text:
# 1 Introduction & Relational Model

## Data Models

### Hierarchical Model

• one-to-many relationships
• tree

### Network Model

• many numbers of many-to-many relationships
• introduced pointers
• acyclic directed graph
• records (vertices) and sets (edges)
• several items with the same kind of records will use linked lists

### Relational Model

• many-to-many relationships
• decomposing many-to-many relationship into two one-to-many relationships using intersection record

## Relational Model

### General Terminology

• tuple: a row in a table (usually considered to be unordered and non-identical)
• relation: a two-dimensional table of data
• attribute: a column in a table (usually considered as unordered)
• domain: the set of permissible values for a given attribute
• schema: $R(A_1, A_2, \cdots, A_n)$ consisting of the name of the relation followed by its attributes
• component: an atomic single value of an attribute (structure NG)
• instance of a schema: the current contents of a relation

### Keys in Relational Model

• primary key: one key that uniquely identify tuples in the relation
• secondary key: any key which is not primary
• superkey: any set of attributes which satisfies uniqueness condition
• key: a minimal superkey, no proper subset of the key is a superkey

## Relational Algebra

### ordinary set operations

#### projection

the relation obtained by selecting attributes from the relation and remove others

#### selection

the set of tuples in R which satisfy the condition C

#### join

Let $R(A_1, A_2, \cdots, A_m, B_1, B_2, \cdots, B_n)$ and $S(A_1, A_2, \cdots, A_m, C_1, C_2, \cdots, C_k)$ be two relations that has some attributes in common.

Then the natural join of R and S $R \Join S$ is:

The theta join of R and S on condition C $R \Join_C S$ is: the set of elements of the cartesian product which satisfy the join condition C.

#### renaming

rename R to S, with attributes rename to $A_1, A_2, \cdots, A_n$.

### bags

A bag is similar to a set, but allows multiple occurrences of an element.

#### duplicate elimination

If R is a relation then $\delta (R)$ denotes duplicate elimination.

#### aggregation

operators: COUNT, SUM, MAX, MIN, AVG

Example: MAX(width)(Artworks)

#### grouping

If R is a relation, then an expression using grouping is:

Example:

#### extended projection

If R is a relation, then an expression using extended projection is:

Example:

#### sorting

Ordered by primary sort key then secondary sort key and so on.

#### outerjoin

Sometimes there is no matching tuple for some tuples and they got disappeared in the joined relation. So outerjoin would keep them leave the unmatched part a null value $\perp$.

For R and S:

• in left outerjoin, only dangling tuples from R are added
• in right outerjoin, only dangling tuples from S are added

You could also define a outer theta-join using the same way as before.


text:
# Seconds, anyone? .....

Geometry Level 4

A concentric circle, (radius $$\dfrac{2}{\sqrt{3}}$$), and square, (side length $$2$$), have a region of overlap with an area of

$$\dfrac{a}{b} \sqrt{b} + \dfrac{a}{c} \pi$$,

where $$a$$ is coprime with both $$b$$ and $$c$$. Find $$a + b + c$$.

×

format: plain


text:
CoCalc Public Fileswww / talks / padic_heights / e2heights / cycloht.tex
Author: William A. Stein
1\documentclass[11pt]{article}
2\bibliographystyle{amsalpha}
3\include{macros}
4\title{Computing $p$-adic Sigma Functions and Cyclotomic Heights}
5\author{Barry Mazur, William Stein, John Tate}
6\begin{document}
7\maketitle
8
9\section{Introduction}
10Fix an elliptic curve $E$ over $\Q$, given by a minimal
11Weierstrass equation
12\begin{equation}\label{eqn:weq}
13 y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
14\end{equation}
15Suppose $p$ is a prime
16of good ordinary reduction for $E$, so $$p \nmid N_E \cdot a_{E,p},$$
17where $N_E$ is the conductor of $E$ and $a_{E,p}=p+1-\#E(\F_p)$.
18
19\section{The Canonical $p$-adic sigma function}
20Let
21\begin{equation}\label{eqn:sigma}
22  \sigma(t) = t + s_1t^2 + s_2 t^3 + s_3 t^4 + \cdots
23       \in t (1+t\Z_p[[t]])
24\end{equation}
25be the canonical $p$-adic $\sigma$-function attached to $E$
26as in \cite{mazur-tate:sigma}.
27
28\subsection{Computing $\sigma(t)$}
29
30[[Describe the resultin algorithm here -- method to compute $\sigma$.]]
31
32
33Theorem 3.1 of \cite{mazur-tate:sigma} asserts that if $m\in \Z$ and
34$Q\in E^f(\overline{R})$, then
35\begin{equation}\label{eqn:sigma_mq}
36\sigma(m Q) = \sigma(Q)^{m^2} \cdot f_m(Q).
37\end{equation}
38We use this relation
39theorem to determine $s_1$ and $s_2$ in terms of the formal
40endomorphism $[p]$ and the $p$th division polynomial.  [[Idea -- maybe
41we could just use $m=2$ instead of $m=p$?!  It's probably easier to
42compute $[2]$ and $f_2$ than to compute $[p]$ and $f_p$; if not $m=2$,
43then $m=3$?]]  Then we use knowledge of $s_1$ and $s_2$ and the
44differential equation satisfied by $\sigma$ to determine the constant
45$c\in\Z_p$ such that $x(t) - c$ is the canonical eks'' (pronounced
46like the letter x'') function of
47\cite{mazur-tate:sigma}.  Finally, given $x(t) - c$, we find the
48unique series $\sigma(t) \in t(1+t\Z_p[[t]])$
49that satisfies the differential equation
50\begin{equation}\label{eqn:de1}
51  x(t) - c = -\frac{d}{\omega}\left(\frac{1}{\sigma}
52       \frac{d\sigma}{\omega}\right)
53\end{equation}
54of \cite[pg.~665]{mazur-tate:sigma}.
55
56Let $\omega = dx/(2y+a_1x + a_3)$ be the invariant differential
57on $E$.  Inverting the power series in equation 15 of
58\cite{tate:arithmetic}, we have
59$$60 \frac{dt}{\omega} = 611 - a_1 t - a_2 t^2 - 2a_3 t^3 + (-2 a_1 a_3 - 2 a_4)t^4 + \cdots 62 = \sum b_n t^n, 63$$
64where the $a_i$'s are as in (\ref{eqn:weq}).
65\comment{
66> K<a1,a2,a3,a4> := FieldOfFractions(PolynomialRing(Q,4));
67> R<t> := PowerSeriesRing(K);
68> f := 1 + a1*t + (a1^2+a2)*t^2 + (a1^3+2*a1*a2+2*a3)*t^3 + (a1^4+3*a1^2*a2+6*a1*a3+a2^2+2*a4)*t^4 + O(t^5);
69> 1/f;
701 - a1*t - a2*t^2 - 2*a3*t^3 + (-2*a1*a3 - 2*a4)*t^4 + O(t^5)
71}
72Using the chain rule, etc., and writing
73$\sigma(t) = \sum_{n\geq 1} s_{n-1} t^n$ (as in (\ref{eqn:sigma})),
74we see that
75the differential equation (\ref{eqn:de1}) written in terms of~$t$
76is
77\begin{equation}\label{eqn:de_big}
78\frac{d}{dt}\left(
79 \frac{\left(\sum_{n\geq 1} n s_{n-1} t^{n-1}\right) \cdot
80       \left(\sum_{n\geq 0} b_n t^n\right)}{\sum_{n\geq 1} s_{n-1} t^n}
81  \right)\cdot \sum_{n \geq 0} b_n t^n  = c - x(t).
82\end{equation}
83Here
84$$85 x(t) = \frac{1}{t^2} - \frac{a_1}{t} - a_2 - a_3 t - (a_4+a_1 a_3)t^2 + \cdots 86$$
87is the formal expansion of $x$ in terms of $t$.
88\comment{
89> K<a1,a2,a3,a4,s1,s2,s3,s4,s5> := FieldOfFractions(PolynomialRing(Q,9));
90> R<t> := LaurentSeriesRing(K);
91> w := 1 - a1*t - a2*t^2 - 2*a3*t^3 + (-2*a1*a3 - 2*a4)*t^4 + O(t^5);
92> sig := t + s1*t^2 + s2*t^3 + s3*t^4 + s4*t^5 + s5*t^6 + O(t^7);
93> sigprime := Derivative(sig);
94> Derivative(((sigprime*w)/sig))*w;
95-t^-2 + a1*t^-1 + -a1*s1 - s1^2 + 2*s2 + (a1^2*s1 + a1*a2 +
96    3*a1*s1^2 - 6*a1*s2 - 2*a2*s1 - 2*a3 + 2*s1^3 - 6*s1*s2 +
97    6*s3)*t + (-2*a1^2*s1^2 + 4*a1^2*s2 + 3*a1*a2*s1 - 5*a1*s1^3
98    + 15*a1*s1*s2 - 15*a1*s3 + a2^2 + 4*a2*s1^2 - 8*a2*s2 -
99    6*a3*s1 - 4*a4 - 3*s1^4 + 12*s1^2*s2 - 12*s1*s3 - 6*s2^2 +
100    12*s4)*t^2 + O(t^3)
101}
102Expanding the left hand of (\ref{eqn:de_big})
103we see that
104\begin{align*}
105&-\frac{1}{t^2} + \frac{a_1}{t} + (-a_1 s_1 - s_1^2 + 2s_2)
106      +\\
107&\quad (a_1^2 s_1 + a_1 a_2 +
108    3 a_1 s_1^2 - 6 a_1 s_2 - 2 a_2 s_1 - 2 a_3 + 2 s_1^3 - 6 s_1 s_2 +
109    6 s_3) t +\\
110&\quad (-2 a_1^2 s_1^2 + 4 a_1^2 s_2 + 3 a_1 a_2 s_1 - 5 a_1 s_1^3
111    + 15 a_1 s_1 s_2 - 15 a_1 s_3 + a_2^2 + 4 a_2 s_1^2 - \\
112&\qquad8 a_2 s_2 -
113    6 a_3 s_1 - 4 a4 - 3 s_1^4 + 12 s_1^2 s_2 - 12 s_1 s_3 - 6 s_2^2 +
114    12 s_4) t^2 + \cdots
115 \\
116&=
117c - \left(\frac{1}{t^2} - \frac{a_1}{t} + - a_2 - a_3 t - (a_4+a_1 a_3)t^2 + \cdots\right).
118\end{align*}
119Equating constant coefficients, we see that
120$$121 c = a_2 - a_1 s_1 - s_1^2 + 2s_2. 122$$
123We will not use the higher terms yet...
124
125Next we use the relation (\ref{eqn:sigma_mq}) to compute
126$s_1$ and $s_2$ in terms of $[p](t)$ and $f_p(t)$.
127Write the formal multiplication by $p$ as
128$$129 [p](t) = pt + r_2 t^2 + r_3 t^3 + \cdots \in \Z_p[[t]] 130$$
131and the formal division polynomial'' $f_p(t)$ as
132$$133 f_p(t) = pt^{1-p^2} + d_1 t^{2-p^2} + d_2 t^{3-p^2} + \cdots 134 \in (\Z_p[[t]])[1/t]. %\left[\frac{1}{t}\right]. 135$$
136The identity (\ref{eqn:sigma_mq})
137is
138$$139\sum_{n\geq 1} s_{n-1}\cdot 140\left(pt + r_2 t^2 + r_3 t^3 + \cdots \right)^n 141 =\qquad\qquad\qquad\qquad$$
142$$\qquad\qquad\qquad\left(\sum_{n \geq 1} s_{n-1} t^n\right)^{p^2} \cdot 143 \left(pt^{1-p^2} + d_1 t^{2-p^2} + d_2t^{3-p^2} + \cdots\right). 144$$
145Multiply both sides by $t^{p^2-1}$, so that both sides are
146in $\Z_p[[t]]$.
147$$148\sum_{n\geq 1} s_{n-1}\cdot 149\left(pt + r_2 t^2 + r_3 t^3 + \cdots \right)^n\cdot t^{p^2-1} 150 =\qquad\qquad\qquad\qquad$$
151$$\qquad\qquad\qquad\left(\sum_{n \geq 1} s_{n-1} t^n\right)^{p^2} \cdot 152 \left(p + d_1 t + d_2 t^2 + \cdots\right). 153$$
154
155
156[[Since we have this for all $m$, I'm going to replace $p$ by $2$ and
157see what happens.  Also, $f_2(t) = 2t^{1-2^2} + \cdots$ and
158$[2](t)=2t+r_2t^2+\cdots$.]]
159$$160t^3 \cdot \sum_{n\geq 1} s_{n-1}\cdot 161\left(2t + r_2 t^2 + r_3 t^3 + \cdots \right)^n 162 =\qquad\qquad\qquad\qquad$$
163$$\qquad\qquad\qquad\left(\sum_{n \geq 1} s_{n-1} t^n\right)^{4} \cdot 164 \left(2 + d_1 t + d_2 t^2 + \cdots\right). 165$$
166Expanding both sides we get
167$$168 2 t^4 + (4 s_1 + r_2) t^5 + (4 s_1 r_2 + 8 s_2 + r_3) t^6 + \cdots = 169\qquad\qquad\qquad 170$$
171$$172\qquad\qquad\qquad 1732 t^4 + (8 s_1 + d_1) t^5 + (12 s_1^2 + 4 s_1 d_1 + 8 s_2 + d_2) t^6 + \cdots. 174$$
175Equating coefficients of $t^5$ we see that
176$$177 4 s_1 + r_2 = 8 s_1 + d_1 178$$
179so
180$$181 s_1 = \frac{r_2 - d_1}{4}. 182$$
183Equating coefficients of $t^6$ shows that
184$$1854 s_1 r_2 + 8 s_2 + r_3 = 186 12 s_1^2 + 4 s_1 d_1 + 8 s_2 + d_2, 187$$
188so we get NO INFORMATION ABOUT $s_2$ from this.
189
190[[Observation: Actually one could compute $\sigma$ directly from
191equating coefficients!!!  See the commented MAGMA session below, where
192it appears to me that one can simply read off the other coefficients
193of $\sigma$ from (\ref{eqn:sigma_ma}) with $m=2$.]]
194
195However, we do get information from the coefficient of $t^7$.
196Taking differences, we have that
197\begin{verbatim}
1980 = (-4*s1 - d1 + r2)*t^5 + (-12*s1^2 - 4*s1*d1 + 4*s1*r2 - d2 +
199    r3)*t^6 + (-8*s1^3 - 6*s1^2*d1 - 24*s1*s2 - 4*s1*d2 + s1*r2^2
200    + 4*s1*r3 - 4*s2*d1 + 12*s2*r2 + 8*s3 - d3 + r4)*t^7 +
201    (-2*s1^4 - 4*s1^3*d1 - 24*s1^2*s2 - 6*s1^2*d2 - 12*s1*s2*d1 -
202    24*s1*s3 - 4*s1*d3 + 2*s1*r2*r3 + 4*s1*r4 - 12*s2^2 - 4*s2*d2
203    + 6*s2*r2^2 + 12*s2*r3 - 4*s3*d1 + 32*s3*r2 + 24*s4 - d4 +
204    r5)*t^8 + O(t^9)
205\end{verbatim}
206Thus
207$$208-8 s_1^3 - 6 s_1^2 d_1 - 24 s_1 s_2 - 4 s_1 d_2 + s_1 r_2^2 209 + 4 s_1 r_3 - 4 s_2 d_1 + 12 s_2 r_2 + 8 s_3 - d_3 + r_4 = 0. 210$$
211Unfortunately we have both $s_1$ and $s_3$ together
212in this formula, so this doesn't determine $s_2$.
213Using $m=-1$ we get
214\begin{verbatim}
2150 = (2*s1 - d1 + r2)*t^2 + (-s1*d1 - 2*s1*r2 - d2 + r3)*t^3 + (-s1*d2
216    + s1*r2^2 - 2*s1*r3 - s2*d1 + 3*s2*r2 + 2*s3 - d3 + r4)*t^4 +
217    (-s1*d3 + 2*s1*r2*r3 - 2*s1*r4 - s2*d2 - 3*s2*r2^2 + 3*s2*r3
218    - s3*d1 - 4*s3*r2 - d4 + r5)*t^5 + O(t^6).
219\end{verbatim}
220The coefficient of $t^4$ gives a different linear equation for $s_2$
221and $s_3$, and probably taking two of these together we'll be able to
222solve for both.
223
224For $m=3$ it's:
225\begin{verbatim}
226(-18*s1 - d1 + r2)*t^10 + (-108*s1^2 - 9*s1*d1 + 6*s1*r2 - d2 +
227    r3)*t^11 + (-252*s1^3 - 36*s1^2*d1 - 216*s1*s2 - 9*s1*d2 +
228    s1*r2^2 + 6*s1*r3 - 9*s2*d1 + 27*s2*r2 + 54*s3 - d3 +
229    r4)*t^12 + (-378*s1^4 - 84*s1^3*d1 - 756*s1^2*s2 - 36*s1^2*d2
230    - 72*s1*s2*d1 - 216*s1*s3 - 9*s1*d3 + 2*s1*r2*r3 + 6*s1*r4 -
231    108*s2^2 - 9*s2*d2 + 9*s2*r2^2 + 27*s2*r3 - 9*s3*d1 +
232    108*s3*r2 + 216*s4 - d4 + r5)*t^13 + O(t^14)
233\end{verbatim}
234
235\subsection{Computing formal multiplication by $p$}
236
237\subsection{Computing division polynomials}
238
239\section{The $p$-adic cyclotomic height}
240Let $E$ and $p$ be as in the introduction.
241
242Suppose $Q \in E(\Q)$ is a point that reduces to the point at infinity
243modulo $p$, and lies in the identity component modulo every bad prime.
244For example, if $M$ is the least common multiple of $\#E(\F_p)$ and
245the Tamagawa numbers $c_\ell$ of $E$ at bad primes, then every element
246of $M E(\Q)$ satisfies our hypothesis on $Q$.
247Since ...??, we can write
248$$249 Q=(x,y)=\left(\frac{a}{e^2},\frac{b}{e^3}\right), 250$$
251with $a,b,e\in\Z$ and $\gcd(a,e)=1$.  Let $t=-x/y$. Then the $p$-adic
252cyclotomic height $h_p(Q)$ is
253\begin{equation}\label{eqn:hpq}
254   h_p(Q) = \log_p\left(\frac{\sigma(t)}{e}\right).
255 \end{equation}
256[[I think exactly this formula appears on the upper right page of a preprint
257Tate was holding yesterday, which perhaps I don't have a copy.]]
258Here
259$$260 \log_p : \Q_p^* \to \Q_p 261$$
262is the the unique extension of the usual $p$-adic logarithm
263on $1+p\Z_p$ to a homomorphism from the multiplicative group
264of $\Q_p^*$ to the additive group of $\Q_p$.  Thus if $au \in \Q_p^*$
265with $u\in 1 + p\Z_p$, then $\log_p(au) = \log_p(u)$, and
266one computes $\log_p(u)$ using the formula
267$$268 \log_p(1-x) = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots\right) 269$$
270with $x=1-u$.
271[[Again, there's a discussion of this in a paper Tate was holding,
272that I don't seem to have.]]
273
274The cyclotomic height $h_p(Q)$ extends uniquely to a $\Z_p$-valued
275function on $E(\Q)$ that satisfies $h_p(nQ) = n^2 Q$ for all
276integers $n$.   If $M$ is as above, then
277we can compute $h_p(P)$ for any $P\in E(\Q)$ by computing
278$$279 h_p(P) = \frac{h_p([M]P)}{M^2} = \frac{1}{M^2}\cdot 280\log_p\left(\frac{\sigma(t)}{e}\right), 281$$
282where $t=-x/y$ and $e$ are for $Q=[M]P=(x,y)=(a/e^2,b/e^3)$.
283
284
285
286\bibliography{biblio}
287\end{document}
288
289
290Using $m=2$ still,
291taking the difference of the two sides and setting the
292$s_{2n}=0$, since $\sigma$ is odd, we see that
293\begin{verbatim}
2940 = (-d1 + r2)*t^5 + (-d2 + r3)*t^6 + (-4*s2*d1 + 12*s2*r2 - d3 +
295    r4)*t^7 + (-12*s2^2 - 4*s2*d2 + 6*s2*r2^2 + 12*s2*r3 + 24*s4
296    - d4 + r5)*t^8 + O(t^9)
297\end{verbatim}
298\comment{
299> K<s1,s2,s3,s4,s5,d1,d2,d3,d4,d5,r2,r3,r4,r5> := FieldOfFractions(PolynomialRing(Q,14));
300> R<t> := LaurentSeriesRing(K);
301> sig := t + 0*t^2 + s2*t^3 + 0*t^4 + s4*t^5 + 0*t^6 + O(t^7);
302> m:=2; mult := m*t + r2*t^2+r3*t^3+r4*t^4+r5*t^5 + O(t^6); divpol := m+d1*t+d2*t^2+d3*t^3+d4*t^4+d5*t^5+O(t^6);t^(m^2-1)*Evaluate(sig,mult) - sig^(m^2)*divpol;
303(-d1 + r2)*t^5 + (-d2 + r3)*t^6 + (-4*s2*d1 + 12*s2*r2 - d3 +
304    r4)*t^7 + (-12*s2^2 - 4*s2*d2 + 6*s2*r2^2 + 12*s2*r3 + 24*s4
305    - d4 + r5)*t^8 + O(t^9)
306}
307The coefficient of $t^7$ determines $s_2$, so long as
308$r_2 - 3d_1\neq 0$.  Since $r_2 = d_1$, we're OK, at least as
309long as $d_1\neq 0$.
310So we get
311$$312 s_2 = \frac{d_3-r_4}{8d_1}. 313$$
314Now that we know $s_2$, the coefficient of $t^8$ determines
315$s_4$.  Etc., and it seems likely we can easily determine $\sigma(t)$
316from the formal $[2]$ and $f_2(t)$, without having to mess with
317any differential equations.   Surprising if right.
318
319Here is a relevant MAGMA session, which suggests that the
320information coming from (\ref{eqn:sigma_ma}) for different $m$ is
321equivalent:
322
323\begin{verbatim}
324> K<s1,s2,s3,s4,s5,d1,d2,d3,d4,d5,r2,r3,r4,r5> :=
325  FieldOfFractions(PolynomialRing(Q,14));
326> R<t> := LaurentSeriesRing(K);
327> sig := t + 0*t^2 + s2*t^3 + 0*t^4 + s4*t^5 + 0*t^6 + O(t^7);
328> m:=2; mult := m*t + r2*t^2+r3*t^3+r4*t^4+r5*t^5 + O(t^6);
329  divpol := m+d1*t+d2*t^2+d3*t^3+d4*t^4+d5*t^5+O(t^6);
330  t^(m^2-1)*Evaluate(sig,mult) - sig^(m^2)*divpol;
331(-d1 + r2)*t^5 + (-d2 + r3)*t^6 + (-4*s2*d1 + 12*s2*r2 - d3 +
332    r4)*t^7 + (-12*s2^2 - 4*s2*d2 + 6*s2*r2^2 + 12*s2*r3 + 24*s4
333    - d4 + r5)*t^8 + O(t^9)
334> m:=3; mult := m*t + r2*t^2+r3*t^3+r4*t^4+r5*t^5 + O(t^6);
335  divpol := m+d1*t+d2*t^2+d3*t^3+d4*t^4+d5*t^5+O(t^6);
336  t^(m^2-1)*Evaluate(sig,mult) - sig^(m^2)*divpol;
337(-d1 + r2)*t^10 + (-d2 + r3)*t^11 + (-9*s2*d1 + 27*s2*r2 - d3 +
338    r4)*t^12 + (-108*s2^2 - 9*s2*d2 + 9*s2*r2^2 + 27*s2*r3 +
339    216*s4 - d4 + r5)*t^13 + O(t^14)
340> m:=5; mult := m*t + r2*t^2+r3*t^3+r4*t^4+r5*t^5 + O(t^6);
341  divpol := m+d1*t+d2*t^2+d3*t^3+d4*t^4+d5*t^5+O(t^6);
342  t^(m^2-1)*Evaluate(sig,mult) - sig^(m^2)*divpol;
343  (-d1 + r2)*t^26 + (-d2 + r3)*t^27 + (-25*s2*d1 + 75*s2*r2 - d3 +
344    r4)*t^28 + (-1500*s2^2 - 25*s2*d2 + 15*s2*r2^2 + 75*s2*r3 +
345    3000*s4 - d4 + r5)*t^29 + O(t^30)
346> m:=1; mult := m*t + r2*t^2+r3*t^3+r4*t^4+r5*t^5 + O(t^6);
347  divpol := m+d1*t+d2*t^2+d3*t^3+d4*t^4+d5*t^5+O(t^6);
348  t^(m^2-1)*Evaluate(sig,mult) - sig^(m^2)*divpol;
349(-d1 + r2)*t^2 + (-d2 + r3)*t^3 + (-s2*d1 + 3*s2*r2 - d3 +
350    r4)*t^4 + (-s2*d2 + 3*s2*r2^2 + 3*s2*r3 - d4 + r5)*t^5 +
351    O(t^6)
352\end{verbatim}
353Thankfully, we can't use $m=1$ since $d_1=0$ there.
354


text:
# Bitcoin cloud hashing review

Posted by on Jan 7, 2014 in Bitcoin, HeadLines | 1 comment

# Bitcoin cloud hashing review

Over the last few weeks, and of course, because of the latest frenzy on bitcoins due to its price, I decided to try to get into it a bit. People will comment it’s a bubble, other it’s not, but the fact is that it has a very good momentum that’s likely to continue for quite a while. Of course nobody knows what will happen, that’s the only sure fact, but I rather have some and lose some money that not have any and missed a good opportunity to earn some more.

Of course, I kind of knew that although buying bitcoins at around $1000 might not be the best idea at that point (although it reached$1200 later on but it’s now around $650), the first thing I started to look into was bitcoin mining. I won’t go into details of what that is or how it works because that’s not what I was planning on writing. I looked for mining rigs, its setup, its configuration, etc. and because I am not based in the US, purchasing and bringing those equipments here to make it profitable was not an option. From there I resolved to look for people offering hosted solutions, and of course, I found one: Cloudhashing.com. They were offering, last November, a platinum contract of 20GH/s for two years for$950. Without thinking much about it I gave it a try. Of course bitcoin mining is not a sure deal but they point you to an estimator that supposedly gives you a good estimate of what you might get. See below what it says now for 20GH/s:

So this says, that at the current difficulty you can get around 0.0085BTC per day. So in two years:

0.00853652=6.205

After trying the first 20GH/s and seeing that I was getting to around that estimate (it was more back in November), I decided to purchase an additional one, so I am now with 40GH/s for two years.

Cloudhashing.com also takes a 10% management fee and, depending on how you have it configured some more for a reinvestment program I am not currently using, so following that estimate I would get:

6.2050.9=5.5845

So that’s about 5.5BTC for $950 per contract, not so bad! Although…. read below! So what does “the current difficulty” means? In a nutshell, the difficulty is a parameter set by the bitcoin network that makes it more (or less) difficult to mine bitcoins. The bigger this value is, the less amount of bitcoins you will get for the same amount of GH/s I researched a bit on how the difficulty was increased in the past few months: This image is a bit scary. It makes sense though, that as bitcoins reached an all extreme high of over$1000 a lot of people (like me) wanted to get a piece of it and started mining, increasing the bitcoin difficulty exponentially.

By looking at the past few months, you can see that on average you got an increase of roughly 50% monthly, that’s a lot!

You can also check BlockChain’s nice difficulty interactive chart.

Now let’s look at another estimate, from cex.io, which takes into the estimate difficulty increases:

This picture, shows your approximate earnings over one year assuming that the difficulty increases in average 50% per month. Although this is a lot, is at least what’s been happening the past few months.

So with this estimate, it says that you will earn approximately 0.73BTC in 1 year, and although the estimate is for one year, you will see that the last months of the estimation you hardly gain any bitcoin, so it’s likely that the rest of the two years contract the 20GH/s will be rendered useless.

## My conclusions

At the current prices of the contracts, with the current average monthly difficulty increase, purchasing any of the available contracts I found below is really a bad investment, and bear in mind, as I said, I have two contracts of 20GH/S for two years! You are better off just purchasing BTCs at the current price, simpler, easier and your only risk is its value, at least you don’t have to worry on the difficulty.

Below is a small review of the services I checked and their current prices for cloud mining. Don’t take my word for it, just do the math, I think mine are pretty accurate though, and also, again, these are all based on estimates.

Cloudhashing.com

This was my first find and the one I purchased. One good thing about this one is that you are allowed to pay with credit card. They recently revamped their homepage and update their prices scheme, as the time of writing this a 50GH/s/year contract costs $3999. They also charge a 10% management fee taken out from your BTC payments. Running the cex.io estimator mentioned above with the 50% monthly increase of difficulty it gives an approximate earning of 1.8BTC over one year. The current USD/BTC price is around$650. With $3999 you currently buy 6.15BTC. bit-mining.co.com I found this one recently, their prices schemes are in BTC, so it’s even easier to do the maths. They are currently selling 10GH/s at 0.5 BTC for life, with 4 years guaranteed. Don’t be too excited about the timeless contract, after 1 year and for sure after 2, with the current increases of difficulty, those GH/s will be close to having nothing. They charge 8% fee for the first year and 5% for the rest of the years. Running the cex.io estimator mentioned above with the 50% monthly it gives an approximate earning of 0.37 BTC over one year. So you are basically are probably going to get less than what you are paying to mine. cex.io This one is a bit different, is more of an exchange site. You can both buy and sell GH/s that you own, and while you are owning them, they work for you as they mine the same as above. The only difference is that if the price of the GH/s goes up, you might make a profit of it. If you think about it for a sec, although this will be more driven by offer and demand, the fact that the GH/s gets more useless as difficulty increases, it’s likely that they will loose price as well. Now, without considering any profit of the price fluctuation of the GH/s you purchased and just taking into account the mining power of it, the current price for 1GH/s is 0.069BTC. Say you bought 10 for 0.69BTC. Their current management fee is around 3.95%, so running their own estimator mentioned above with these values for 10GH/s I will get 0.34BTC, again, not a great investment. butterfly labs This is probably the “Apple Inc.” of bitcoin mining. They sell different equipments for running your own mining hardware and they are preparing a hosted solution at$10.83 per GH/s/year. I said preparing, so this is on pre-order. Their mentioned schedule is that this will launch towards the end of this year and they will take each order on a first ordered first serve basis, so you might have to wait until you get it if you buy it right now. Also, the price might change once they lunch it publicly. So for 10GH/s you will pay $108.30. They don’t say anything of any management fee as of now, so I will assume 0%. I am going to estimate the values with the current price and as if you would start mining immediately. Running those values on the same estimator as the others you will approximately get 0.35BTC after one year. At todays price of$650BTC that’s $227.50, so at least here we are able to see some profit from what would have been purchasing those BTC right now. One final note, I am not saying mining bitcoins is not profitable, purchasing your own equimentment at a good price, running it properly with good maintenance, cheap bandwidth and OK electricity, it is probably profitable. But if you, like me, are unable to do that because of where you live, money or time, then you are better off purchasing BTC directly assuming you want to invest on it or wait until any of these services have a more reasonable price Taken from Here ### One Comment 1. 9-10-2014 Cloudhashing is awful. I purchased a$999 contract in January 2014. So far my earnings are 0.18 BTC (or $81 currently). I wrote to them a few days ago asking why my ROI is so exceptionally poor. Have not heard back. 8 months in, I’ve made$80. After my 1 year contract is up, at this rate, I will have ‘made’ $120. A net loss of$880.

I don’t work for another company, etc.. just very disappointed on the ROI and feel they have cheated me. Their advertising states that their mining cpus generate many more BTC than I am seeing in my account.

Do not purchase a contract from Cloudhashing.


text:
Lemma 29.26.2. A flat closed immersion of finite presentation is the open immersion of an open and closed subscheme.

Proof. The affine case is Algebra, Lemma 10.108.5. In general the lemma follows by covering $X$ by affines. Details omitted. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Can't see added bibliography on latex file, what is wrong?

I am not able to include the bib file I have (file is cool.bib), it works in another latex project I have, but on this one, something is just not working, I am using TexMaker for it. Please help me understand why it doesn't show anything, and why if I use cite it shows [?]. thanks

\documentclass[conference]{IEEEtran}
\IEEEoverridecommandlockouts
% The preceding line is only needed to identify funding in the first footnote. If that is unneeded, please comment it out.
\usepackage{cite}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{algorithmic}
\usepackage{graphicx}
\usepackage{textcomp}
\usepackage{xcolor}
\usepackage[super]{natbib}
\bibliographystyle{IEEEtranN}
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\begin{document}

\title{name of paper\\
\thanks{Identify applicable funding agency here. If none, delete this.}
}

\author{\IEEEauthorblockN{other guy}
\IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
\textit{name of organization (of Aff.)}\\
City, Country \\
\and
\IEEEauthorblockN{name me}
\IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
\textit{name of organization (of Aff.)}\\
City, Country \\
}

\maketitle
\nocite{*}

\begin{abstract}
abstract
\end{abstract}

\section{Introduction}
text bla bla \cite{graph} bla.

\section*{Acknowledgment}

template

\section*{Literature}
\begingroup
\renewcommand{\section}[2]{}%
\bibliography{cool}

\endgroup

\end{document}

blg

This is BibTeX, Version 0.99d
Capacity: max_strings=200000, hash_size=200000, hash_prime=170003
The top-level auxiliary file: paper4.aux
Reallocating 'name_of_file' (item size: 1) to 5 items.
I found no \bibstyle command---while reading file paper4.aux
You've used 2 entries,
0 wiz_defined-function locations,
86 strings with 497 characters,
and the built_in function-call counts, 0 in all, are:
= -- 0
> -- 0
< -- 0
+ -- 0
- -- 0
* -- 0
:= -- 0
add.period$-- 0 call.type$ -- 0
change.case$-- 0 chr.to.int$ -- 0
cite$-- 0 duplicate$ -- 0
empty$-- 0 format.name$ -- 0
if$-- 0 int.to.chr$ -- 0
int.to.str$-- 0 missing$ -- 0
newline$-- 0 num.names$ -- 0
pop$-- 0 preamble$ -- 0
purify$-- 0 quote$ -- 0
skip$-- 0 stack$ -- 0
substring$-- 0 swap$ -- 0
text.length$-- 0 text.prefix$ -- 0
top$-- 0 type$ -- 0
warning$-- 0 while$ -- 0
width$-- 0 write$ -- 0
(There was 1 error message)
• have you run bibtex to generate the bibliography? Jul 17 at 17:05
• @DavidCarlisle if you mean on the build, I am doing quickbuild: pdflatex->biblatex->pdflatex x2 -> view pdf Jul 17 at 17:27
• that is your error then, your document is not using biblatex, you need pdflatex-bibtex-pdflatex Jul 17 at 17:28
• @DavidCarlisle I think that it is the same since it Texmaker it says: "PdfLatex + Bib(la)tex + PdfLatex(2x) + View Pdf" which means it probably accepts both Jul 17 at 17:38
• if bibtex failed to produce a bibliography it will leave errors in the bibtex log (.blg) so you could post that. Jul 17 at 20:07

I'm not sure why doing the \begingroup and \endgroup business. You just want to modify \refname.

\begin{filecontents*}{\jobname.bib}
@article{graph,
author={A. Uthor},
title={Title},
journal={Journal},
year={2021},
}
\end{filecontents*}

\documentclass[conference]{IEEEtran}
\IEEEoverridecommandlockouts
% The preceding line is only needed to identify funding in the first footnote. If that is unneeded, please comment it out.
\usepackage{cite}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{algorithmic}
\usepackage{graphicx}
\usepackage{textcomp}
\usepackage{xcolor}
\usepackage[super]{natbib}

\bibliographystyle{IEEEtranN}
\renewcommand{\refname}{Literature}

\begin{document}

\title{name of paper\\
\thanks{Identify applicable funding agency here. If none, delete this.}
}

\author{\IEEEauthorblockN{other guy}
\IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
\textit{name of organization (of Aff.)}\\
City, Country \\
\and
\IEEEauthorblockN{name me}
\IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
\textit{name of organization (of Aff.)}\\
City, Country \\
}

\maketitle
\nocite{*}

\begin{abstract}
abstract
\end{abstract}

\section{Introduction}
text bla bla \cite{graph} bla.

\section*{Acknowledgment}

template

\bibliography{\jobname}

\end{document}

I used filecontents* just to make the example self-contained, you can continue to use your .bib file.

You can notice that I removed the lines

\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}

They're completely useless (and wrong).


text:
# Can baking soda or citric acid help clean a pot with burned-in milk rice?

The other day I was cooking milk rice and forgot about it so it burned a little on the bottom. This lower layer was very hard to clean. A lot of websites recommend boiling water in the pot with baking soda (Sodium bicarbonate) or citric acid to dissolve the burned residue and make it easier to clean the pot. I tried this but it did not help. I doubt there is even a chemical reaction between these reagents that would help in any way.

Is this another myth or is there scientific basis for given recommendation?

• Not sure about acid, but soda does help sometimes, and that's for a reason. – Ivan Neretin Dec 25 '15 at 8:19

The hotter the water, the more likely it will be to remove burnt-on starch. Not only does $\ce{NaHCO3}$ raise the boiling point of the water, but it also helps saponify fats (actually, "washing soda", $\ce{Na2CO3}$ would be better).

However, you still need to do a lot of scrubbing afterwards to remove the carbonized layer.

• 1.) I suppose the carbonized layer is by far the hardest part to remove so this is what sodium bicobonate and acid should work on. 2.) Isn't burnt-on-starch the same as carbonized layer. Your answer implies them to be different, if I understand it correctly. – problemofficer Dec 27 '15 at 6:59
• 1. Use either sodium bicarbonate or acid, not both at once. 2. You can't clean the hard bottom layer off until you remove the partially burnt porridge. 3. No, the alkali will not significantly remove pure baked-on carbon -- use abrasive & elbow grease. – DrMoishe Pippik Dec 27 '15 at 7:18
• Then is there any significant advantage to using NaHCO3 instead of regular dish liquid to solve the fats? If there isn't then the recommendation is non-effective and a myth. Since as you already stated removing the carbonized layer does not become easier with NaHCO3 or acid. – problemofficer Dec 27 '15 at 9:18
• Boiling with liquid detergent will cover the stove in foam. If you prefer, use detergent for dish washing machines, which is alkaline and does not foam. – DrMoishe Pippik Dec 27 '15 at 23:05

Since an abrasive action is required to remove the burnt protein material after a significant amount of baking soda is covered, I suggest using a stiff plastic scraper and pressure to lift layers of the adhered material. The peeling tensions, shear, of a dry stiff material help with removal.

I found that a 1/8 to 1/4 inch thick application of baking soda with a spray of water without stirring or puddling of water encourage the ions to migrate and bond at the metal interface.

After the application is dried, a spray of water will encourage more ions to combined at the metal interface.

We might view this as a way of drying the burnt material like we might dry meat. The tension and bonding at the metal surface is releasing and favoring the burnt protein structure.

So the adhesion of the protein to the material surface is altered by changing the properties of the Burt material to create more stress or tension or contraction by shortening the bonds of the protein and changing the bond strength at metal or ceramic surface.

Adding water relieves the stress. So a dry mechanical scraper acts like a knife lifting encouraging the dry tensile material to pull away from the pan.

Take the wet pan with no liquid in it. Coat the bottom area with thick Baking Soda Spray mist the baking soda with water Let dry for 4 hrs Spray mist with water again Let dry over night Scrape the dry material with a strong force of a hard plastic kitchen scraper blade Persist with strong force from clean surface across material coating.

This applies to stainless steel counters and stoves

• Welcome to ChemSE. Initially I assumed putting three times the same image into the reply was an accident. Reading your answer now leads me to the speculation you either were distracted, stressed or/and tired. If so, stay safe in your doing now, and be welcome to polish your answer in about a day. – Buttonwood Dec 18 '19 at 21:24

Soaking burnt on food with baking soda - I do it all the time! I think boiling the pot is counter productive. Just let it sit covered with water and baking soda - lots of baking soda. It's very cheap. B Putting the pot back on the stove just heats the burnt food more potentially making it even harder to remove. Just soak it - overnight for a serious build up. You still might have to abrasively remove the last of the residue. I just cleaned a Pyrex casserole from dinner that had cheese scalloped potatoes burnt all around the top edge. Soaked it in q larger bowl covered with a water and a generous shake of baking soda. A few hours later it all came off with very little effort.


text:
Article| Volume 130, ISSUE 1, P141-152, July 13, 2007

# Stability and Nuclear Dynamics of the Bicoid Morphogen Gradient

Open Archive

## Summary

Patterning in multicellular organisms results from spatial gradients in morphogen concentration, but the dynamics of these gradients remain largely unexplored. We characterize, through in vivo optical imaging, the development and stability of the Bicoid morphogen gradient in Drosophila embryos that express a Bicoid-eGFP fusion protein. The gradient is established rapidly (∼1 hr after fertilization), with nuclear Bicoid concentration rising and falling during mitosis. Interphase levels result from a rapid equilibrium between Bicoid uptake and removal. Initial interphase concentration in nuclei in successive cycles is constant (±10%), demonstrating a form of gradient stability, but it subsequently decays by approximately $30%$. Both direct photobleaching measurements and indirect estimates of Bicoid-eGFP diffusion constants (D ≤ 1 μm2/s) provide a consistent picture of Bicoid transport on short (∼min) time scales but challenge traditional models of long-range gradient formation. A new model is presented emphasizing the possible role of nuclear dynamics in shaping and scaling the gradient.

## Introduction

The most widely accepted conceptual framework for pattern formation in developing multicellular organisms is the morphogen gradient: a spatially varying concentration of a transcription factor or extracellular signaling molecule across a field of developing cells in which cells acquire information about their spatial location from local measurements of morphogen concentration (
• Wolpert L.
Positional information and the spatial pattern of cellular differentiation.
). Although molecular studies have identified candidate morphogens and demonstrated that cells can respond differently to different concentrations of these molecules (
• Lawrence P.A.
The Making of a Fly: The Genetics of Animal Design.
,
• Driever W.
• Nüsslein-Volhard C.
The Bicoid protein determines position in the Drosophila embryo.
,
• Zecca M.
• Basler K.
• Struhl G.
Sequential organizing activities of engrailed, hedgehog and decapentaplegic in the Drosophila wing.
,
• Ariizumi T.
• Asashima M.
Control of the embryonic body plan by activin during amphibian development.
), the mechanisms that produce the gradients and read out positional information are not understood.
One of the best studied morphogen gradients is the Bicoid (Bcd) gradient in early Drosophila melanogaster embryos (
• Driever W.
• Nüsslein-Volhard C.
A gradient of Bicoid protein in Drosophila embryos.
,
• Struhl G.
• Struhl K.
• Macdonald P.M.
The gradient morphogen Bicoid is a concentration-dependent transcriptional activator.
,
• Ephrussi A.
• St Johnston D.
Seeing Is Believing: The Bicoid Morphogen Gradient Matures.
). Bcd is a maternal anterior determinant in the developing embryo that controls anterior-posterior (AP) patterning. Its mRNA is deposited during oogenesis at the anterior pole of the egg and is translated soon after fertilization. Antibody staining in fixed embryos demonstrates that Bcd concentration decays approximately exponentially with distance from the anterior pole. The exponential shape is consistent with the gradient being formed by a balance of localized synthesis, diffusion, and spatially uniform degradation (
• Crick F.
Diffusion in embryogenesis.
,
• Lacalli T.C.
• Harrison L.G.
From gradient to segments: models for pattern formation in early Drosophila.
), which we will refer to as the SDD model. Important unmeasured parameters in the SDD model are the diffusion coefficient ($D$) of Bcd and its lifetime ($τ$) in the cytoplasm. In general, the model remains untested, and alternatives, such as active transport, have not been ruled out.
The dynamic properties of the Bcd gradient have not previously been characterized. For example, it is unknown how quickly the gradient is established and whether or not its amplitude and shape evolve or fluctuate with time. Characterizing these dynamics is important both for defining the plausibility of mechanisms implied by the SDD model and for defining what positional information actually is available for morphogen readout. In the simplest SDD model implementation, the concentration profile of Bcd reaches a steady state, and this requires some minimum time that depends on the Bcd diffusion coefficient. Is the actual development time of the Bcd gradient consistent with its diffusion coefficient? Similarly, the mechanism of gradient readout might necessarily be more complex than a single time point measurement of concentration if the gradient is found to change shape or fluctuate significantly.
Our understanding of the development and stability of the Bcd gradient has been limited by the use of fixed tissue and antibody staining techniques that provide only static snapshots of the morphogen distribution. In addition, the analysis is complicated by the fact that the gradient arises during a stage when the embryo is undergoing rapid syncytial nuclear mitoses. As expected for a transcription factor, Bcd protein localizes to nuclei during interphase, but during mitosis nuclear envelopes break down, and it is unknown if the gradient is affected. Are events inside the nucleus responding passively to a gradient established in the surrounding cytoplasm, or do intranuclear processes help to shape the gradient itself? Along these lines, it recently has been suggested that Bcd degradation in nuclei contributes to the mechanism for scaling across embryos of different sizes (
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
).
Here we address these issues with a dynamic measurement of the Bcd gradient in single embryos. Fly transformants that encode a fusion gene of bcd and the coding region of the enhanced green fluorescent protein (eGFP) were generated. By observing these transformants with time-lapse two-photon fluorescence microscopy and photobleaching methods, the dynamics of the Bcd gradient were measured and perturbed during the first three hours of embryonic development. We show that the gradient is formed within the first hour after egg fertilization and that it then is stably maintained during blastodermal stages. After each mitotic division, the concentration of Bcd in nuclei at a given position along the AP axis of the egg reaches the same value it had in the previous nuclear cycle with a precision of $10%$, despite changes in nuclear size and density. The nuclear and cytoplasmic concentrations are shown to be in a dynamic equilibrium on the ∼1 min time scale. Direct and indirect measurements of the Bcd diffusion constant are all consistent with D ∼ 0.3 μm2/s, which is much smaller than expected and raises problems for understanding how it is possible for the gradient to be established so rapidly. Finally, we propose a variant of the SDD model, involving nuclear degradation, that captures the Bcd dynamics revealed in our measurements and emphasizes a role for those dynamics in solving the scaling problem.

## Results

### Bcd-GFP Construct: Initial Characterization

To visualize the spatiotemporal dynamics of Bcd concentration we made transgenic Drosophila embryos in which endogenous Bcd was replaced with a fluorescent eGFP-Bcd fusion protein (called Bcd-GFP hereafter). Flies were generated utilizing a transcript coding for eGFP (
• Tsien R.Y.
The green fluorescent protein.
) fused to the N terminus of Bcd. As in previous work with a GFP (rather than eGFP) fusion protein (
• Hazelrigg T.
• Liu N.
• Hong Y.
• Wang S.
GFP expression in Drosophila tissues: time requirements for formation of a fluorescent product.
), the construct contained endogenous bcd 5′ and 3′ UTRs, which are known to mediate anterior localization and translation of bcd mRNA.
Embryos expressing Bcd-GFP demonstrated an intricate spatial and temporal pattern of concentration dynamics that was captured by time-lapse two-photon excitation laser scanning microscopy (
• Denk W.
• Strickler J.H.
• Webb W.W.
Two-photon laser scanning fluorescence microscopy.
,
• Svoboda K.
• Denk W.
• Kleinfeld D.
• Tank D.W.
In vivo dendritic calcium dynamics in neocortical pyramidal neurons.
). A typical image stack of three focal planes from a Bcd-GFP embryo during nuclear cycle 12 is shown in Figure 1A. The fluorescence consisted of two components: bright nuclei and dispersed cytoplasmic fluorescence of lower intensity. The bright nuclei are consistent with previous antibody stainings of Bcd and the fact that Bcd is a transcription factor that should, at some point, be targeted to nuclei. A gradient in fluorescence intensity from anterior to posterior is observed in both the nuclear and cytoplasmic components, which is also consistent with previous work (
• Driever W.
• Nüsslein-Volhard C.
A gradient of Bicoid protein in Drosophila embryos.
; Movie S1).
Fluorescence from Bcd-GFP was first detected in anterior surface nuclei during nuclear cycle 9, approximately 90 min after fertilization. As shown in Figure 1B, during each successive nuclear division 9–14, bright nuclei reappear during interphase in the cortical layer. Qualitatively, these images reveal that in any given perimeter region, the total amount of fluorescence increases with each division. However, the intensity within individual nuclei at a given spatial location appears, qualitatively, to be the same. To provide initial quantification of these effects, at each time point fluorescence was measured within a rectangular region (containing several nuclei and surrounding cytoplasm) that was slid along the embryo perimeter (
• Wieschaus E.
• Leibler S.
Establishment of developmental precision and proportions in the early Drosophila embryo.
). As shown in Figure 1C, a fluorescence gradient that is approximately an exponential function of distance along the embryo can first be measured about one hour after fertilization. The overall intensity increases progressively with development time. In contrast, as shown in the inset to Figure 1C, the intensity measured in a very small box within individual nuclei maintains a fixed profile as a function of distance along the embryo. Both the shape of the spatial decay and the absolute concentration do not appear to change, within measurement error, as a function of nuclear cycle.
A series of control experiments was performed to test the biological relevance of these measurements from two different perspectives. First, we asked the question: Does Bcd-GFP functionally substitute for endogenous Bcd, and do flies expressing Bcd-GFP develop normally? Second, we asked: Does the GFP fluorescence quantitatively represent the distribution of Bcd-GFP, and how does this distribution compare to that of endogenous Bcd?
First, the P[egfp-bcd] transgene completely rescues the headless anterior defect normally observed in embryos from bcd mutant mothers; qualitatively, no developmental defects are observed throughout the entire life cycle. Quantitatively, the position along the AP axis of the cephalic furrow, a strong indicator of the total amount of active Bcd protein, was measured. In embryos with the Bcd-GFP construct in a bcd null mutant background, the furrow location at 32.8 ± 1.3% egg length (mean and standard deviation [SD] over n = 10 embryos) was identical to that of wild-type embryos (33.8 ± 0.7%; n = 12); these results are consistent with previous qualitative estimates of approximately $35%$ (
• Driever W.
• Nüsslein-Volhard C.
The Bicoid protein determines position in the Drosophila embryo.
). Thus, by both qualitative and quantitative criteria, fluorescent egfp-bcd behaves indistinguishably from wild-type bcd in the early embryo, indicating that its localization and translational regulation are conserved and that the protein is folded properly despite the attached eGFP epitope.
Second, to determine the faithfulness with which eGFP fluorescence reports both Bcd-GFP and endogenous Bcd concentration, fixed embryos were colabeled with fluorescent antibodies against Bcd and GFP, and the fluorescence intensities at locations around the perimeter were compared to each other and to endogenous GFP fluorescence. As summarized in the scatter plots in Figure 2, the intensity of one probe always is linearly related to the intensity of a different probe. These results demonstrate that, except for differences in background levels, the fluorescence intensity in each case is proportional to the protein concentration. Further implications of these studies, such as the linearity of antibody staining as a method of quantifying relative protein concentration, are discussed in Supplemental Data.
Together, these control experiments reveal that the biological and physical properties of the Bcd-GFP construct reproduce the properties of the endogenous bcd gene. Thus our time-lapse observations of dynamics in Bcd-GFP-expressing embryos with a bcd null mutant background quantitatively represent wild-type behavior of the Bcd morphogen gradient.

### Constancy of Nuclear Bcd Concentration

The low magnification time-lapse movies in Figure 1 provide initial support for the idea that the gradient in intranuclear Bcd concentration along the embryo develops early and is approximately constant in amplitude and shape over time during nuclear cycles 9–14. This stability is, however, somewhat surprising, since several processes are occurring simultaneously that would be expected to produce local spatial and temporal variations in Bcd concentration: The number of nuclei doubles with each mitotic cycle, the size (diameter) of individual nuclei changes during interphase and between adjacent cycles, and Bcd concentrations must rise and fall as nuclear membranes form and dissolve. To further characterize and understand these expected changes in Bcd concentration and what kind of stability may exist at the local level, individual nuclei were tracked over time at higher magnification.
High-resolution time-lapse images were taken of nuclei in the anterior half of the embryo, with the field of view reduced to 50 × 50 μm2. Figure 3A depicts two nuclei during interphase 12; the same field of view during mitosis 13 is shown in Figure 3B. The concentration profiles of a thin horizontal rectangle across the center of the nuclei in Figure 3A and the corresponding position in Figure 3B, depicted in Figure 3C, clearly show that the cytoplasmic concentration increases during mitosis at the location between the two nuclei, as expected if Bcd-GFP is freed to diffuse into the cytoplasm as the nuclei enter mitosis.
To measure the complete time course of nuclear Bcd-GFP concentration from time-lapse movies during cycles 10–14, a fixed-size region of interest (ROI) within the nuclear membrane boundary of a single nucleus was determined during interphase. The average intensity within this ROI was determined for a sequence of images starting with the cytoplasm before nuclear membrane formation, through interphase, and continuing in the cytoplasm immediately following nuclear envelope breakdown. To continue the time course into the succeeding cycle, the ROI was switched to an area within the newly forming nucleus that was closest to the position of the first nucleus in order to maintain positional constancy along the AP axis of the egg. Figure 3D shows two time courses of nuclear concentration, determined by this procedure, from the same embryo.
These data demonstrate a stereotyped response in which peak intranuclear concentrations occur early in each syncytial nuclear cycle. They further reveal that these peaks are nearly constant between nuclear cycles 10 and 14. This is quantified by plotting, in Figure 3E, the concentration in one cycle against that in the preceding cycle. When monitoring three to four nuclei in each of seven different embryos, in most cases the peak is reproducible to better than $10%$ accuracy. More quantitatively, the ratio of fluorescence intensities from one cycle to the next has a mean of $1.02$ and a standard deviation of just $7.9%$ (n = 77). These results hold over a wide range of absolute concentrations, corresponding to positions from $5%$ to $35%$ of the distance from anterior to posterior pole.
Figure 3D also shows the cytoplasmic Bcd-GFP concentration across nuclear cycles 10–14. As for the nuclear concentration, a highly reproducible pattern between interphase and mitosis is observed with cytoplasmic concentration peaks during mitosis representing the equilibration of Bcd-GFP concentration after release from the nuclei, as shown in Figure 3C.
Our data show a surprising coexistence of highly dynamic behavior and precise constancy. Over the course of a single nuclear cycle, the concentration of Bcd varies systematically over a factor of four at the location corresponding to a single nucleus. When the next cycle starts, however, the nuclear Bcd concentration is restored to the same value with ∼10% accuracy. This reproducibility validates a dynamic version of the positional information hypothesis: From one nuclear cycle to the next, Bcd concentration provides information about the spatial location of the nucleus, and the mapping from position to concentration is invariant across cycles. This stability in the nuclear concentration gradient occurs despite the fact that the total concentration of Bcd in a local cortical region containing multiple nuclei is increasing with each cycle.

### Nucleo-Cytoplasmic Bcd Concentration Equilibrium

Although peak Bcd concentrations are reproduced from one nuclear cycle to the next, closer inspection of the nuclear concentration time courses reveals a highly dynamic pattern, both in relative and in absolute terms. We identify three distinct behaviors of nuclear Bcd-GFP concentration during each mitotic cycle, marked as intervals I (nuclear envelope breakdown), II (refilling), and III (interphase) during nuclear cycle 13 in Figure 4A. Note that during interphase interval III, a concentration drop of approximately $30%$ is observed.
A fundamental question raised by our results is whether the high intranuclear concentration of Bcd results from simple trapping of Bcd or from a dynamic equilibrium of Bcd molecules exchanging between cytoplasm and the nuclei. To address this question, we bleached the fluorescence of Bcd-GFP in single nuclei during interphase. Nuclear fluorescence recovered, demonstrating that Bcd-GFP is being transported across the nuclear membrane. A typical recovery trace of a single bleached nucleus in cycle 14 is shown in Figure 4B. If it is repetitively bleached during the slow interphase decay (interval III), the recovering concentration asymptotically approaches the intensity level of adjacent control nuclei. These data suggest that there is a dynamic equilibrium between transport into the nucleus and a loss, either via outward transport or intranuclear protein degradation.
To interpret the results of the photobleaching experiments more quantitatively, a simple model is considered in which molecules are transported into the nucleus at a rate proportional to the concentration outside in the cytoplasm ($kinCout$) and leave the nucleus either from transport back into the cytoplasm (at rate $kout$) or by degradation (at rate $kd−nuc$). The dynamics for the number of molecules within a nucleus $n(t)$ is given by
$dn(t)dt=kinCout−1τnn(t),$
(1)

$1τn=kout+kd−nuc,$
(2)

where $τn$ is the effective lifetime of the molecule in the nucleus. If the cytoplasmic concentration is approximated as being constant during the measurement (consistent with experiment, see Figure 3D), then if the photobleaching pulse reduces the number of fluorescent molecules by an amount $Δn0$, the model predicts an exponential recovery
$n(t)=n∞−Δn0exp(−t/τn),$
(3)

where $n∞=kinτnCout$ is the steady state number of molecules in the nucleus, and the observable fluorescence is proportional to the number of molecules $n(t)$. Notice that although recovery of fluorescence after photobleaching provides evidence for Bcd transport into the nucleus, the actual time constant for exponential recovery is related to the rate at which molecules leave the nucleus, either by export or degradation.
Following Equation 3, exponential functions were fit to 19 fluorescence recovery curves from four embryos in nuclear cycle 14. The time constant determined from these fits is τn = 68.9 ± 17.6 s (mean ± SD). Thus, any disequilibrium in Bcd concentration between the nucleus and the surrounding cytoplasm will be corrected within roughly 1 or 2 min, which is quite fast on the scale of the nuclear cycles. In particular, this time is short enough that at each moment during interphase interval III (cf. Figure 4A), the nuclear concentration should be in steady state. Why, then, is it drifting downward by ∼30%?

### Geometric Factors Correlate with Nuclear Bcd Concentration Reduction during Interphase

The sizes of nuclei during interphase are not constant; their diameters approximately double in size, as shown quantitatively in Figure 4C. Then perhaps the simplest hypothesis, which could explain the drop in intranuclear Bcd concentration that is evident during interphase interval III, is that as the nuclei increase in volume, the intranuclear concentration is reduced by dilution. Indeed, the model above predicts that the steady state number of molecules inside the nucleus is given by $n∞=kinτnCout$, and hence the concentration will be given by $Cin=n∞/(4πrn3/3)$, where $rn$ is the radius of the nucleus, or $Cin/Cout=3kinτn/(4πrn3)$; if all other parameters are fixed, the product $Cinrn3$ should be constant. In fact this prediction is not consistent with the data: $Cinrn3$ is an increasing function of time during interphase interval III (green curve in Figure 4D).
To understand the concentration decrease during interval III it may be useful to look more carefully at the physical factors that determine the rate of transport into the nucleus, $kin$. The largest possible value for $kin$ can be calculated by assuming that the entire spherical surface of the nuclei is a perfect absorber for Bcd molecules that arrive via diffusion. Consideration of this limit is inspired by ligand binding to receptors on a bacterial cell surface (
• Berg H.C.
• Purcell E.M.
Physics of chemoreception.
) and is analogous to the diffusion-limited rate constant for an enzymatic reaction (
• Fersht A.
Enzyme Structure and Mechanism.
). Transport occurs through discrete nuclear pores, but the nuclei have of the order of 2500 nuclear pore complexes on their surface (
• Kiseleva E.
• Rutherford S.
• Cotter L.M.
• Allen T.D.
• Goldberg M.W.
Steps of nuclear pore complex disassembly and reassembly during mitosis in early Drosophila embryos.
) or 20–40 pores per square micron. If each pore acts as an absorber with a radius of a few nanometers, then this density of pores may be sufficient to make the entire nucleus act as an almost perfect absorber (
• Berg H.C.
• Purcell E.M.
Physics of chemoreception.
). If this is true, the rate of Bcd transport into the nucleus should be given by the maximal, diffusion-limited rate $kin=4πrnD$, where $D$ is the diffusion constant of Bcd in the surrounding cytoplasm. Diffusion-limited transport means that, as the nuclei increase in size, the flux into the nucleus becomes larger, and this should partially offset the effect of dilution. More quantitatively, with $kin=4πrnD$ our simple model predicts that
$CinCout=3Dτnrn2.$
(4)

We have seen that the cytoplasmic concentration $Cout$ and the recovery time $τn$ both are approximately constant during interphase, so we predict that the product $Cinrn2$ should be constant, which is consistent with our data, as Figure 4D shows.
The combination of this result with the photobleaching experiment supports a picture in which nuclear Bcd is in rapid dynamic equilibrium with cytoplasmic Bcd and the transport into the nucleus is as fast as possible, being limited by cytoplasmic diffusion. All of the quantities in Equation 4 are directly measured except for the diffusion constant $D$. Thus this simple model predicts a cytoplasmic diffusion constant in terms of our other data, $D=(Cin/Cout)(rn2/3τn)$. The result, as shown in Figure 4D, is D = 0.37 ± 0.05 μm2/s (mean ± SD). Although our results in Figure 4D are consistent with a simple model, there are several caveats. In particular, the nuclear pores might not act as perfect absorbers, either because the kinetics of transport through the pore itself are too slow (but this is unlikely; see
• Kubitscheck U.
• Grunwald D.
• Hoekstra A.
• Rohleder D.
• Kues T.
• Siebrasse J.P.
• Peters R.
Nuclear transport of single molecules: dwell times at the nuclear pore complex.
) or because competition among many protein species lowers the efficiency for transport of Bcd. We thus view the diffusion constant which emerges from the analysis of Figure 4D not as a measurement, but as a prediction to be tested by more direct experiments.

### Bcd-GFP Diffusion in the Cortical Cytoplasm

To directly measure the diffusion constant of Bcd-GFP in the cortical cytoplasm, we measured the dynamics of recovery after pattern photobleaching. A rectangular volume 16 × 16 × 7 μm3 in the anterior cytoplasm of Bcd-GFP-expressing embryos was photobleached during mitosis 13, when Bcd-GFP is uniformly distributed (locally) throughout the cortical cytoplasm and the cytoplasmic concentration is relatively high. A typical recovery curve is shown in Figure 5. Recovery curves were fit to solutions of the diffusion equation to estimate $D$ (see Experimental Procedures). From a total of 21 recovery curves in four embryos, our analysis provided a cytoplasmic diffusion constant D = 0.30 ± 0.09 μm2/s (mean ± SD) during mitosis 13, which is very close to our estimate from the transport model.
This cortical diffusion constant is surprisingly small, especially given earlier measurements (
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
), which showed significantly higher diffusion constants for biologically inert molecules injected into eggs. As we discuss below, it is very difficult to reconcile the small diffusion constant measured for Bcd-GFP with the fact that the gradient forms in about one hour and remains stable over nuclear cycles 10–14. Within the SDD model, the gradient would never reach steady state with our measured small diffusion constant for Bcd-GFP, given the obvious developmental time constraints.

### Cortex versus Core

The diffusion constants for Bcd protein we measure on the surface of the embryos would not be representative of the embryo as a whole if the cortex contains structures that trapped the protein or slowed its diffusion. The classic example of this effect is the slowing of calcium ion diffusion in neurons (
• Hodgkin A.L.
• Keynes R.D.
Movements of labelled calcium in giant axons.
). In the Supplemental Data, we show mathematically how the presence of binding sites would affect diffusion and observed Bcd concentration in the context of the SDD model. The result is that the presence of fixed binding sites should both slow the diffusion and enhance the total (bound + free) concentration of Bcd, both in proportion to the density of sites.
In sectioned cycle 14 embryos (Figure 6A), we do in fact see a significantly higher concentration of Bcd in the cortex relative to the central core of the egg. The intensity ratio between the cortex and the core increases in an approximately exponential fashion with progressing nuclear cycles and the increasing number of nuclei in the cortical cytoplasm (see Figure 6D). This suggests that structures which bind Bcd in the cortex are associated with the nuclei themselves. The roughly 10-fold enhancement of Bcd concentration, presumably as a result of binding to an enhanced density of fixed sites near the cortex, would be consistent with diffusion constants being an order of magnitude larger in the core of the embryo (see Equation S9 in Supplemental Data), although we are unable to measure this directly.

### Bcd Diffusion and Degradation in Unfertilized Eggs

To evaluate whether the presence of nuclei or some nuclear-related cytoplasmic structure changes the diffusive behavior of Bcd protein in the cortex, we examined the movement of Bcd in the cortex of unfertilized eggs. Although such eggs have only a single female pronucleus and do not undergo the progressive reorganization of the cortical cytoplasm associated with normal development, they still initiate translation of bcd RNA when the egg is laid. In contrast to the syncytial blastoderm stages described above, Bcd protein in unfertilized eggs shows no obvious enrichment in the cortex (see Figure 6A). To determine the Bcd diffusion constant in such eggs, we performed photobleaching experiments identical to those described earlier for fertilized eggs. In seven measurements on three eggs, the diffusion constant was D = 0.35 ± 0.12 μm2/s, which is nearly identical to that measured in the cortical cytoplasm of embryos. We conclude that the diffusive behavior measured in the cortex is not dependent on the presence of nuclei or associated cytoplasmic structure and may in fact represent the general behavior of all regions of the egg. In particular, to the extent that the unorganized cortex of the unfertilized egg approximates the environment in the central core of the fertilized egg, our results argue against the possibility that the central cytoplasm would provide a faster path for Bcd diffusion (see Discussion).
Although diffusion constants appear to be identical, the fertilized or unfertilized state of the egg may still impact the final distribution of Bcd protein. To assess this possibility, we measured the average Bcd-GFP fluorescence intensity in 400 μm2 regions near the anterior and posterior poles and compared the results with those from similar measurements in fertilized eggs. (A more detailed comparison of full in vivo gradients in unfertilized and fertilized eggs will be presented elsewhere.) In the fertilized eggs, intensities were 3.8 ± 0.4 times larger at the anterior pole than at the posterior (mean and SD over 13 embryos). This ratio is not as large as the full dynamic range of the gradient (cf. Figure 1C) because the averaging region is chosen relatively large so as to have a robust signal from the unfertilized eggs.
In the unfertilized eggs, we found that Bcd intensities at the posterior portion of unfertilized eggs are reproducibly greater than the near background levels observed in that region of fertilized eggs (data not shown), which is in agreement with earlier findings by
• Driever W.
• Nüsslein-Volhard C.
A gradient of Bicoid protein in Drosophila embryos.
. At the anterior pole, intensities were lower and the overall ratio was 1.5 ± 0.3 (mean and SD over 13 embryos). At 5 hr, when the anterior intensity of Bcd-GFP in unfertilized eggs was more nearly equal to that in fertilized eggs at cycle 14, the anterior/posterior ratio had not changed (1.3 ± 0.3; n = 13 embryos).
These observations suggest that gradient formation in unfertilized eggs differs from that of fertilized eggs in two significant respects: The levels achieved at the posterior pole are higher, and the overall shape is flatter. Since the level of Bcd at the posterior pole would be expected to be greatly influenced by degradation, this provides support for considering models in which the nuclei present in fertilized eggs contribute to the degradation process itself (see Discussion).

## Discussion

Our principal results, together with those of previous work (
• Driever W.
• Nüsslein-Volhard C.
A gradient of Bicoid protein in Drosophila embryos.
,
• Driever W.
• Nüsslein-Volhard C.
The Bicoid protein determines position in the Drosophila embryo.
,
• Wieschaus E.
• Leibler S.
Establishment of developmental precision and proportions in the early Drosophila embryo.
,
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
), provide the following foundation for any mechanistic model for the formation or read out of the Bcd gradient:
• 1.
The gradient is approximately (but not necessarily exactly) an exponential decay in intranuclear (and cytoplasmic) concentration that is established rapidly (less than 90 min).
• 2.
Bcd diffuses relatively slowly (D = 0.3 μm2/s) in the cortical cytoplasm containing the nuclei.
• 3.
The Bcd gradient is stable over nuclear cycles 10–14, when the number of nuclei is growing by a factor of two with each division and Bcd is concentrated and released from nuclei in a dynamic process. In particular, the initial postinterphase concentration in nuclei in successive cycles is constant to at least 10% but decays by approximately 30% during the subsequent interphase period.
• 4.
Bcd is not simply trapped in nuclei; rather, it is in dynamic equilibrium between influx and efflux with the cytoplasm and, possibly, intranuclear degradation.
• 5.
The total (bound + free) concentration of Bcd in the cortical cytoplasm is distinctly higher than in the yolk.
• 6.
The spatial shape of the Bcd gradient (the length constant for an exponential decay) scales with embryo length over a factor of five range in lengths in different dipteran species (
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
).

### Problems with the Simplest Gradient Model

The most widely considered and simplest model for Bcd gradient formation—namely the SDD model—consists of a source of protein synthesis localized at the anterior pole, intracellular Bcd diffusion away from the source, and spatially uniform degradation by a first-order reaction. If Bcd molecules diffuse throughout the embryo with diffusion constant $D$ and are degraded with lifetime $τ$, the concentration of Bcd protein in this simple model obeys
$∂CBcd(r→,t)∂t=D∇2CBcd(r→,t)−1τCBcd(r→,t).$
(5)

The source is described as a boundary condition at the anterior pole ($x=0$).
This model predicts the steady state concentration profile, and its principal virtue is that it produces the experimentally observed exponential decay, $C(x)=C(0)exp(−x/λ)$, with $λ=Dτ$. Steady state is reached, however, only after a time much greater than the protein lifetime $τ$, assuming (as observed) that the length constant $λ$ is much smaller than the length of the embryo. Since we have no direct measurement of $τ$, it is more useful to say that the time required to reach steady state is τss >> λ2/D. The observed concentration profiles have λ ∼ 100 μm, which means that to reach a steady state within 90 min requires that the diffusion constant of Bcd must be D >> 2 μm2/s. This is an order of magnitude larger than what we measure directly in photobleaching experiments (Figure 5) or what we infer indirectly from the dynamics of nuclear-cytoplasmic exchange (Figure 4D).
It could have been that the diffusion constants were slow but that correspondingly the gradient did not reach steady state during the blastodermal stages of development. Alternatively, the apparent steady state behavior of the gradient for cycles 10 through 14 would not be surprising if diffusion constants were large. The paradox is that we observe both slow diffusion and the rapid establishment of a steady state. Simply put, the diffusion processes that we observe cannot be the ones which generate the steady state profile because they are too slow.
There are, broadly speaking, three possible ways in which the paradox of slow diffusion and rapid arrival at the steady state could be resolved: Diffusion in the bulk of the embryo could be faster than we observe in the cortex, diffusion could be faster in the first hour after fertilization (where we were unable to measure Bcd), or the effective diffusion constants could be different on different time scales (∼1 min versus ∼1 hr) because of active transport mechanisms. These possibilities are discussed in the Supplemental Data.

Perhaps the most striking result in the present work is the dynamic exchange of Bcd from cytoplasm to nuclei. In cycle 14, the number of nuclei, the enhancement of cytoplasmic concentration near the cortex, and the further enhancement of concentration above this in the nuclei all combine to mean that ∼40% of the Bcd molecules in the entire embryo are localized in the nuclei (see Experimental Procedures for details). Thus, rather than passively sampling a large excess of molecules in the cytoplasm, the nuclei must perturb the gradient significantly. In particular, it is natural to ask if events inside the nucleus could influence or even dominate Bcd degradation (
• Dhananjayan S.C.
• Ismail A.
• Nawaz Z.
Ubiquitin and control of transcription.
,
• Collins G.A.
• Tansey W.P.
The proteasome: a utility tool for transcription?.
). Although our results do not directly prove this assertion, it is intriguing that there is a smaller anterior-to-posterior ratio of Bcd concentration in unfertilized eggs than in fertilized eggs undergoing nuclear divisions.
• Driever W.
• Nüsslein-Volhard C.
A gradient of Bicoid protein in Drosophila embryos.
also noted a pronounced increase in Bcd concentration in unfertilized eggs after the first hour, which is consistent with our observations (data not shown).
What is particularly interesting about the possibility of nuclear involvement in degradation is that it can naturally solve the scaling problem. To illustrate how this can work, it is useful to consider an extreme (and admittedly unrealistic) model. Suppose that there is no degradation of Bicoid in the bulk of the embryo, but the degradation rate inside nuclei is so high that once molecules diffuse into the cortical layer of nuclei they never escape. Then rather than a model in which degradation is a specific term in the dynamics of the Bcd concentration, as in Equation 5, one can think of the degradation process as a boundary condition near the embryo surface. In this limit, no lifetimes or rate constants appear in the equations, and so the steady state is determined by balancing boundary conditions—a source at the anterior pole and a distributed “sink” along the cortical surface. As a result, the steady state profile depends only on the geometry of the embryo itself and hence automatically scales as one considers embryos of different sizes.
As a more realistic model, imagine a two-dimensional array of nuclei at positions near the embryo surface, and let us assume that degradation of protein within the nuclei occurs with lifetime $τnuc$, in contrast to the lifetime $τ$ in the cytoplasm outside the nuclei. If τ >> τnuc, then almost all the degradation will happen in the nuclei, and the effective rate of degradation, averaged over some reasonable volume, can be shown to be proportional to the density of nuclei in that volume, $τeff∝1/ρnuc$, so that the length constant $λ=Dτ∝1/ρnuc$. Importantly, the number of nuclei at a given cycle is constant across embryos of very different size (
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
) so that $ρnuc=Nnuc/(Aℓ)$, where $A$ is the surface area of the egg and $ℓ$ is the thickness of the cortical layer ($ℓ∼30μm$ in cycle 14). Different size embryos from different flies seem to have similar aspect ratios, so we expect AGL2, where G ∼ 0.3 reflects this geometry. The result is that $ρnuc=NnucGL2ℓ$ so that $λ∝1/ρnuc$ is equivalent to $λ∝L$. Thus, if trapping of Bcd in the nuclei dominates the dynamics and if nuclei are ordered in space and time as observed, then we have scaling automatically. Within this model, scaling is achieved because the number of nuclei is conserved in differently sized eggs, and hence the nuclear density is reduced in larger eggs.
One central problem of models in which the nuclei play an essential role is that the number of nuclei is changing, doubling every 9 to 12 min during the crucial period of development. There is no evidence that the Bcd profile changes shape during this time. In the simplest view, if nuclear processes dominate the degradation of Bcd, then the effective lifetime should change by a factor of two with each successive cycle, thus changing the length constant $λ$ by a factor of four from cycle 10 to 14. But this simple view breaks down if the other relevant parameters are changing. Over the course of these four doublings in the number of nuclei, the radius of the nuclei changes by nearly a factor of two (Figure 4C), and the thickness $ℓ$ of the cortical layer changes by a factor of three (
• Foe V.E.
• Odell G.M.
• Edgar B.A.
Mitosis and morphogenesis in the Drosophila embryo: point and counterpoint.
). In addition, the partitioning of Bcd molecules between the cortical layer and the interior bulk of the egg changes substantially (Figure 6D). In a coarse-grained picture, if degradation is dominated by events inside the nuclei, then the effective lifetime of a Bcd molecule in the embryo should depend on the fraction of molecules localized in nuclei; the different factors listed above in fact conspire so that this fraction remains constant across cycles 11 through 14, within the ∼20% accuracy of our estimates (see Experimental Procedures).

### Perspective

Our earlier work (
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
) established that molecular motion in the embryo is described well by the diffusion equation on the time (∼1 hr) and space (∼$0.1mm$) scales of relevance for morphogenesis. The present work shows that diffusion is an equally good description of Bicoid transport on the scale of minutes and microns. The difficulty is that the relevant diffusion constants differ by more than an order of magnitude. To explain our observation that the Bcd gradient reaches a nuclear steady state very quickly we need the larger diffusion constants, but the dynamics of transport into and out of the nuclei are consistent with the smaller diffusion constant, which we also measure directly. We have discussed possible resolutions of this paradox, but new experiments will be required to decide which of these is correct.
The most dramatic qualitative feature that we see in watching the development of the embryos expressing Bcd-GFP is the filling and emptying of the nuclei. Quantitatively, this results in a startling juxtaposition of dynamics and stability. Thus, although Bcd concentrations vary in time over a factor of four during the course of a mitotic cycle, the nuclear concentration near the start of interphase is reproducible with ∼10% accuracy from cycle to cycle. Although the number of nuclei is changing by a factor of 16 from cycle 10 to cycle 14, the total number of Bcd molecules that are localized in nuclei changes hardly at all. At our present level of understanding, both these examples of stability in the presence of change seem to be the result of cancellation among several independent processes, which is implausible. One way of summarizing the problem is that the simplest model looks like it works, but this is only because many parameters have been adjusted to make it work, leaving the simplest model as an effective description of the dynamics after the mechanisms responsible for this adjustment have done their job. This layer of mechanisms remains to be discovered.

## Experimental Procedures

### Construction of P[egfp-bcd]

P[egfp-bcd] transcription is driven by known bcd enhancers upstream of the 5′ UTR and in the first intron. This transcript contains bcd's natural 5′ and 3′ UTRs which are known to mediate the localization and translation of normal bcd mRNA. Thus by using the endogenous regulates of bcd transcription and translation, our construct was designed to be maternally expressed and localized to the anterior pole.
pNBGA1 containing a 6.5 kb D. melanogaster genomic fragment of the entire bcd gene, with the GFP coding region inserted directly upstream of and in frame with the first exon of bcd, was kindly provided by Tulle Hazelrigg (
• Hazelrigg T.
• Liu N.
• Hong Y.
• Wang S.
GFP expression in Drosophila tissues: time requirements for formation of a fluorescent product.
). This plasmid was digested with NheI and SphI to remove GFP and replace it with either eGFP generated by PCR from pStinger (
• Barolo S.
• Carver L.A.
• Posakony J.W.
GFP and beta-galactosidase transformation vectors for promoter/enhancer analysis in Drosophila.
) using primers appended with NheI and SphI restriction sites. The resulting fusion gene was subsequently subcloned into the BamHI and EcoRI sites of pCaSpeR4 (
• Thummel C.S.
• Pirrotta V.
New pCaSpeR P element vectors.
) to generate P[egfp-bcd].

### Fly Transformation

We followed standard P element transformation procedures (
• Rubin G.M.
Transposition of cloned P-elements into Drosophila germ line chromosomes.
). We injected 0.4 mg/ml of construct DNA together with 0.1 mg/ml helper plasmid pTURBO, which expresses the transposase into D. melanogaster yw embryos. We scored several different P[egfp-bcd] insertions, including insertions on chromosomes I, II, and X. Stocks were established for each of these.

### Fly Stocks and Genetics

For all experiments with flies expressing egfp-bcd we used a stock with an X-chromosomal insertion of P[egfp-bcd]. For substitution of endogenous bcd we conducted the mutant crosses of egfp-bcd with bcdE1, pp/TM3, Sb to generate egfp-bcd;bcdE1, pp. Unfertilized eggs of Oregon R wild-type D. melanogaster flies and of bcdE1;egfp-bcd flies were produced using sterile males generated from C(1,Y),yB/0.

### Immunostaining of Embryos

All D. melanogaster embryos were collected at 25°C. Embryos expressing Bcd-GFP were formaldehyde fixed (for 20 min in 3.7% formaldehyde/PBS:heptane and devitellinized in heptane:methanol), all other embryos were heat fixed, and the eggs were subsequently labeled with fluorescent probes following previously published protocols (
• Wieschaus E.
• Nüsslein-Volhard C.
Looking at embryos.
). We used rat anti-Bcd and rabbit anti-HB antibodies, gifts of J. Reinitz (
• Kossman D.
• Small S.
• Reinitz J.
Rapid preparation of a panel of polyclonal antibodies to Drosophila segmentation proteins.
), as well as guinea pig anti-GFP (Molecular Probes) and rabbit anti-GFP (Chemicon). Secondary antibodies were conjugated with Alexa-488, Alexa-546, and Cy5-647 (Molecular Probes). Nuclei were labeled with DNA probes YOYO1 or TOTO3 (Molecular Probes). Eggs were mounted in AquaPolymount (Polysciences, Inc.).

### Cut-Sections of Fixed Eggs

Heat-fixed embryos were embedded in $50%$ glycerol and hand-cut with a sharp razor blade. The embryos employed were stained with Bcd antibodies (see above), and the cut-sections were imaged focusing at the top-most surface of the section. To exclude a misrepresentation of our experiment due to problems with permeabilization and antibody penetration into the embryo, we tested and quantified two versions of this experiment: in the first version we heat-fixed embryos stained with Bcd antibodies and hand-cut the eggs after the staining procedure was completed; in the second version of the experiment we heat-fixed the embryos and hand-cut them before running the staining procedure. Both procedures led to the same quantitative results.

### Imaging of Fixed Tissue by Confocal Microscopy

High-resolution digital images (1024 × 1024, 12 bits per pixel) of fixed eggs were obtained on a Zeiss LSM 510 confocal microscope with a Zeiss 20× (NA 0.45) A-plan objective. Images were focused at midembryo to avoid geometric distortion for gradient extraction experiments. For measurements of nuclear fluorescence intensities eggs were flattened with a cover slip to maximize the number of nuclei at the focal top surface.

### Live Imaging by Two-Photon Microscopy

Living embryos were collected 0–15 min after egg deposition, dechorionated, glued onto a cover slip, and immersed in halocarbon oil (Sigma-Aldrich) or $dH2O$. Two-photon microscopy (
• Denk W.
• Strickler J.H.
• Webb W.W.
Two-photon laser scanning fluorescence microscopy.
) was performed using a custom-built scanning microscope similar in design to that of reference (
• Svoboda K.
• Denk W.
• Kleinfeld D.
• Tank D.W.
In vivo dendritic calcium dynamics in neocortical pyramidal neurons.
). Time-lapse image sequences were taken with a Zeiss Plan-Neofluar multi-immersion objective (25×, NA 0.8) and an excitation wavelength of 900–920 nm, minimizing embryonic yolk autofluorescence. Average laser power at the specimen was between 10 and 30 mW.

### Photobleaching Experiments

Bcd-GFP-expressing embryos were mounted as described above. Photobleaching of individual nuclei was performed using our scanning two-photon microscope by reducing the scan-field 8-fold while keeping the laser power constant, effectively increasing the power per pixel 64-fold for a duration of 5 s. Subsequently the recovery curve was recorded in 0.5 s time bins with the original scan-field, where the recording power has been adjusted to avoid further photobleaching. The bleach-volume (16 × 16 × 7 μm3) generated with the 25× (NA 0.8) Zeiss Plan-Neofluar multi-immersion objective was measured separately using fluorescently-labeled 40 kDa-dextran $dH2O$ solution diluted in polyethylene. Cytoplasmic photobleaching was performed by rapid modulation of the beam with a KDP Pockels Cell (model 350-50; Conoptics, Danbury, CT) to generate the bleaching and monitoring intensities (
• Axelrod D.
• Koppel D.E.
• Schlessinger J.
• Elson E.
• Webb W.W.
Mobility measurement by analysis of fluorescence photobleaching recovery kinetics.
). In these experiments the power modulation at the specimen was 2-fold, the bleach pulse was 6 s, and the bleach-volume was 16 × 16 × 7 μm3. Data was fitted to the exact solution of the 3D diffusion equation (
• Brown E.B.
• Wu E.S.
• Zipfel W.
• Webb W.W.
Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery.
).

### Profile Quantification

Bcd protein profiles were extracted from images of stained embryos using software routines (MATLAB, MathWorks, Natick, MA) that allowed a circular window of the size of a nucleus to be systematically moved along the outer edge of the embryo (
• Wieschaus E.
• Leibler S.
Establishment of developmental precision and proportions in the early Drosophila embryo.
). At each position, the average pixel intensity within the window was plotted versus the projection of the window center along the AP-axis of the embryo. Measurements of the Bcd concentration were made separately along the dorsal and ventral sides of the embryo. All embryos were prepared, and images were taken under the same conditions: (i) all embryos were formaldehyde fixed for 20 min, (ii) embryos were stained and washed together in the same tube, and (iii) all images were taken with the same microscope settings in a single acquisition cycle.

### Comparing Gradients of Antibody Staining and GFP Fluorescence

Gradients of double stained embryos expressing Bcd-GFP were compared in a scatter plot with the different gradient visualization methods (antibody against Bcd, antibody against GFP and endogenous GFP fluorescence) on each axis. Scatter plots of gradients of identical shape should only have data points on the diagonal. We tested this method for comparing intensity profiles using artificially generated distributions of gradients with different mean shapes, and we obtained significant power-law dependencies for distributions that differed by more than $4%$ of their mean (the mean is characterized by the average length constant of artificially-generated, exponentially decaying intensity profiles).

### Fraction of Bcd Molecules in Nuclei

To estimate the relative number of Bcd molecules in nuclei, cortical cytoplasm and the core of an embryo of length L ∼ 490 μm, we consider a cross-sectional slice of thickness $Δx$ with volume $Vtot=πR2Δx$, where R ∼ 90 μm is the radius of the slice (see Figure 2A). The number of nuclei in the slice is given by $(N/L)Δx$, where $N$ is the total number of nuclei in the embryo, typically ∼6000 during nuclear cycle 14; hence the total nuclear volume is $Vnuc=43πrn3NΔxL$, with nuclear radius $rn$ (see Figure 1C for values). The core volume is given by $Vcore=π(R−ℓ)2Δx$, with $ℓ$ being the thickness of the cortical cytoplasm. The relative number of Bcd molecules in nuclei, cortical cytoplasm, and the core of an embryo are then respectively given by $ncore=CcoreVcore$, $nnuc=CnucVnuc$ and $ncort.−cyt.=Ccort.−cyt.(Vtot−Vcore−Vnuc)$, with core, nuclear, and cortical Bcd concentrations $Ccore$, $Cnuc$ and $Ccort.−cyt.$, respectively. Finally, the fraction of Bcd molecules in the nuclei is given by
$fnuc=nnucncore+nnuc+ncort.−cyt..$
(6)

To estimate $fnuc$ for nuclear cycles 11 to 14, we use cortical thicknesses $ℓ$ of 11 μm, 15 μm, 24 μm, and 31 μm, respectively (
• Foe V.E.
• Odell G.M.
• Edgar B.A.
Mitosis and morphogenesis in the Drosophila embryo: point and counterpoint.
). Reading the respective concentrations and nuclear radii from our data (see Figures 2D and 1C), we obtain the fraction of nuclear Bcd molecules in the nucleus: $33%$, $39%$, $41%$, and $41%$ for nuclear cycles 11 to 14, respectively. Our estimated accuracy on these values is ∼20%.

## Acknowledgments

We thank N. Bisaria, M. Coppey, J. Drocco, T. Hazelrigg, J. Kinney, R. Samantha, and G. Tkačik. This work was supported in part by the Howard Hughes Medical Institute, the MRSEC Program of the National Science Foundation under Award Number DMR-0213706, and by NIH grants P50 GM071508 and R01 GM077599.

## Supplemental Data

• Document S1. Extended Figure 2 Legend, Supplemental Discussion, and Supplemental References
• Movie S1. Two-Photon Time-Lapse Movie of a Drosophila Embryo Expressing Bcd-GFP

Typical time-lapse movie of a Drosophila embryo expressing Bcd-GFP in a bcd null mutant background, as in Figure 1A. Three focal planes have been acquired every 30 seconds; the movie shows the developing embryo from 40 min after fertilization until gastrulation. See legend of Figure 1 and Experimental Procedures for more information.

## References

• Ariizumi T.
• Asashima M.
Control of the embryonic body plan by activin during amphibian development.
Zoolog. Sci. 1995; 12: 509-521
• Axelrod D.
• Koppel D.E.
• Schlessinger J.
• Elson E.
• Webb W.W.
Mobility measurement by analysis of fluorescence photobleaching recovery kinetics.
Biophys. J. 1976; 16: 1055-1069
• Barolo S.
• Carver L.A.
• Posakony J.W.
GFP and beta-galactosidase transformation vectors for promoter/enhancer analysis in Drosophila.
Biotechniques. 2000; 29: 726-732
• Berg H.C.
• Purcell E.M.
Physics of chemoreception.
Biophys. J. 1977; 20: 193-219
• Brown E.B.
• Wu E.S.
• Zipfel W.
• Webb W.W.
Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery.
Biophys. J. 1999; 77: 2837-2849
• Collins G.A.
• Tansey W.P.
The proteasome: a utility tool for transcription?.
Curr. Opin. Genet. Dev. 2006; 16: 197-202
• Crick F.
Diffusion in embryogenesis.
Nature. 1970; 225: 420-422
• Denk W.
• Strickler J.H.
• Webb W.W.
Two-photon laser scanning fluorescence microscopy.
Science. 1990; 248: 73-76
• Dhananjayan S.C.
• Ismail A.
• Nawaz Z.
Ubiquitin and control of transcription.
Essays Biochem. 2005; 41: 69-80
• Driever W.
• Nüsslein-Volhard C.
A gradient of Bicoid protein in Drosophila embryos.
Cell. 1988; 54: 83-93
• Driever W.
• Nüsslein-Volhard C.
The Bicoid protein determines position in the Drosophila embryo.
Cell. 1988; 54: 95-104
• Ephrussi A.
• St Johnston D.
Seeing Is Believing: The Bicoid Morphogen Gradient Matures.
Cell. 2004; 116: 143-152
• Fersht A.
Enzyme Structure and Mechanism.
Freeman, San Francisco, CA1985
• Foe V.E.
• Odell G.M.
• Edgar B.A.
Mitosis and morphogenesis in the Drosophila embryo: point and counterpoint.
in: Bate M. Martinez-Arias A. The Development of Drosophila. Cold Spring Harbor Laboratory Press, Plainview, NY1993: 149-300
• Gregor T.
• Bialek W.
• de Ruyter van Steveninck R.R.
• Tank D.W.
• Wieschaus E.F.
Diffusion and scaling during early embryonic pattern formation.
Proc. Natl. Acad. Sci. USA. 2005; USA102: 18403-18407
• Hazelrigg T.
• Liu N.
• Hong Y.
• Wang S.
GFP expression in Drosophila tissues: time requirements for formation of a fluorescent product.
Dev. Biol. 1998; 199: 245-249
• Hodgkin A.L.
• Keynes R.D.
Movements of labelled calcium in giant axons.
J. Physiol. 1957; 138: 253-281
• Wieschaus E.
• Leibler S.
Establishment of developmental precision and proportions in the early Drosophila embryo.
Nature. 2002; 415: 798-802
• Kiseleva E.
• Rutherford S.
• Cotter L.M.
• Allen T.D.
• Goldberg M.W.
Steps of nuclear pore complex disassembly and reassembly during mitosis in early Drosophila embryos.
J. Cell Sci. 2001; 114: 3607-3618
• Kossman D.
• Small S.
• Reinitz J.
Rapid preparation of a panel of polyclonal antibodies to Drosophila segmentation proteins.
Dev. Genes Evol. 1998; 208: 290-298
• Kubitscheck U.
• Grunwald D.
• Hoekstra A.
• Rohleder D.
• Kues T.
• Siebrasse J.P.
• Peters R.
Nuclear transport of single molecules: dwell times at the nuclear pore complex.
J. Cell Biol. 2005; 168: 233-243
• Lacalli T.C.
• Harrison L.G.
From gradient to segments: models for pattern formation in early Drosophila.
Semin. Dev. Biol. 1991; 2: 107-117
• Lawrence P.A.
The Making of a Fly: The Genetics of Animal Design.
Blackwell Scientific, Oxford, UK1992
• Rubin G.M.
Transposition of cloned P-elements into Drosophila germ line chromosomes.
Science. 1982; 218: 341-347
• Struhl G.
• Struhl K.
• Macdonald P.M.
The gradient morphogen Bicoid is a concentration-dependent transcriptional activator.
Cell. 1989; 57: 1259-1273
• Svoboda K.
• Denk W.
• Kleinfeld D.
• Tank D.W.
In vivo dendritic calcium dynamics in neocortical pyramidal neurons.
Nature. 1997; 385: 161-165
• Thummel C.S.
• Pirrotta V.
New pCaSpeR P element vectors.
Drosoph. Inf. Serv. 1992; 71: 150
• Tsien R.Y.
The green fluorescent protein.
Annu. Rev. Biochem. 1998; 67: 509-544
• Wieschaus E.
• Nüsslein-Volhard C.
Looking at embryos.
in: Roberts D.B. Drosophila, a Practical Approach. IRL Press, Oxford, UK1986: 199-227
• Wolpert L.
Positional information and the spatial pattern of cellular differentiation.
J. Theor. Biol. 1969; 25: 1-47
• Zecca M.
• Basler K.
• Struhl G.
Sequential organizing activities of engrailed, hedgehog and decapentaplegic in the Drosophila wing.
Development. 1995; 121: 2265-2278