Sodium Phosphate Buffer step was repeated three times.

Step 5c: IgG Cross-Linking. 1 mL of cross linking buffer (0.2 M Triethanolamine, pH 8.2) was added to magnetic bead and antibody and vortexed to re-suspend. The magnet was applied and aspirated. The beads were washed and re-suspended by adding 1 mL of cross linking buffer with cross linker (25 mM DMP). The beads were mixed, incubated, applied to magnet, and aspirated. 1 mL blocking buffer (0.1 M Ethanolamine, pH 8.2) was added, vortexed, applied to magnet, and aspirated. The above step was repeated and incubated at room temp. for 1 hr. Further, it was washed with 1 ml of phosphate-buffered saline (PBS), vortexed, applied to magnet, and aspirated. The beads were washed and 1 mL of elution buffer (0.1 M Glycine-HCL, pH 2.5) was added, vortexed, applied to magnet, and supernatant was removed. Beads were stored in 100 µL of PBS storage buffer.

Step 6: Immunoprecipitation using Protein A Beads

Step 6a: Cell lysis. PBS were rinsed with a dish of confluent cells. The cells were lysed with 0.5 ml of immunoprecipitation buffer (150 mM NaCl, 10 mM Tris-HCl (pH 7.4), 1 mM EDTA, 1 mM EGTA (pH 8.0), 0.2 mM sodium orthovanadate, 0.2 mM PMSF, 1% Triton X-100, 0.5% NP-40) and maintained at constant agitation for 30 mins at 4°C. The cells were scraped, sonicated on ice for 5 mins, and was repeated 4 times. The cells were centrifuged for 5 mins at 4°C to get crude cell lysate.

Step 6b: Immunoprecipitation. 25 μL of Protein A/G Magnetic Beads was added to 200μL of crude cell lysate. This was gently vortexed and incubated at 4°C for 2 hrs. A magnetic field was applied for 30 secs to pull the beads to the side of the tube. The protein was gently removed from beads.

Step 7: Protein Quantification Working Solution was prepared by adding 20μL of Copper Reagent to 1000μL of BCA Reagent. Eight tubes were labeled 1-8. Stock Solution was created by adding 40μL of BSA Standard and 360μL of immunoprecipitation buffer. 256μL of Stock Solution was added to 144μL of immunoprecipitation buffer. From here, a 1:2 dilution was performed. 50μL from tube 8 was pipetted into a well of a 96 well plate. The same was done with the other tubes and protein. 100 μL of Working Solution was added into each well containing a sample. This was incubated for an hour at 37°C. OD was measured at 562 nm using a ToxCast Assay Network (TCAN), as shown in Graph 2 in results.

Step 8: Combining Fatty Acid and DAB1 Protein by BSA 18.4% fatty acid-free BSA was dissolved in lauric acid (room temp, 3 hrs). 9mM sodium palmitate was added and agitated (37°C), 6 hrs or overnight. DAB1 protein was treated with FFA/BSA mix (up to 48 hrs).

Step 9: Combine DAB1 protein/ fatty acid mixture and Cadherin Protein and FRET From here, 1 mL of DAB1 protein was added to 0.5 mL of cadherin proteins. A Varian Cary Eclipse was later used to carry out FRET. The above sample was prepared three times to carry out three different samples. A control sample of 1.5 mL of DAB1 protein and was also prepared three times to measure three different samples. Results from the FRET Machine are shown in Graph 3.

Results

Graph 1: One can see the difference of Gibbs free energy with respect to various acids. The lower the Gibbs free energy, the more spontaneous the reaction is. However in the diagram displayed values are negative change of Gibbs free energy. Therefore, Lauric Acid displays the most spontaneous reaction. Made by Siya Goel using Excel.

Graph 2: A series of absorbance values were given based on the protein concentration in the assay from Step 7. A line of best fit was discovered and DAB1 protein concentration was 0.16 mg/ml based on absorbance. Made by Siya Goel using Excel. Equation 1 adresses entripy. Entropy is a commonly used metric for the measurement of brain complexity and addresses randomness and predictability. Greater entropy is associated with more complexity, more randomness, and less system order. The randomness and complexity has been suggested to decrease the risk of Alzheimer's. Higher entropy also leads to a lower Gibbs free energy, thus leading to more spontaneity and decreased risk of Alzheimer's. Made by Siya Goel using Excel.

Graph 4 shows the ratio of change of entropy with and without Lauric Acid with respect to concentration of the DAB1 protein. Due to Equation 1, the ratio should be less than 1 as B2>B1. Therefore, the negative change in entropy with Lauric Acid is more than negative change without. Made by Siya Goel using Excel.

Discussion and Conclusion

The purpose of this research was to examine if adding Lauric Acid to DAB1 protein increases tyrosine phosphorylation thus reducing the risk of Alzheimer's. Both computational and experimental studies were conducted to test the hypothesis. The computational study suggested the most spontaneity between Lauric Acid and DAB1 complex when bounded to the cadherin proteins (see figure 6). During experimental study, when Lauric Acid was combined with DAB1 protein, the results suggested that increasing the concentration of Lauric Acid to DAB1 protein increases the intensity of fluorescence. Previous research has shown that an increase in fluorescence shows an increase in phosphorylation [START_REF] Wang | Fluorescence spectroscopy as a probe of the effect of phosphorylation at serine 40 of tyrosine hydroxylase on the conformation of its regulatory domain[END_REF]. One can also conclude that this intensity of fluorescence is measuring tyrosine phosphorylation because tyrosine fluoresces around the wavelengths of 290-320 nm [START_REF] Guzow | Photophysical properties of tyrosine and its simple derivatives studied by time-resolved fluorescence spectroscopy, global analysis, and theoretical calculations[END_REF]. The peak of fluorescence measured here has a wavelength of 306 nm. This further shows that tyrosine phosphorylation is increased when Lauric Acid is added to DAB1 protein, thus supporting the first part of hypothesis.

Entropy is a common used metric for the measurement of brain complexity and addresses randomness and predictability. Greater entropy is associated with more complexity, more randomness, and less system order. The randomness and complexity is associated with one's ability to adapt to a changing environment. Loss of randomness and complexity has been suggested as the signal of aging and increase in risk of Alzheimer's (Li et al., 2018). Higher entropy also leads to a lower Gibbs free energy, thus leading to more spontaneity and decreased risk of Alzheimer's.

The second part of the hypothesis was also supported. Based on intensity of fluorescence, change in entropy was calculated. The results suggest the negative change in entropy using DAB1 and Lauric Acid (ΔS 2 ) was more when compared with negative change in entropy using DAB1 (ΔS 1 ), thus suggesting more randomness and reduced risk of Alzheimer's.

Errors, Future Experiments, and Implication

Some errors were observed throughout this research. The first was not conducting enough trials in the Carry Eclipse due to the lack of protein which may have skewed the results. Secondly, the only form of protein purification used was from the beads and an SDS-PAGE was not carried out. This further leads to the possibility that the protein was not fully purified. In the future, this research could be conducted again with more trials and an SDS-PAGE for more precise results. The experiment could also be tested using circular dichroism to show the precise structural changes, strengthening the current conclusion. In addition, experimenting on a cellular level using CRISPR to see if the conclusion still holds, would be relevant. If it does, then Lauric Acid can be tested on mice and later patients with Alzheimer's, thus leading to an improved treatment of this deadly disease. Furthermore, Lauric Acid is found in coconuts. Food experiments could potentially be carried out to examine if coconut consumption decrease the symptoms of Alzheimer's.

Figure 1 :

 1 Figure 1: Comparison between a healthy brain and Alzheimer's brain. Adapted from Mayo Clinic News Network by Liza Torborg, 2016

Figure 2 :

 2 Figure 2: Symptoms of Alzheimer's disease. Adapted from MedIndia

Figure 4 :

 4 Figure 4: VLDLR deficiency leading to Alzheimer. Made by Siya Goel using GEO2RFurther, string-db, a bio-informatics tool was used to examine a connection between the three different genes -VLDLR, LRP8, and DAB1. The result suggests that DAB1 binds to the LRP8 and VLDLR protein on their cadherin tails and issues tyrosine phosphorylation of the DAB1 protein. This then sends signals to maintain neuron function in human body (see figure5).

Figure 5 :

 5 Figure 5: Connection between VLDLR, LRP8, and DAB1. Made by Siya Goel using STRING.

Graph 3 :

 3 Through results given by the FRET microscope, the average intensity of fluorescence was based on the concentration of protein. The intensity of DAB1 and Lauric Acid is higher than the intensity of just the DAB1 protein. THe results shown are also statistically different due to the fact that the SEM error bars do not overlap. Made by Siya Goel using Excel.

Acknowledgements:

I would like to thank Dr. Clark Gedney for providing me the resources and materials needed for this project. I would also like to thank Ms. Brittany Croy for teaching me key assets needed for this project and sponsoring me for JSHS.