text:
1
MCQ (Single Correct Answer)

AIPMT 2010 Mains

Some of the properties of the two species, $NO_3^ -$ and ${H_3}{O^ + }$ are described below. Which one of them is current?
A
Dissimilar in hybridization for the central atom with different structures.
B
Isostructural with same hybridization for the central atom.
C
Isostructural with different hybridization for the central atom
D
Similar in hybridization for the central atom with different structures.

Explanation

No. of electron pairs = No. of atoms bonded to it + 1/2[Gp. no. of central atom - Valency of central atom $\pm$ No. of electrons]

For NO3- = 3 + 1/2[5 - 6 + 1] = 3 (sp2 hybridisation)

For H3O+ = 3 + 1/2[6 - 3 - 1] = 4 (sp3 hybridisation)
2
MCQ (Single Correct Answer)

AIPMT 2010 Mains

In which of the following molecules the central atom does not have sp3 hybridization?
A
CH4
B
SF4
C
BF4$-$
D
NH4+

Explanation

For neutral molecules

No. of electron pairs = No. of atoms bonded to it + 1/2[Gp. no. of central atom - Valency of central atom]

$\therefore$For CH4, no. of e- pairs = 4 + 1/2[4 - 4] = 4 (sp3 hybridisation)

For SF4, no. of e- pairs = 4 + 1/2[6 - 4] = 5 (sp3d hybridisation)

and For ions,

No. of electron pairs = No. of atoms bonded to it + 1/2[Gp. no. of central atom - Valency of central atom $\pm$ No. of electrons]

$\therefore$For BF4-, no. of e- pairs = 4 + 1/2[3 - 4 + 1] = 4 (sp3 hybridisation)

For NH4+, no. of e- pairs = 4 + 1/2[5 - 4 - 1] = 4 (sp3 hybridisation)
3
MCQ (Single Correct Answer)

AIPMT 2010 Prelims

In which one of the following species the central atom has the type of hybridization which is not the same as that present in the other three ?
A
SF4
B
I3$-$
C
SbCl52$-$
D
PCl5

Explanation

The hybridisation of the central atom can be calculated as

H = ${1 \over 2}\left[ \matrix{ \left( \matrix{ No.\,of\,electrons \hfill \cr in\,valence\,shell \hfill \cr of\,atom \hfill \cr} \right) + \left( \matrix{ No.\,of\,monovalent \hfill \cr atoms\,around \hfill \cr central\,atom \hfill \cr} \right) \hfill \cr - \left( \matrix{ Ch{\mathop{\rm arge}\nolimits} \,on \hfill \cr cation \hfill \cr} \right) + \left( \matrix{ Ch{\mathop{\rm arge}\nolimits} \,on \hfill \cr anion \hfill \cr} \right) \hfill \cr} \right]$

Applying that formula we find that all the given species except [SbCl5]2- have central atom with sp3d ( corresponding to H = 5) hybridisation. In [SbCl5]2- Sb is sp3d2 hybridized.
4
MCQ (Single Correct Answer)

AIPMT 2010 Prelims

In which of the following pairs of molecules/ ions, the central atoms have sp2 hybridisation ?
A
NO$_2^{ - }$ and NH3
B
BF3 and NO$_2^{ - }$
C
NH$_2^{ - }$ and H2O
D
BF3 and NH$_2^{ - }$

Explanation

The hybridisation of the central atom can be calculated as

H = ${1 \over 2}\left[ \matrix{ \left( \matrix{ No.\,of\,electrons \hfill \cr in\,valence\,shell \hfill \cr of\,atom \hfill \cr} \right) + \left( \matrix{ No.\,of\,monovalent \hfill \cr atoms\,around \hfill \cr central\,atom \hfill \cr} \right) \hfill \cr - \left( \matrix{ Ch{\mathop{\rm arge}\nolimits} \,on \hfill \cr cation \hfill \cr} \right) + \left( \matrix{ Ch{\mathop{\rm arge}\nolimits} \,on \hfill \cr anion \hfill \cr} \right) \hfill \cr} \right]$

$\therefore$ For BF3, H = ${1 \over 2}\left[ {(3) + (3) - (0) + (0)} \right] \Rightarrow 3 \Rightarrow$ sp2 hybridisation

$\therefore$ For NO2-, H = ${1 \over 2}\left[ {(5) + (0) - (0) + (1)} \right] \Rightarrow 3 \Rightarrow$ sp2 hybridisation

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text:
# Double sequences with GCD

Given a sequence called $$A$$ including $$50$$ terms ( from $$1$$ to $$50$$), which can be described as ${ A }_{ n }=20+{ n }^{ 2 }$

Given another sequence called $$B$$ including $$49$$ terms, which have a property that ${ B }_{ n }=GCD({ A }_{ n },{ A }_{ n+1 })$

Given that the largest term in $$B$$ is $${B}_{x}$$. Find the value of $$\sqrt { { B }_{ x }+x }$$.

Note: $$GCD(a,b)$$ is denoted as the greatest common divisor of $$a$$ and $$b$$.

×

format: plain


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# Just another sequence

Hey guys, I've been into brainteasers since I was little and recently decided to make a puzzle-related website. I figured that this will be the best place to share some of my problems and see if they are any fun. I guess they are too easy, but hope that will entertain you for a bit.

Just another sequence

1, 3, 7, 12, 18, 26, 35, 45, 56, ?

P.S. Please, let me know if you have seen identical puzzle elsewhere, don't want to take someone's credit, even accidentally.

• mathworld.wolfram.com/HofstadterFigure-FigureSequence.html – qwertylpc Jul 16 '15 at 19:14
• @qwertylpc Aha! That's the same pattern as I found, only perhaps a more elegant way of expressing it. – Rand al'Thor Jul 16 '15 at 19:30
• Thank you guys. As a mathematician, I feel a little bit embarrassed... Should have expected it to be well-known already. – Puzzle Prime Jul 16 '15 at 19:32
• @ArturKirkoryan I had a similar experience with my very first question here :-) – Rand al'Thor Jul 16 '15 at 19:33
• Yeah, I feel there are not many more fun sequences to be discovered:) – Puzzle Prime Jul 16 '15 at 19:35

• The first term is $1$.
• For all $n$, the $(n+1)$th term of the sequence is the sum of the $n$th term and the smallest number which hasn't appeared among the first $n$ terms and the first $n-1$ differences between consecutive terms.
real terms: $1,3,7,12,18,26,35,45,56,69,\dots$
differences: $2,4,5,6,8,9,10,11,13,14,\dots$


text:
# What is the lower bound and upper bound on time for inserting n nodes into a binary search tree?

So given a $n$ array of few numbers(say $n$) we can sort them using the binary search tree (BST) as a black box . In order to that we first build a BST out of the array taking all the elements in order and then do an in order traversal .

So the steps will be :

1. Let the empty BST be $\phi$
2. Take each element of the array and insert into the BST $\phi$ so that after insertion the BST property of the tree still holds.
3. Do an inorder traversal of the BST.

So steps 1 is $O(1)$ and step 3 is $\Theta(n)$ . I want to find out the lower and upper bounds on step 2.

-
Hint:we will get upper bound while inserting at the deepest node and lower bound while inserting at the shallowest node.Note that BST's are not height balanced. –  Saurabh Aug 20 '12 at 7:35
It might be more appropriate to use cs.stackexchange.com for such questions. –  Karolis Juodelė Aug 20 '12 at 7:36

The upper bound is $1+2+\dots+n = \frac{n(n+1)}{2}$ operations as the upper bound of insertion into a tree of $m$ nodes is $m$. This bound is reached is your array is sorted already.
The lower bound is $\log(1)+\log(2)+\dots+\log(n) = \log(n!)$ as the lower bound of insertion into a tree of $m$ nodes is $\log m$.. By Stirling's approximation this is $\log (\sqrt{2 \pi n}(\frac{n}{e})^n) = \frac{1}{2}\log(2\pi n) + n(\log(n) - 1)$ which is $O(n\log n)$.
@Geek, yes. Notice that these are not the only ways to reach it though. $2^{n-1}$ out of $n!$ orderings of the array do, I think. –  Karolis Juodelė Aug 20 '12 at 10:09


text:
Home How to bin timestamp data into buckets of n minutes in postgres

# How to bin timestamp data into buckets of n minutes in postgres

user963
1#
user963 Published in April 21, 2018, 3:26 pm
 I have the following query which works, binning timestamped "observations" into buckets whose boundaries are defined by the bins table: SELECT count(id), width_bucket( time :: TIMESTAMP, (SELECT ARRAY(SELECT start_time FROM bins WHERE owner_id = 'some id' ORDER BY start_time ASC) :: TIMESTAMP[]) ) bucket FROM observations WHERE owner_id = 'some id' GROUP BY bucket ORDER BY bucket;  I would like to modify this to allow for querying arbitrary n-minute bins starting from a specified timestamp, rather than having to pull from from an actual "bins" table. That is, given a start time, a "bin width" in minutes, and a number of bins, is there a way I can generate the array of timestamps to pass into the width_bucket function? Alternatively, is there a different/simpler approach to get the same results?
 You need to login account before you can post.
Processed in 0.299579 second(s) , Gzip On .


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Video submitted by User

# Fort Noesis

## Reviews

• ArmoredTier

Posted this review

7

Pros:

+   Map has daylight sections
+/- Huge emphasis on survival horror
+   It is possible to beat the map

Cons:

-Ridiculously hard because of the map layout and confusing escape sequences(Played on Advanced with a friend and two AI teammates)

-Aside from a couple of arrows on the walls, there is no clear sense of direction especially during the mad dashes to the saferooms and helicopter.

-Survival horror elements in the map are a hit-or-miss for a fast-paced game like L4D. Explosives and Tier 2 weapons are few and far between( on Advanced anyway.)

- Most times survival is based NOT on quick reaction time and teamwork but rather previous experience with the map (Dying repeatedly to become familiar with the special infected spawn and safehouse locations)

Conclusion:

It's a challenging map, not all for the right reasons though. It's beatable but you might find yourself more frustrated rather than satisfied by the end of it.

## File Details

noesis.vpk

(97.8 MB)

View Changelog

Version 5 Complete

Featured
• Mod
Ithaca M37
N/A

Ithaca M37 for Left 4 Dead. Includes a HUD icon. To change the hud icon on the weapon panel you will need to edit the file mod_textures.txt manually found in ..\left 4 dead\left4dead\scripts. "icon_equip_pumpshotgun" { "file"...

3,406
• Mod
N/A

I don't own this mod I just collected beta files from the internet and put it in one vpk file. Enjoy!

1,907
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L4D1 Zoey - The Aftermath
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A scrounged-together, though modern look for Zoey, in which she dons a tough bomber jacket, a somewhat abraded body stocking and short, practical boots -- all in addition to a sturdier belt on which she can rely for protection. Also pres...

• Updated
10,195
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Sirgibsalot's Globally Offensive M16
N/A

Globally Offensive M16 slightly altered from the original mod, i removed the zip ties and made the flashlight smaller. Includes a HUD icon. To change the hud icon on the weapon panel you will need to edit the file mod_textures.txt man...

• New
155
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text:
# CF887A Div. 64

• 174通过
• 538提交
• 题目来源
• 评测方式 RemoteJudge
• 标签 字符串 数论,数学 进制
• 难度 入门难度
• 时空限制 1000ms / 256MB
• 提示：收藏到任务计划后，可在首页查看。

## 题目描述

超模$Izabella$ 参加了比赛。她想要给评委留下深刻的印象并且展示她的数学技能。

她的问题如下：对于一个只由$0$ 和$1$ 组成的字符串，判断是否能删掉一些数字，使得得到的二进制数能被$64$ 整除。

## 输入输出格式

### 输入格式：

一行，为一个长度不超过$100$ 的$01$ 字符串。

### 输出格式：

如果可以，输出$yes$ ，否则输出$no$ 。

感谢@M_sea 提供的翻译

## 题目描述

Top-model Izabella participates in the competition. She wants to impress judges and show her mathematical skills.

Her problem is following: for given string, consisting of only 0 and 1, tell if it's possible to remove some digits in such a way, that remaining number is a representation of some positive integer, divisible by 64, in the binary numerical system.

## 输入输出格式

输入格式：

In the only line given a non-empty binary string $s$ with length up to $100$ .

输出格式：

Print «yes» (without quotes) if it's possible to remove digits required way and «no» otherwise.

## 输入输出样例

输入样例#1： 复制
100010001

输出样例#1： 复制
yes
输入样例#2： 复制
100

输出样例#2： 复制
no

## 说明

In the first test case, you can get string $1\ 000\ 000$ after removing two ones which is a representation of number $64$ in the binary numerical system.


text:
# confused ..... y=x-|x|

by mikebc
Tags: confused, yx|x|
 P: 20 The problem is to graph this equation y=x-|x|. From what I understand of absolute values, this x would be positive. If it is positive then y=0 and there would be no points to graph. Is there something that I am missing? The question is worth 4 points, so I can't see the answer just being 0. Thanks for any suggestions.
 Mentor P: 13,654 What happens when x is negative?
 P: 91 What I was taught to do when dealing with absolute value, is to rewrite the equation so that the absolute value is isolated, then find the 2 equations. so you'll have: x=x-y y=o and also: x=-x+y 2x=y
P: 20

## confused ..... y=x-|x|

The absolute value of -x would be x. But I remember with inequalities there are 2 possible answers with absolute value (-,+). From what you are explaining it sounds like that is what you are saying to do, use both possible values. Then I would graph by beginning with y at 0 and continue by substituting values into 2x=y? That seems to make sense to me. Thank you both for your help!
 P: 639 Also just try plugging in some numbers: For 2 -> Y = 2 - |2| = 0 For -2 - > Y = -2 -|-2| = -2 -2 = -4 see? So for negatives you have Y = 2X, X<0
 HW Helper P: 2,692 Solve or graph y = x - |x|. If x>0, then y = x - x, meaning y=0. If x<0, then y = x - (-x) [ notice those are parentheses, not absolute value notation symbols ], meaning y = x + x = 2x.
PF Patron
Thanks
Emeritus
P: 38,429
 Quote by mikebc The problem is to graph this equation y=x-|x|. From what I understand of absolute values, this x would be positive.
Surely you didn't mean to say that! x itself can be any number. |x| is always positive (or 0- don't forget that!

 Quote by dranseth What I was taught to do when dealing with absolute value, is to rewrite the equation so that the absolute value is isolated, then find the 2 equations. so you'll have: x=x-y y=o
Why did you switch to x=? If x$\ge 0$ y= x- x= 0. The graph is just the x axis from x= 0 to the right.

 and also: x=-x+y 2x=y
If x< 0, don't forget that. Then y= x- (-x)= 2x.

 Quote by mikebc The absolute value of -x would be x.
No, no, no! |-x|= |x| which may be eigther x or -x depending upon what x is.

 But I remember with inequalities there are 2 possible answers with absolute value (-,+). From what you are explaining it sounds like that is what you are saying to do, use both possible values. Then I would graph by beginning with y at 0 and continue by substituting values into 2x=y? That seems to make sense to me. Thank you both for your help!
Draw the graph of y= 2x, to the left of x= 0. To the right, the graph is just y= 0, the x-axis.
 P: 91 I rearranged the formula to isolate the absolute value.
 P: 20 Wow, you guys couldn't have made it any more clear for me. Thanks alot!

 Related Discussions Calculus 13 Calculus & Beyond Homework 2 General Math 4 Introductory Physics Homework 1 Introductory Physics Homework 2


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The Basic Strategy for the Blackjack Game Basic Strategy The most blackjack 5 card rules thing to learn about playing blackjack, and I can not stress this enough, is to learn basic strategy.
Basic strategy was created by using a computer simulation of millions of blackjack hands.
One of the first blackjack simulations was done by Julian Braun of IBM.
Basically, these computer simulations showed the mathematical probability of improving the blackjack hand or beating the dealer, by a certain playing strategy.
In other words, telling the blackjack player what blackjack 5 card rules he or she should take hit, split, double-down, etc.
The important thing to remember is that nearly all basic strategy rules are indisputable in the long run, the finite nature of mathematical science dictates that a particular playing decision, based on the player's cards and the dealer's exposed card, will yield a predictable outcome after millions of hands of play.
It is imperative as a blackjack player to learn this basic strategy, because by using it, you will eliminate the normal 5 or so percent advantage the casino has over the unskilled player.
That is less than a 1 percent disadvantage to the player.
This is why blackjack can be one of the most profitable games to play in a casino.
Basic strategy varies according to the type of playing conditions that you are exposed to.
Namely, how many decks that are in the blackjack 5 card rules you are playing.
Below I have listed the basic strategy rules for single deck games and for multiple deck shoes.
Another fact that I must stress is to play exactly the way listed below.
Do NOT alter your play based on some hunch that you have, or by a perceived idea that you always lose when you split eights for example.
You may think that you lose more often than you win in certain basic strategy situations, but this is NOT the case over the long run.
The only time you should vary from basic strategy is under certain circumstances where you are counting cards.
I will explain this in more detail under the card counting section of this web site.
But always remember, even the most skilled card counter will still use basic strategy for the vast majority of his or her playing decisions.
So I urge you to study the rules below and practice them until they do not require any thought on your part.
There are many inexpensive blackjack games which you can buy for your PC, and several shareware ones which you can easily obtain off of the internet.
The best commercial version that I have found is called "Play to Win" by Wizard Works, and it is available now in any major computer software supply store.
This game also has a coaching option where it will tell you what decision to make based on basic strategy.
In fact, if you attempt to make a wrong decision a dialog box will pop up and suggest to you the proper playing decision to make.
Since it opens a new window you can ALT-TAB back to Blackjack Center whenever you want to review the basic strategy rules listed below.
Regardless on what you use to practice, whether it be a computer blackjack game or simply a deck of cards, you MUST learn basic strategy to have any hope of being a consistent winner at the blackjack tables.
The charts below should be self explanatory.
On the left hand column is the total of the cards you have in your hand and the proper playing strategy based on the dealers up card.
SINGLE DECK - BASIC STRATEGY Your Hand vs Dealer's Upcard 8 Double on 5 to 6.
A,2 to A,5 Double on 4 to 6.
A,6 Double on 2 to 6.
A,7 Double on 3 to 6.
Stand on 2,7,8 or A.
Hit on 9 or 10.
A,8 Double on 6.
Stand on 7,10 or A.
The above chart assumes the casino doesn't allow doubling down after pair splitting.
If the casino allows doubling down after pair splitting then use the following pair splitting rules.
Your Hand vs Dealer's Upcard 2,2 Split on 2 to 7.
FOUR, SIX, EIGHT DECK - BASIC STRATEGY Your Hand vs Dealer's Upcard 5 to 8 Always Hit.
Hit on 10, A.
A,2 Double on 5,6.
A,3 Double on 5,6.
A,4 Double blackjack 5 card rules 4 to 6.
A,5 Double on 4 to 6.
A,6 Double on 3 to 6.
A,7 Double strip casinos with \$5 blackjack 3 to 6.
Stand on 2,7 or 8.
Hit on 9,10 or A.
A,8 to A,10 Always stand.
Treat as 10 above.
Stand on 7,10, or Https://yournaughtystory.com/blackjack/blackjack-basic-strategy-quiz.html />The above multiple deck basic strategy is valid if the players are allowed to double down after pair splitting.
If doubling down is blackjack 5 card rules allowed after pair splitting, then use the following pair splitting rules.
Your Hand vs Dealer's Upcard 2,2 Split on 4 to 7.


text:
# [XeTeX] No-Break Space

enrico.gregorio at univr.it enrico.gregorio at univr.it
Wed Mar 2 17:04:12 CET 2011

BPJ wrote:

> I just realized that the solution to a problem I've had may lurk
> here: I've been using interpuncts U+00B7 inside some words, and
> XeTeX won't hyphenate those words.  However it's not only that I
> want them hyphenated -- I don't want them hyphenated too close
> to the interpunct!  Never closer than two ordinary letters and
> preferably more if possible.  When there are many of those words
> it gets quite annoying to insert soft hyphens by hand.

You can try with

\newunicodechar{·}{·\nobreak\hspace{0pt}}

Thus the MIDDLE DOT will typeset itself, there will no line break
after it and the \hspace{0pt} allows the following word to be
hyphenated. You can't control the number of letters before the
hyphen, but (Xe)TeX is generally not abundant with hyphens.

Note that you can use the defined character in the second argument
as itself, this is a felicitous consequence of the implementation of
\newunicodechar. :)

Ciao
Enrico

--
Enrico Gregorio          + Dipartimento di Informatica          + Tel: +39 045 8027937
Enrico.Gregorio at univr.it + Università degli Studi di Verona     +
(gregorio at math.unipd.it) + Strada le Grazie 15 / I-37134 Verona + Fax: +39 045 8027928


text:
3 deleted 7 characters in body

I'd think the construction in question is the homotopy coimage of $f$ (it's unfortunately called "coimage" even though it behaves like the image).

First one forms the homotopy Cech nerve

$$C(f) = \left( \cdots X \times_Y X \times_Y X \stackrel{\to}{\stackrel{\to}{\to}}X \times_Y X \stackrel{\to}{\to} X \right)$$

This is the internal groupoid object that encodes the $\infty$-equivalence relation on the elements of $X$ "is equivalent in $Y$"

Forming its homotopy colimit

$$coim(f) := \lim_{\leftarrow} lim_{\to} C(f)$$

produces the homotopy quotient of $X$ by this equivlence relation.

As an example, take $\mathbf{B}G$ the one-object groupoid of a group $G$ and $* \to \mathbf{B}G$ the point inclusion. One wants to see that the homotopy image of the point inclusion $* \to \mathbf{B}G$ is not just the point, but is $*//G$, i.e. $\mathbf{B}G$ itself, because there is $G$ worth of ways for the point to be equivalent to itself after inclusion into $\mathbf{B}G$.

So one computes the homotopy Cech nerve and finds the familiar

$$C(* \to \mathbf{B}G) = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\times G \stackrel{\to}{\to} G \right)$$

but now regarded as a simplicial object in $\infty Grpd$. This is the diagram that encodes the action of $G$ on the point. Its homotopy colimit is indeed again $\mathbf{B}G$, so that we find

$$coim(* \to \mathbf{B}G) = \mathbf{B}G$$

That this comes out this way is an example of Giraud's axioms at work, since $\infty Grpd$ happens to be an $\infty$-topos: this implies that every $\infty$-groupoid object (simplicial diagram as above) in $\infty Grpd$ is effective .

2 added 10 characters in body

The

I'd think the construction in question is the homotopy coimage of $f$ (it's unfortunately called "coimage" even though it behaves like the image).

First one forms the homotopy Cech nerve

$$C(f) = \left( \cdots X \times_Y X \times_Y X \stackrel{\to}{\stackrel{\to}{\to}}X \times_Y X \stackrel{\to}{\to} X \right)$$

This is the internal groupoid object that encodes the $\infty$-equivalence relation on the elements of $X$ "is equivalent in $Y$"

Forming its homotopy colimit

$$coim(f) := \lim_{\leftarrow} C(f)$$

produces the homotopy quotient of $X$ by this equivlence relation.

As an example, take $\mathbf{B}G$ the one-object groupoid of a group $G$ and $* \to \mathbf{B}G$ the point inclusion. One wants to see that the homotopy image of the point inclusion $* \to \mathbf{B}G$ is not just the point, but is $*//G$, i.e. $\mathbf{B}G$ itself, because there is $G$ worth of ways for the point to be equivalent to itself after inclusion into $\mathbf{B}G$.

So one computes the homotopy Cech nerve and finds the familiar

$$C(* \to \mathbf{B}G) = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\times G \stackrel{\to}{\to} G \right)$$

but now regarded as a simplicial object in $\infty Grpd$. This is the diagram that encodes the action of $G$ on the point. Its homotopy colimit is indeed again $\mathbf{B}G$, so that we find

$$coim(* \to \mathbf{B}G) = \mathbf{B}G$$

That this comes out this way is an example of Giraud's axioms at work, since $\infty Grpd$ happens to be an $\infty$-topos: this implies that every $\infty$-groupoid object (simplicial diagram as above) in $\infty Grpd$ is effective .

1

The construction in question is the homotopy coimage of $f$ (it's unfortunately called "coimage" even though it behaves like the image).

First one forms the homotopy Cech nerve

$$C(f) = \left( \cdots X \times_Y X \times_Y X \stackrel{\to}{\stackrel{\to}{\to}}X \times_Y X \stackrel{\to}{\to} X \right)$$

This is the internal groupoid object that encodes the $\infty$-equivalence relation on the elements of $X$ "is equivalent in $Y$"

Forming its homotopy colimit

$$coim(f) := \lim_{\leftarrow} C(f)$$

produces the homotopy quotient of $X$ by this equivlence relation.

As an example, take $\mathbf{B}G$ the one-object groupoid of a group $G$ and $* \to \mathbf{B}G$ the point inclusion. One wants to see that the homotopy image of the point inclusion $* \to \mathbf{B}G$ is not just the point, but is $*//G$, i.e. $\mathbf{B}G$ itself, because there is $G$ worth of ways for the point to be equivalent to itself after inclusion into $\mathbf{B}G$.

So one computes the homotopy Cech nerve and finds the familiar

$$C(* \to \mathbf{B}G) = \left( \cdots G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\times G \stackrel{\to}{\to} G \right)$$

but now regarded as a simplicial object in $\infty Grpd$. This is the diagram that encodes the action of $G$ on the point. Its homotopy colimit is indeed again $\mathbf{B}G$, so that we find

$$coim(* \to \mathbf{B}G) = \mathbf{B}G$$

That this comes out this way is an example of Giraud's axioms at work, since $\infty Grpd$ happens to be an $\infty$-topos: this implies that every $\infty$-groupoid object (simplicial diagram as above) in $\infty Grpd$ is effective .


text:
# Question #80ebf

Mar 27, 2017

See the entire solution process below:

#### Explanation:

The $\sqrt{1025} = 32.02$

${1}^{2} = 1$

${2}^{2} = 4$

Therefore all numbers from $2$ through $32$ when squared will be between $2$ and $1025$

There are $31$ numbers from $2$ to $1025$ which are squares of whole numbers.


text:
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# Varied Length Stokes Shift BODIPY-Based Fluorophores for Multicolor Microscopy

## Abstract

Multicolor microscopy tools necessary to localize and visualize the complexity of subcellular systems are limited by current fluorophore technology. While commercial fluorophores cover spectral space from the ultraviolet to the near infrared region and are optimized for conventional bandpass based fluorescence microscopy, they are not ideal for highly multiplexed fluorescence microscopy as they tend to have short Stokes shifts, restricting the number of fluorophores that can be detected in a single sample to four to five. Herein, we synthesized a library of 95 novel boron-dipyrromethene (BODIPY)-based fluorophores and screened their photophysical, optical and spectral properties for their utility in multicolor microscopy. A subset of our BODIPY-based fluorophores yielded varied length Stokes shifts probes, which were used to create a five-color image using a single excitation with confocal laser scanning microscopy for the first time. Combining these novel fluorophores with conventional fluorophores could facilitate imaging in up to nine to ten colors using linear unmixing based microscopy approaches.

## Introduction

Multicolor microscopy has gained in popularity for the detection and visualization of multiple subcellular entities in a single sample. Specificity in highly multiplexed microscopy routinely relies on small molecule fluorophore tagged affinity reagents such as antibodies, antibody fragments, nanobodies, affibodies, and lectins for specific biomolecular labeling. Imaging using conventional bandpass based microscopy is common, where spectral separation is limited to four to five colors that span the ultraviolet (UV) to near-infrared (NIR) spectra. Unique fluorophore detection in a single sample can be improved using spectral imaging and linear unmixing based fluorescence microscopy to simultaneously identify overlapping emission spectra, increasing multicolor imaging capability1,2. Fluorescence linear unmixing studies have demonstrated up to five color imaging with overlapping subcellular entities using two excitation wavelengths and emission distributed over 150 nm of spectral space3.

Optimal fluorophore technology for bandpassed based- and linear unmixing based-multicolor microscopy different substantially. Ideal fluorophores for bandpass-based multicolor microscopy have relatively short Stokes shifts (15–25 nm) and narrow emission spectra, minimizing spectral overlap for unique color detection. In contrast, fluorophores with varied length Stokes shifts are ideal for linear unmixing-based microscopy as they permit increased spectral detection using each excitation wavelength. Spectral separation using linear unmixing is also improved when emission spectra are distinct, where separate emission maxima and narrow or distinctly shaped emission spectra are desirable for a high degree of multiplexing3. Additionally, ideal fluorophores for both bandpass-based and linear unmixing-based multicolor microscopy generate high contrast, making it strategic to choose probes with high quantum yield and low signal to background ratio (SBR) for the untargeted small molecule fluorophore, permitting high SBR upon affinity reagent tagging4.

Imaging with fluorophore tagged affinity reagents enables versatility in fluorophore choice based on photophysical and optical properties, where affinity reagents are typically conjugated to small molecule fluorophores consisting of core chemical scaffolds including xanthene, rhodamine, Boron-dipyrromethene (BODIPY), cyanine, and oxazine5,6. While these fluorophore classes cover a broad range of excitation and emission spectra, they tend to have short Stokes shifts. For example, commercially available BODIPY based fluorophores are mostly limited to those with Stokes shifts less than 15 nm5. AlexaFluor dyes, one of the most popular brands of fluorophores, are based on a variety of scaffolds and have a typical Stokes Shift of 20 nm or less5. While these conventional fluorophores are ideal for bandpass-based multicolor imaging and have been used in for a variety of imaging studies, the need for complementary fluorophore technology optimized for linear unmixing-based multicolor microscopy persists. The development of a suite of fluorophores with both varied length Stokes shifts and the ability to generate high contrast would fulfill this need.

BODIPY is a widely utilized fluorophore scaffold due to its valuable photophysical and optical properties including its spectral stability at varied polarity, pH and physiologic conditions, resistance to photobleaching, relatively high quantum yield, and narrow emission spectra7,8. Furthermore, the BODIPY scaffold is particularly advantageous for novel fluorophore development since small modifications to its structure enables tuning of its fluorescence characteristics and the scaffold can be readily customized using a variety of synthetic routes8,9,10,11,12,13,14. Fluorophore libraries originating from various core BODIPY scaffolds have resulted in probes with an array of Stokes shifts which have demonstrated utility as chemical sensors and probes15,16,17,18. However, the previously synthesized BODIPY fluorophore libraries did not contain chemical moieties suitable for conjugation chemistries, limiting their utility for affinity tag labeling. Other BODIPY syntheses have successfully created long Stokes shift fluorophores for bioconjugation, but have been restricted to the NIR spectral region19.

Herein, we report the first synthesis of a BODIPY-based fluorophore library specifically designed to generate varied length Stokes shift probes that each contain a chemical moiety compatible with conjugation to affinity reagents for multicolor imaging applications. The novel BODIPY-based library was characterized for its SBR, photophysical, optical and spectral properties to enable selection of a subset of fluorophores most appropriate for linear unmixing-based multicolor microscopy imaging applications. The BODIPY-based library was also screened for organelle specificity in fixed, permeabilized cells and characterized for its in vitro biodistribution. Novel BODIPY-based fluorophores that targeted the nucleus and nucleolar regions, vesicular structures, and cytosolic regions in fixed cells were identified from the library. Through conjugation of our varied length Stokes shift BODIPY-based fluorophores to affinity reagents for five distinct subcellular structures, we demonstrated accurate labeling and imaging of five colors in a single sample with confocal laser scanning microscopy using spectral imaging and linear unmixing. Excitingly, our five-color image was obtained on spatially overlapping structures using a single excitation wavelength and emission spectra distributed over just 60 nm of spectral space. Utilization of our novel BODIPY-based fluorophores in combination with conventional short Stokes shift probes and other long Stokes shift fluorophores, such as the NIR BODIPY probes12,19, will permit highly multiplexed linear unmixing-based microscopy studies.

## Results

### Solid Phase Synthesis of BAA Library

A library of BODIPY-based fluorophores was synthesized using solid phase synthesis to add structurally diverse styryl moieties (Fig. 1), with the goal of creating fluorophores with varied length Stokes shifts for multicolor imaging applications. Commercially available BODIPY FL was loaded onto chlorotrityl chloride polystyrene resin (CTC-PS) through its carboxylic acid moiety (1), where the CTC-PS acted as a protecting group for the carboxylic acid20. The styryl modification of BODIPY FL was carried out by reacting the core fluorophore with 79 structurally diverse aromatic aldehydes via Knoevenagel condensation (b)16. Each aromatic aldehyde reaction with BODIPY FL resulted in novel fluorophores with a single styryl modification (BAA-a), two styryl modifications (BAA-b) or a mixture of the single and double styryl addition (Table S1). The novel BAA fluorophores were cleaved from the CTC-PS resin (c) yielding BAA-a, BAA-b or a mixture of BAA-a and BAA-b. After purification of the cleaved material by high performance liquid chromatography (HPLC), a library of 95 novel BAA fluorophores with a minimum of 80% purity (average purity = 99%) was obtained (Table S1). 60 BAA fluorophores had one styryl modification and 35 fluorophores had two styryl modifications. NMR and MS spectra of five selected BAA fluorophores (BAA-37a, BAA-22a, BAA-5a, BAA-2a and BAA-39a) revealed single addition of the aromatic aldehyde occurred exclusively at the α-methyl group (Figs S110).

### BAA Library Photophysical and Optical Properties

Addition of the aromatic aldehydes to the core BODIPY FL structure (λmax ABS = 503 nm, λmax EM = 512 nm) resulted in substantially red-shifted absorbance (ABS) and emission (EM) spectra of all novel BAA fluorophores (Fig. 2A). The BAA library ranged in maximum absorbance wavelengths from 480–614 nm and maximum emission wavelengths from 578–714 nm. Notably, 84% of the BAA fluorophores had emission wavelengths in a narrow range (578–610 nm), shifting the average emission wavelengths 75–100 nm from the original BODIPY FL compound. The BAA fluorophores ranged in Stokes shifts from 2–112 nm, where the novel fluorophores generally had longer Stokes shifts than BODIPY FL (Stokes shift = 9 nm), with an average Stokes shift of 18 nm (Table S2). The quantum yields of the BAA fluorophores varied widely, ranging from low at 0.04 to high at > 0.99, with an average of 0.38 (Fig. 2B). While the average quantum yield was lower than the original BODIPY FL (quantum yield = 0.81), it will still provide bright signal for fluorescence imaging applications. The full-width-at-half-maximum (FWHM) was quantified to determine the spectral spread in emission wavelengths for multicolor microscopy. The FWHM of the BAA library had a broad range (30–102 nm) with an average of 44 nm (Fig. 2C), which was generally wider than BODIPY FL (34 nm) as would be expected from longer Stokes shift fluorophores. The BAA fluorophores also had limited background staining in cells, which would permit high contrast when affinity conjugated for staining of specific structures (Fig. 2D).

### Organelle Specificity and SBR of BAA Library

The BAA library was screened in permeabilized, fixed U2OS cells at 10 µM to assess any organelle specificity of the novel fluorophores (Tables S3 and S4). At the selected concentration, all BAA fluorophores showed cell staining patterns with higher intensity than background autofluorescence demonstrating some level of organelle specificity. Interestingly, the cellular staining pattern varied across the library (Table S3), where three BAA fluorophores were found to localize only to the nucleus and nucleolar regions of the cell including BAA-2a, BAA-30a, and BAA-77a (Fig. 3A), vesicular structures were stained by 22 of the BAA fluorophores (Fig. 3B), cytosolic staining was seen from 34 of the BAA fluorophores (Fig. 3C), and the remaining 36 BAA fluorophores showed a combination of vesicular and cytosolic structural staining.

The BAA library was also screened for fluorescence staining in fixed U2OS cells at 100 nM to determine if nonspecific binding of the BAA fluorophores would be detectable at fluorophore concentrations typically used for immunofluorescence staining. The BAA fluorophores had limited staining at 100 nM concentration, where more than half of the compounds had a SBR less than 2 and ~75% had a SBR less than 3, while the average SBR for all 95 BAA compounds was 2.65 (Fig. 2D and Table S4). Background staining of 2–3 was minimal compared to the SBR seen in organelle specificity screening, where SBR ranged from 9–20, while immunofluorescence images had a minimum SBR of 10, demonstrating the utility of the BAA fluorophores for specific staining studies.

Molecular properties calculations for the BAA fluorophore library were completed where the partition coefficient (LogD), number of rotatable bonds, number of hydrogen bond donors and acceptors (HBD and HBA), polar surface area and pKa were characterized (Table S4). Addition of the aromatic aldehydes to BODIPY FL generated a wide range of partition coefficients ranging from −4.22 to 7.12, with an average of 2.25, displaying a general increase from BODIPY FL (−2.13). The number of rotatable bonds was also highly variable in the BAA library, ranging from 5–22 with an average of 8.5, where all BAA fluorophores showed an increase of at least 3 rotatable bonds over BODIPY FL (rotatable bonds = 2) due to the increase in size of the BAA fluorophores compared to BODIPY FL. The number of hydrogen bond donors was similar across the library (1–3, with an average HBD = 1.1) and to BODIPY FL (HBD = 1), while the number of hydrogen bond acceptors showed a fairly large range from 2–12 with an average HBA = 4, differentiating the derivatives from BODIPY FL (HBA = 4) with both higher and lower values. The polar surface area of the BAA fluorophores was also widely variable (45–153, with an average of 72), where polar surface area was relatively increased in comparison to BODIPY FL (polar surface area = 45). The calculated pKa of the carboxylic acid moiety was not substantially different across the BODIPY library ranging from 3.52–4.77 with an average of 4.21, which was also not substantially different than the pKa of BODIPY FL (pKa = 4.49).

### BAA Fluorophores Selected for Immunofluroescence Staining

Five BAA fluorophores were selected to create a multicolor immunofluorescence image with linear unmixing-based laser scanning confocal microscopy (LSM), including BAA-37a, BAA-22a, BAA-5a, BAA-2a, and BAA-39a (Fig. 4A). The BAA fluorophores were selected due to their relatively high quantum yield, narrow FWHM and photostability (Figs 2B,C, 4A–D) permitting five color simultaneous fluorescence microscopy, where all fluorophores could be optimally excited using a single 561 nm laser. The minimum spectral separation between the five selected BAA fluorophores was 14 nm, facilitating simultaneous spectral detection. Density functional theory (DFT) calculations were used to determine the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy levels of the five selected BAA fluorophores (Fig. S11). The HOMO-LUMO energy gaps ranged from 2.31–2.50 eV (Table 1). As expected, BAA fluorophores with lower energy gaps tended to have red shifted maximum absorption wavelengths.

### Multicolor Linear Unmixing-Based Microscopy Imaging with BAA Fluorophores

Five subcellular structures in fixed U2OS cells were labeled with the selected BAA fluorophores and imaged using linear unmixing-based LSM. The carboxylic acid moiety of the BAA fluorophores was converted to an N-hydroxysuccinimide (NHS) ester before conjugation to the labeling proteins used to specifically target five subcellular structures with indirect immunofluorescence and lectin labeling strategies (Table 2). The inner nuclear membrane was labeled with an anti-Lamin B1 antibody via indirect immunofluorescence labeling with BAA-37a, the outer mitochondrial membrane was labeled with an anti-Tomm20 antibody via indirect immunofluorescence labeling with BAA-22a, tubulin was labeled with an anti-tubulin antibody via indirect immunofluorescence labeling with BAA-5a, the extracellular membrane was labeled with the lectin, wheat germ agglutinin conjugated to BAA-2a, and vimentin was labeled with an anti-vimentin antibody via indirect immunofluorescence labeling with BAA-39a.

Each subcellular structure was labeled and imaged individually to confirm the accuracy of immunofluorescence staining (Fig. 5A–E) and to assess the in situ spectral emission of the five BAA fluorophores for linear unmixing of multicolor images (Fig. 5F). The in situ spectral emission was collected at 8.9 nm step size and was similar to the emission spectra collected in solution (Figs 2B and 5F). The only differences between the in solution and in situ spectra were that the emission peaks of BAA-5a and BAA-39a were blue shifted 7 and 15 nm, respectively from the in solution spectral emission. A five-color image, with all five subcellular structures labeled simultaneously, allowed for correct identification and visualization of each structure upon linear unmixing (Fig. 5G–L). Notably, labeling of tubulin (Fig. 5J) and the extracellular membrane (Fig. 5K) showed lower contrast in the multicolor image than in their corresponding individually labeled images (Fig. 5C,D). Additionally, there was minimal cross talk between the channels, determined based on the visible structures in each linearly unmixed image (Fig. 5H–L).

## Discussion

In summary, we designed and developed a library of BODIPY-based fluorophores with varied length Stokes shifts compatible with conjugation to affinity reagents for multicolor microscopy imaging applications. The diverse styryl modifications of the core BODIPY structure extended its π-conjugation resulting in substantially red-shifted absorbance and emission wavelengths21,22. The maximum absorbance and emission of each unique BAA fluorophore was influenced by the electron withdrawing or donating tendencies of the aromatic aldehyde used for styryl modification23,24.

The judicious design of steric hindrance and resonance effects in fluorophore architectures lead to enhanced Stokes shifts. For example, the relatively sterically encumbered BAA-16a had a Stokes shift of 58 nm without any heteroatoms in its styryl modification (Fig. 6A). All of the other fluorophores in the BAA library that exhibited Stokes shifts >30 nm possessed amine groups with lone electron pairs that delocalize into the styryl π-system. This was in agreement with a previously observed, strong effect of an electron-donating aniline on the Stokes shift of BODIPY dyes25,26. In the case of styryl and related BODIPYs, such as those described herein, the delocalization of the amine group lone pairs lead to charge-separated structures that were relatively strong contributors to excited state geometries. This resulted in dipole moment differences between the ground and excited states, enhancing the Stokes shifts. Compound BAA-39a showed two types of amine groups as part of the extended styryl framework (Fig. 6B). Only the nitrogen in the pyrrolydine ring could delocalize its lone electron pair and be part of the π-conjugated system. In contrast, the lone pair nitrogen in the pyridine ring was perpendicular to the orbitals in the aromatic system. Calculations of the molecular electrostatic potential surface showed that the positive charge was delocalized along the styryl extended π-conjugated system, while the negative charge was localized between the boron and the fluorine atoms (Fig. 6B).

DFT calculations of the dipole moment (μ) for optimized geometries in the ground and excited states for BAA compounds with varied length Stokes shifts including BAA-2a, BAA-5a, BAA-28a, BAA-37a and BAA-39a revealed a strong linear correlation between the absolute value of the difference in the dipole moments and Stokes shifts (Fig. 6C). These results suggest that significant changes in the geometry and charge distribution in the fluorophore excited state, can indeed produce larger Stokes shifts. Notably, the exceptional correlation between the simulations and the experimental data can potentially be used in the future design of fluorophores, where a known Stokes shift value is required, saving valuable time and synthetic resources.

Although the photophysical and optical properties varied among the novel BAA fluorophores, most compounds were compatible with excitation using a standard 561 nm laser. This enabled production of a multicolor image with a single excitation source by choosing fluorophores with different length Stokes shifts from the library including, BAA-37a, BAA-22a, BAA-5a, BAA-2a, and BAA-39a. Linear unmixing-based multicolor microscopy demonstrated simultaneous visualization of five subcellular structures (Figs 5G–L), facilitated by bright, spectrally distinct novel BODIPY-based fluorophores for detection and spectral separation. Tubulin labeled with BAA-5a and the extracellular membrane labeled with BAA-2a, showed lower contrast when imaged simultaneously (Figs 5J,K) compared to when imaged individually (Figs 5C,D). This was likely due to significant spectral overlap between the two fluorophores, which caused signal loss during linear unmixing (Figs 5F). Using a fluorophore with an increased red-shifted emission in place of BAA-2a in the five-color imaging scheme, such as BAA-30a, could improve multicolor imaging contrast by further increasing the spectral separation, thus improving linear unmixing.

While five fluorophores were selected for the BAA library for the multicolor microscopy imaging example demonstrated herein, additional fluorophores from the BAA library will find utility in future fluorescent imaging applications. For example, additional BAA fluorophores could be chosen to increase multiplexing capacity using a standard 561 nm laser for excitation, in addition to other common lasers such as 594 and 647 nm, where BAA fluorophores from the library showed absorbance. Likewise, BAA fluorophores could also be combined with commercially available fluorophores to create alternative color combinations and to further increase multiplexing capacity by taking advantage of both long and conventional short Stokes shift fluorophores. For example, using a combination of our five selected BODIPY fluorophores and conventional short Stokes shift fluorophores excited with 488 nm, 561 nm, 594 nm and 647 nm lasers, could result in an 8–10 color image from simultaneous fluorescence labeling.

Interestingly, we found that most BAA fluorophores had some intracellular labeling affinity, where many of our novel fluorophores labeled multiple areas of the cells, but unexpectedly some BAA fluorophores appeared to have organelle specificity (Fig. 3, Tables S3 and S4). Notably, we found three BAA fluorophores that specifically labeled the nucleus and nucleolar regions, indicating potential affinity for nucleic acids. Other BAA fluorophores appeared specific to particular vesicles, possibly localizing to the adiposomes, lysosomes or peroxisomes. Additionally, there were BAA fluorophores that appeared specific to a particular structure within the cytosol, possibly localizing to the endoplasmic reticulum, Golgi complex or mitochondria. Recent studies have shown that pH can have a strong effect on observed fluorescence signal in cellular organelles27. The pKa of our novel BODIPY compounds was calculated to determine if pH could explain the organelle staining differences seen across our library, however the pKa was not substantially different across our 95 BODIPY compounds, demonstrating the pH was not the sole reason for varied organelle accumulation. The reasons behind the precise intracellular location of the BAA fluorophores is under further investigation.

In summary, we designed, synthesized and characterized a BODIPY-based library to generate novel probes with varied length Stokes shifts for multicolor microscopy imaging applications. We found that individual BAA fluorophores have potential as organelle specific fluorophores for labeling fixed cells, but overall staining at immunofluorescence concentrations resulted in low SBR ideal for targeted imaging. We demonstrated the application of our BAA fluorophores to generate a representative five-color image over a narrow 60 nm spectral range using a single excitation wavelength. Linear unmixing-based microscopy with fluorophores in a narrow spectral window will advance multicolor imaging, where our novel probes could be combined with conventional short Stokes shift fluorophores to generate an 8–10 color multiplexed image from a simultaneously stained sample.

## Materials and Methods

### Chemicals and Fluorophores

All commercially available starting materials were used without further purification unless otherwise stated. 4,4-Difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-propionic acid (BODIPY FL) was used as the starting material for all novel fluorophore synthesis (Thermo Fisher Scientific, Waltham, MA). All novel BODIPY based fluorophores were synthesized through the addition of styrl based aromatic aldehyde building blocks (Thermo Fisher Scientific or Sigma-Aldrich, St. Louis, MO). Solid phase synthesis was performed using 2-Chlorotrityl chloride polystyrene (CTC-PS) 100–200 mesh resin (EMD-Millipore, Billerica, MA). Solvents for synthesis, purification and analysis were purchased at liquid chromatography/mass spectroscopy (LCMS) purity grade (Thermo Fisher Scientific).

### Synthesis of BODIPY Aromatic Aldehyde (BAA) Fluorophore Library

The BODIPY FL was loaded onto the CTC-PS resin as follows; 500 mg of BODIPY FL in 17.2 ml of dichloromethane (DCM) was added to 5 g of CTC-PS resin in 14.9 ml of diisopropylethylamine (DIEA). The mixture was protected from light and mixed on a shaker overnight. The mixture was washed while filtering the CTC-PS resin using DCM, dimethylformamide (DMF), methanol (MeOH), DMF, and DMC, with three washes per solvent. The BODIPY FL loaded CTC-PS resin was then dried with ethyl ether, resulting in ~90% BODIPY FL loaded onto the resin, as quantified by absorbance spectroscopy and the extinction coefficient of BODIPY FL (80,000 M−1cm−1). BODIPY FL loaded CTC-PS resin (100 mg in 3 ml of 1:1 dimethylsulfoxide (DMSO):acetonitrile (ACN)) and 15 equivalents of the selected aromatic aldehyde (Table S1, 0.45 mmol in 1 ml 1:1 DMSO:ACN) were mixed with 65 equivalents of acetic acid (105 µl, 2 mmol) and 65 equivalents of pyrrolidine (150 µl, 2 mmol) at 85 °C for 15 min. The BODIPY FL loaded CTC-PS resin was washed three times each with DCM, DMF, DCM, DMF, DCM, and DMF, followed by washing three times with ethyl ether. The synthesized BODIPY aromatic aldehyde (BAA) fluorophores were then cleaved with 0.5% trifluoroacetic acid (TFA) in DCM for 10 min at room temperature. The BAA fluorophores were purified using high performance liquid chromatography (Agilent 1250 Infinity HPLC) with a C18 column (150 × 21.2 mm) using solvents A: water-formic acid (CH2O2): 99.9:0.1 and B: ACN-CH2O2: 99.9:0.1, with the gradient 90% A to 50% A (6 min) followed by 50% B to 10% B (14 min) at a flowrate of 8 ml/min. The purified samples were lyophilized for analysis (Labconco, Kansas City, MO).

### BAA Library Structural & Purity Assessment

The mass to charge (m/z) ratio and purity of all BAA fluorophores was assessed by tandem LCMS (Agilent 6244 time of flight LCMS with diode array detector VL+) using a C18 column (4.6 × 50 mm), where purity was determined through area under the curve (AUC) analysis of the absorbance at 254 nm (Table S1). Physiochemical properties of the BAA library (Table S4) were calculated using Marvinsketch (ChemAxon, Cambridge, MA)28. Density functional theory (DFT) calculations at the B3LYP/6-31 G(d,p) level were performed using Gaussian 09 (Gaussian, Inc., Wallingford, CT)29. 1H NMR spectra were recorded for the five BAA fluorophores selected for multicolor microscopy (BAA-37a, BAA-22a, BAA-5a, BAA-2a and BAA-39a) on (ARX-400 Avance spectrometer, Bruker, Billerica, MA). All chemical shifts (δ) reported in ppm are relative to the signals of residual solvent DMSO-d6 (2.50 ppm), and coupling constants are given in Hz. High-resolution mass spectra (HRMS) were measured for the five BAA fluorophores selected for multicolor microscopy (BAA-37a, BAA-22a, BAA-5a, BAA-2a and BAA-39a) on a ThermoElectron LTQ-Orbitrap high resolution mass spectrometer (Thermo Fisher) with a dedicated Accela HPLC system (Thermo Fisher).

### BAA Library Optical Property Measurements

Spectroscopic characterization was performed in DMSO in black polystyrene 96 well plates with clear bottoms (Corning, Corning, NY) using a SpectraMax M5 spectrometer with a Microplate reader (Molecular Devices, Sunnyvale, CA). Analysis included solvent corrected measurement of absorbance spectra from 400–800 nm, calculation of maximum absorbance wavelength, measurement of fluorescence emission spectra using the maximum absorbance for excitation from 10 nm above the excitation to 800 nm, calculation of the maximum fluorescence emission wavelength, measurement of quantum yield, and calculation of the full-width-at half maximum (FWHM) (Table S2). Quantum yields (Φfl) were calculated by comparing the area under the emission spectrum of the BAA fluorophore to a reference fluorophore solution of Fluorescein, Rhodamine B, or Cresyl violet, at three concentrations using equation (1), where Grad represents the gradient from the plots of integrated fluorescence intensity vs. absorbance at three concentrations and ɳ is the refractive index of the solvent30. Fluorescein in 0.1 M sodium hydroxide (NaOH) (Φfl = 0.91)31 was used as the reference for all BAA fluorophores with a maximum absorbance of 480–510 nm. For quantum yield measurements using Fluorescein, excitation at 470 nm was used with the emission spectra integrated from 490–800 nm. Rhodamine B in ethanol (Φfl = 0.70)32,33 was used as the reference for all BAA fluorophores with a maximum absorbance of 511–595 nm. For quantum yield measurements using Rhodamine B, excitation at 525 nm was used with the emission spectra integrated from 545–800 nm. Cresyl violet in methanol (Φfl = 0.54)32 was used as the reference for BAA fluorophores with a maximum absorbance above 595 nm. For quantum yield measurements using Cresyl violet, an excitation at 570 nm was used with the emission spectra integrated from 590–800 nm.

$${\Phi }_{fl}^{sample}={\Phi }_{fl}^{reference}(\frac{Gra{d}^{sample}}{Gra{d}^{reference}}){(\frac{{{\rm{\eta }}}^{sample}}{{{\rm{\eta }}}^{reference}})}^{2}$$
(1)

### BAA Photostability Measurements for Select BAA Fluorophores

Photostablity of the five BAA fluorophores selected for multicolor microscopy (BAA-37a, BAA-22a, BAA-5a, BAA-2a and BAA-39a) and the parent BODIPY FL was measured in a 9:1 mixture of phosphate buffered saline (PBS)/DMSO. Fluorescence emission measurements were collected using a quartz cuvette in a SpectraMax M5 spectrometer (Molecular Devices). Fluorescence emission was monitored at the emission maximum of each compound every 5 seconds over a 30 minute period and reported as the normalized fluorescence intensity.

### Cell Culture

The U2OS osteosarcoma human cell line was cultured in phenol red free Dulbecco’s Modified Eagle Medium (Thermo Fisher Scientific) supplemented with 10% fetal bovine serum (VWR, Radnor, PA) and 1% Penicillin-Streptomycin-Glutamine (Thermo Fisher Scientific) at 37 °C and 5% CO2. Cells were plated in a 96-well glass bottom plates (Cellvis, Mountain View, CA) and incubated for 3 days to reach ~50% confluency prior to staining and imaging studies.

### BAA Library Staining for Organelle Specificity & SBR Calculations

Cells were pre-extracted with 0.5% Triton X-100 in 1× phosphate buffered saline (PBS) for 20 s, fixed with 0.4% glutaraldehyde (GA, Electron Microscopy Science, Hatfield, PA) and 0.25% Triton X-100 in PBS for 90 s, washed again with PBS and finally fixed with 3% GA in PBS for 15 min. Cells were washed with PBS (3 × 5 min), reduced with 10 mM sodium borohydride for 10 min, washed again with PBS (3 × 5 min), and blocked with 5% bovine serum albumin (BSA) in PBS for 10 min. BAA fluorophores were diluted to 10 µM in PBS for organelle screening and 100 nM for SBR screening. The diluted BAA fluorophores were incubated with the fixed, permeabilized cells for 30 min after which the fluorophore was removed and the cells were washed with PBS (3 × 5 min). Cells were counterstained with 4′,6-Diamidine-2′-phenylindole dihydrochloride (DAPI) at 1 µg/ml for 5 min and washed with PBS (3 × 5 min) prior to imaging for organelle specificity and SBR measurements.

Cell imaging was completed using a Zeiss AxioObserver (Carl Zeiss Microscopy GmbH, Jena, Germany), with a PhotoFluor II (89 North, Burlington, VT) excitation source and 63 × oil immersion objective. Samples were imaged with the optimal bandpass excitation and emission filter sets (Chroma Technology, Bellows Falls, VT) based on the BAA optical property measurements, where one of three filter sets was used including (1) 470 ± 20 nm excitation and 525 ± 25 nm emission, (2) 545 ± 12.5 nm excitation and 605 ± 35 nm emission, or (3) 620 ± 30 nm excitation and 700 ± 37.5 nm emission. Organelle specificity images were collected at exposure times ranging from 1.5–10 sec. Control images of unlabeled cells were collected at the same exposure times to confirm fluorescence was from BAA fluorophore staining and not from autofluorescence alone. Images for SBR analysis were collected at 1.5, 5, and 10 sec exposure times. Two images were obtained for each exposure time, one image to capture the DAPI using 405 ± 20 nm excitation and 470 ± 20 nm emission, and a second image to capture the BAA fluorophore using its appropriate filter set. Different regions of interest were imaged for each exposure time to ensure photobleaching was not affecting SBR measurements.

Image analysis was performed using QiTissue software (Quantitative Imaging Systems, Pittsburgh, PA). All images were first corrected for camera or scanning sensor bias, as well as dark current introduced at longer exposure times. These values were calculated and subtracted to allow proper computation of the ratios described below. The DAPI images enabled automatic detection of cell nuclei and an estimated location of the surrounding cytoplasm. Regular thresholding would introduce a measurement bias in situations with varied staining results. Instead, non-linear preprocessing, and a-priori knowledge based detection was used to define the organelles, regardless of their variation in intensities. This resulted in a reliable estimate of the area occupied by the target of labeling protocol for each type of image. The calculation then proceeded to determine the average intensity of area occupied (Io) and the average intensity of the background (Ib) for each sample, and baseline intensity for each filter set f with no sample present (Izf). The SBR at each exposure time was calculated using equation 2. The final SBR reported was calculated as the average (avg) SBR measured across the three exposure times for each sample.

$$SBR=(\frac{Io-Izf}{avg(Ibf)-Izf})$$
(2)

### Protein & Antibody Labeling with BAA Fluorophores

BAA fluorophores BAA-2a, BAA-5a, BAA-22a, BAA-37a, and BAA-39a were selected for multicolor cell imaging studies based on their optical property measurement. These five BAA fluorophores were converted from their carboxylate form to an amine-reactive NHS ester prior to protein conjugation. BAA fluorophores were diluted into DMF with 2.2 equivalents of 1-[Bis(dimethylamino)methylene]-1H-1,2,3-triazolo[4,5-b]pyridinium 3-oxid hexafluorophosphate (HATU) and 5.8 equivalents of DIEA and mixed for 30 min. Four equivalents of N-hydroxysulfosuccinimide (sulfo-NHS) were added and mixed for an additional 60 min. The reaction was diluted with ethyl acetate (2 ml) and washed with water (500 µl) to separate the NHS ester functionalized BAA fluorophore from any remaining reactants. The NHS ester functionalized BAA fluorophore in ethyl acetate was dried by lyophilization. LCMS analysis was used to confirm >90% NHS conversion for each BAA fluorophore.

Five targets were selected for multicolor microscopy and labeled using BAA fluorophores conjugated to the appropriate affinity proteins (Table 2). Conjugates were made using standard NHS ester reaction conditions, where the protein or antibody was buffered exchanged into 1× PBS, pH adjusted to 8.3 with 1 M sodium hydroxide, and mixed with the NHS ester functionalized BAA fluorophore in a 3:1 fluorophore to affinity tag molar ratio. The mixture was rocked gently at room temperature protected from light for 3 hours. The conjugation reactions were purified using fast protein liquid chromatography (FPLC, NGC Quest 10 Plus Chromatography System, Bio-Rad, Hercules, CA) by size exclusion (P6 gel filtration column, 40 × 12.6 mm, Bio-Rad). Final fluorophore concentration, protein concentration and conjugation ratios were determined using absorbance spectroscopy (Table 2).

### Immunofluorescence Staining of Cellular Structures

Live cells were incubated with BAA-2a conjugated to WGA to label the cell membrane at 20 µg/ml WGA in PBS for ~10 s before adding paraformaldehyde (PFA, Electron Microscopy Science) to a final concentration of 2% PFA and incubating for 10 min at 37 °C and 5% CO2. The media and fixative mixture was exchanged with fresh 2% PFA and incubated for an additional 20 min at 37 °C and 5% CO2. Cells were pre-extracted with 0.5% Triton X-100 in PBS for 20 s, fixed with 0.4% GA and 0.25% Triton X-100 in PBS for 90 s, washed with PBS and finally fixed with 3% GA in PBS for 15 min. Cells were then incubated with primary antibodies in 5% BSA overnight at 4 °C, protected from light. The primary antibodies were incubated at 10 μg/ml anti-lamin A, 10 μg/ml anti-TOMM20, 10 μg/ml anti-tubulin, and 6.6 μg/ml anti-vimentin. Following overnight incubation, the cells were washed with PBS (3 × 5 min) then incubated with the fluorophore conjugated secondary antibodies at 30 μg/ml BAA-37a-msIgG3, 2 μg/ml BAA-22a-rbIgG, 20 μg/ml BAA-5a-ratIgG, and 50 μg/ml BAA-39a-chIgG as measured by protein concentration in 5% BSA for 30 min at room temperature protected from light. The antibody stained cells were then washed with PBS (3 × 5 min) and post-fixed with 4% PFA in PBS for 10 min before a final PBS wash. For images with single BAA fluorophore labeling, cells without BAA-2a-WGA were used.

### Linear Unmixing-Based Confocal Laser Scanning Microscopy

Images were acquired using a Zeiss LSM 880 with Airyscan using a plan-apochromatic 63 × (NA = 1.4) oil immersion objective. Images were collected in lambda mode with the following settings: laser 561 nm (DPSS 561-10), 0.05%; beam splitter: MBS 458/561; filter: 410–695 nm; pixel dwell: 48.8 µs, average: line 1; master gain: 900; pinhole size: 60 um; and acquisition area: 1384 × 1384 pixels, 16 bit. Spectral data were acquired from 410–695 nm using samples labeled with individual BAA fluorophores as well as samples labeled with all five BAA fluorophore simultaneously. Zen imaging software (Zeiss) was used for spectral unmixing and FIJI ImageJ was used to process the final images34.

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## Acknowledgements

We would like to thank William Liu for experimental assistance and Dr. Nick J. Dolman for insightful discussions. This work was supported by a Collaborative Research Agreement from ThermoFisher Scientific and a Damon Runyon-Rachleff Innovator Award from the Damon Runyon Cancer Research Foundation.

## Author information

Authors

### Contributions

A.M.B. and S.L.G. designed the experiments. A.M.B., A.M.D. and L.W. performed experiments. A.M.B., L.W., M.A.N., J.O.E., R.M.S. and S.L.G. performed data analysis and interpretation. A.M.B. and S.L.G. wrote the manuscript. S.L.G. supervised the project.

### Corresponding author

Correspondence to Summer L. Gibbs.

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### Competing Interests

The authors declare no competing interests.

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Bittel, A.M., Davis, A.M., Wang, L. et al. Varied Length Stokes Shift BODIPY-Based Fluorophores for Multicolor Microscopy. Sci Rep 8, 4590 (2018). https://doi.org/10.1038/s41598-018-22892-8

• Accepted:

• Published:

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• ### BODI-Pt, a Green-Light-Activatable and Carboplatin-Based Platinum(IV) Anticancer Prodrug with Enhanced Activation and Cytotoxicity

• Houzong Yao
• , Shu Chen
• , Zhiqin Deng
• , Man-Kit Tse
• , Yudai Matsuda
•  & Guangyu Zhu

Inorganic Chemistry (2020)

• ### Light-responsive nanomedicine for biophotonic imaging and targeted therapy

• Jihwan Son
• , Gawon Yi
• , Jihye Yoo
• , Changhee Park
• , Heebeom Koo
•  & Hak Soo Choi

• ### Coloring Cell Complexity: The Case for an Expansive Fluorophore Palette

• Kimberly E. Beatty

ACS Central Science (2019)


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You have already agreed to pay $QK$ EUR at $T$ to receive $Q$ units of A. If you sell $Q$ lots of $F^A(t,T)$ then you will receive $Q F^A(t,T)$ EUR and deliver $Q$ units of A. The combined flow is now just in EUR: at $T$ you receive a net of $Q(F^A(t,T)-K)$ EUR. You can hedge that by selling $Q(F^A(t,T)-K)$ of $F^{FX}(t,T).$ Then with both hedges, the net ...

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Machine learning is a very wide field. Most often it is used for classification or regression tasks when you have labelled data to train the model. For example you show thousands of labeled pictures with an apple and computer "learns" what set of features gives high probability that picture contains an apple (for example, round, red etc). Now in your case ...

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QSTK is nice and open source , it is the QuantSciTookKit and it has some good functionality if you are interested in python programming. Here is the link: http://wiki.quantsoftware.org/index.php?title=QuantSoftware_ToolKit

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I really like Philip P's work, but frankly I do not believe this paper is his best one. It is understandable you do not catch how to use it: there is no dataset in the paper, and the orders of magnitude of $\sigma dW$ and $\delta_t x$ are so different. My suggestions: some components are missing, $x$ should be a point process, for instance an Hawkes ...

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I cannot suggest some reference particularly, since the field is going to develop day by day, but, generally, you could take a look to: Engelmann, Bernd, and Robert Rauhmeier, eds. The Basel II risk parameters: estimation, validation, and stress testing. Springer Science & Business Media, 2006. Particularly, look at the chapter 4 and 5; the ...

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If you take the a sample of historical asset returns as model for the risk then you can do two things: You calculate $r_j = \sum_{i=1}^n w_i r_i^j$ thus for each scenario $j$ you aggregate the individual asset returns to get a scenario for the portfolio. Then you can calculate $Var(r_j)$ the variance of the sample of portfolio returns. This is the same as ...

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if you take the variance of a single asset it scales as a quadratic, $$var(\lambda X) = \lambda^2 var(X)$$ so it's not surprising that the general case gives a quadratic form.

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Try to give David Spiegelhalter a read/listen to David Spiegelhalter's work and research. He is a statistician and a Professor of the Public Understanding of Risk at Cambridge England. Rather than new ways of calculating risk, he looks at ways of communicating risk to a general public that doesn't have any knowledge of stats. I Linked an interesting ...

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The retail credit risk management is generally based on models that try to discriminate between good (people that probably will be able to pay back the debt) and bad customers (people that probably will not). Particularly, as the question explicitly asks for, you want to some references to allow to decide which customers, already acquired, to keep and which ...

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 Volume 5, Number 1, 2015, Pages 64-76                                                                DOI：10.11948/2015006 Traveling wavefronts of a delayed lattice reaction-diffusion model Li Shu,Peixuan Weng,Yanling Tian Keywords:Pioneer-climax model, lattice differential system, harmless delay, traveling wave solution, minimal wave speed. Abstract: We investigate a system of delayed lattice differential system which is a model of pioneer-climax species distributed on one dimensional discrete space. We show that there exists a constant $c^*>0$, such that the model has traveling wave solutions connecting a boundary equilibrium to a co-existence equilibrium for $c\geq c^*$. We also argue that $c^*$ is the minimal wave speed and the delay is harmless. The Schauder's fixed point theorem combining with upper-lower solution technique is used for showing the existence of wave solution. PDF      Download reader